how to explain statistics in essay

There are lies, damned lies, and statistics. —Mark Twain

What this handout is about

The purpose of this handout is to help you use statistics to make your argument as effectively as possible.

Introduction

Numbers are power. Apparently freed of all the squishiness and ambiguity of words, numbers and statistics are powerful pieces of evidence that can effectively strengthen any argument. But statistics are not a panacea. As simple and straightforward as these little numbers promise to be, statistics, if not used carefully, can create more problems than they solve.

Many writers lack a firm grasp of the statistics they are using. The average reader does not know how to properly evaluate and interpret the statistics he or she reads. The main reason behind the poor use of statistics is a lack of understanding about what statistics can and cannot do. Many people think that statistics can speak for themselves. But numbers are as ambiguous as words and need just as much explanation.

In many ways, this problem is quite similar to that experienced with direct quotes. Too often, quotes are expected to do all the work and are treated as part of the argument, rather than a piece of evidence requiring interpretation (see our handout on how to quote .) But if you leave the interpretation up to the reader, who knows what sort of off-the-wall interpretations may result? The only way to avoid this danger is to supply the interpretation yourself.

But before we start writing statistics, let’s actually read a few.

Reading statistics

As stated before, numbers are powerful. This is one of the reasons why statistics can be such persuasive pieces of evidence. However, this same power can also make numbers and statistics intimidating. That is, we too often accept them as gospel, without ever questioning their veracity or appropriateness. While this may seem like a positive trait when you plug them into your paper and pray for your reader to submit to their power, remember that before we are writers of statistics, we are readers. And to be effective readers means asking the hard questions. Below you will find a useful set of hard questions to ask of the numbers you find.

1. Does your evidence come from reliable sources?

This is an important question not only with statistics, but with any evidence you use in your papers. As we will see in this handout, there are many ways statistics can be played with and misrepresented in order to produce a desired outcome. Therefore, you want to take your statistics from reliable sources (for more information on finding reliable sources, please see our handout on evaluating print sources ). This is not to say that reliable sources are infallible, but only that they are probably less likely to use deceptive practices. With a credible source, you may not need to worry as much about the questions that follow. Still, remember that reading statistics is a bit like being in the middle of a war: trust no one; suspect everyone.

2. What is the data’s background?

Data and statistics do not just fall from heaven fully formed. They are always the product of research. Therefore, to understand the statistics, you should also know where they come from. For example, if the statistics come from a survey or poll, some questions to ask include:

• Who asked the questions in the survey/poll?
• What, exactly, were the questions?
• Who interpreted the data?
• What issue prompted the survey/poll?
• What (policy/procedure) potentially hinges on the results of the poll?
• Who stands to gain from particular interpretations of the data?

All these questions help you orient yourself toward possible biases or weaknesses in the data you are reading. The goal of this exercise is not to find “pure, objective” data but to make any biases explicit, in order to more accurately interpret the evidence.

3. Are all data reported?

In most cases, the answer to this question is easy: no, they aren’t. Therefore, a better way to think about this issue is to ask whether all data have been presented in context. But it is much more complicated when you consider the bigger issue, which is whether the text or source presents enough evidence for you to draw your own conclusion. A reliable source should not exclude data that contradicts or weakens the information presented.

An example can be found on the evening news. If you think about ice storms, which make life so difficult in the winter, you will certainly remember the newscasters warning people to stay off the roads because they are so treacherous. To verify this point, they tell you that the Highway Patrol has already reported 25 accidents during the day. Their intention is to scare you into staying home with this number. While this number sounds high, some studies have found that the number of accidents actually goes down on days with severe weather. Why is that? One possible explanation is that with fewer people on the road, even with the dangerous conditions, the number of accidents will be less than on an “average” day. The critical lesson here is that even when the general interpretation is “accurate,” the data may not actually be evidence for the particular interpretation. This means you have no way to verify if the interpretation is in fact correct.

There is generally a comparison implied in the use of statistics. How can you make a valid comparison without having all the facts? Good question. You may have to look to another source or sources to find all the data you need.

4. Have the data been interpreted correctly?

If the author gives you her statistics, it is always wise to interpret them yourself. That is, while it is useful to read and understand the author’s interpretation, it is merely that—an interpretation. It is not the final word on the matter. Furthermore, sometimes authors (including you, so be careful) can use perfectly good statistics and come up with perfectly bad interpretations. Here are two common mistakes to watch out for:

• Confusing correlation with causation. Just because two things vary together does not mean that one of them is causing the other. It could be nothing more than a coincidence, or both could be caused by a third factor. Such a relationship is called spurious.The classic example is a study that found that the more firefighters sent to put out a fire, the more damage the fire did. Yikes! I thought firefighters were supposed to make things better, not worse! But before we start shutting down fire stations, it might be useful to entertain alternative explanations. This seemingly contradictory finding can be easily explained by pointing to a third factor that causes both: the size of the fire. The lesson here? Correlation does not equal causation. So it is important not only to think about showing that two variables co-vary, but also about the causal mechanism.
• Ignoring the margin of error. When survey results are reported, they frequently include a margin of error. You might see this written as “a margin of error of plus or minus 5 percentage points.” What does this mean? The simple story is that surveys are normally generated from samples of a larger population, and thus they are never exact. There is always a confidence interval within which the general population is expected to fall. Thus, if I say that the number of UNC students who find it difficult to use statistics in their writing is 60%, plus or minus 4%, that means, assuming the normal confidence interval of 95%, that with 95% certainty we can say that the actual number is between 56% and 64%.

Why does this matter? Because if after introducing this handout to the students of UNC, a new poll finds that only 56%, plus or minus 3%, are having difficulty with statistics, I could go to the Writing Center director and ask for a raise, since I have made a significant contribution to the writing skills of the students on campus. However, she would no doubt point out that a) this may be a spurious relationship (see above) and b) the actual change is not significant because it falls within the margin of error for the original results. The lesson here? Margins of error matter, so you cannot just compare simple percentages.

Finally, you should keep in mind that the source you are actually looking at may not be the original source of your data. That is, if you find an essay that quotes a number of statistics in support of its argument, often the author of the essay is using someone else’s data. Thus, you need to consider not only your source, but the author’s sources as well.

Writing statistics

As you write with statistics, remember your own experience as a reader of statistics. Don’t forget how frustrated you were when you came across unclear statistics and how thankful you were to read well-presented ones. It is a sign of respect to your reader to be as clear and straightforward as you can be with your numbers. Nobody likes to be played for a fool. Thus, even if you think that changing the numbers just a little bit will help your argument, do not give in to the temptation.

As you begin writing, keep the following in mind. First, your reader will want to know the answers to the same questions that we discussed above. Second, you want to present your statistics in a clear, unambiguous manner. Below you will find a list of some common pitfalls in the world of statistics, along with suggestions for avoiding them.

1. The mistake of the “average” writer

Nobody wants to be average. Moreover, nobody wants to just see the word “average” in a piece of writing. Why? Because nobody knows exactly what it means. There are not one, not two, but three different definitions of “average” in statistics, and when you use the word, your reader has only a 33.3% chance of guessing correctly which one you mean.

For the following definitions, please refer to this set of numbers: 5, 5, 5, 8, 12, 14, 21, 33, 38

• Mean (arithmetic mean) This may be the most average definition of average (whatever that means). This is the weighted average—a total of all numbers included divided by the quantity of numbers represented. Thus the mean of the above set of numbers is 5+5+5+8+12+14+21+33+38, all divided by 9, which equals 15.644444444444 (Wow! That is a lot of numbers after the decimal—what do we do about that? Precision is a good thing, but too much of it is over the top; it does not necessarily make your argument any stronger. Consider the reasonable amount of precision based on your input and round accordingly. In this case, 15.6 should do the trick.)
• Median Depending on whether you have an odd or even set of numbers, the median is either a) the number midway through an odd set of numbers or b) a value halfway between the two middle numbers in an even set. For the above set (an odd set of 9 numbers), the median is 12. (5, 5, 5, 8 < 12 < 14, 21, 33, 38)
• Mode The mode is the number or value that occurs most frequently in a series. If, by some cruel twist of fate, two or more values occur with the same frequency, then you take the mean of the values. For our set, the mode would be 5, since it occurs 3 times, whereas all other numbers occur only once.

As you can see, the numbers can vary considerably, as can their significance. Therefore, the writer should always inform the reader which average he or she is using. Otherwise, confusion will inevitably ensue.

2. Match your facts with your questions

Be sure that your statistics actually apply to the point/argument you are making. If we return to our discussion of averages, depending on the question you are interesting in answering, you should use the proper statistics.

Perhaps an example would help illustrate this point. Your professor hands back the midterm. The grades are distributed as follows:

The professor felt that the test must have been too easy, because the average (median) grade was a 95.

When a colleague asked her about how the midterm grades came out, she answered, knowing that her classes were gaining a reputation for being “too easy,” that the average (mean) grade was an 80.

When your parents ask you how you can justify doing so poorly on the midterm, you answer, “Don’t worry about my 63. It is not as bad as it sounds. The average (mode) grade was a 58.”

I will leave it up to you to decide whether these choices are appropriate. Selecting the appropriate facts or statistics will help your argument immensely. Not only will they actually support your point, but they will not undermine the legitimacy of your position. Think about how your parents will react when they learn from the professor that the average (median) grade was 95! The best way to maintain precision is to specify which of the three forms of “average” you are using.

3. Show the entire picture

Sometimes, you may misrepresent your evidence by accident and misunderstanding. Other times, however, misrepresentation may be slightly less innocent. This can be seen most readily in visual aids. Do not shape and “massage” the representation so that it “best supports” your argument. This can be achieved by presenting charts/graphs in numerous different ways. Either the range can be shortened (to cut out data points which do not fit, e.g., starting a time series too late or ending it too soon), or the scale can be manipulated so that small changes look big and vice versa. Furthermore, do not fiddle with the proportions, either vertically or horizontally. The fact that USA Today seems to get away with these techniques does not make them OK for an academic argument.

Charts A, B, and C all use the same data points, but the stories they seem to be telling are quite different. Chart A shows a mild increase, followed by a slow decline. Chart B, on the other hand, reveals a steep jump, with a sharp drop-off immediately following. Conversely, Chart C seems to demonstrate that there was virtually no change over time. These variations are a product of changing the scale of the chart. One way to alleviate this problem is to supplement the chart by using the actual numbers in your text, in the spirit of full disclosure.

Another point of concern can be seen in Charts D and E. Both use the same data as charts A, B, and C for the years 1985-2000, but additional time points, using two hypothetical sets of data, have been added back to 1965. Given the different trends leading up to 1985, consider how the significance of recent events can change. In Chart D, the downward trend from 1990 to 2000 is going against a long-term upward trend, whereas in Chart E, it is merely the continuation of a larger downward trend after a brief upward turn.

One of the difficulties with visual aids is that there is no hard and fast rule about how much to include and what to exclude. Judgment is always involved. In general, be sure to present your visual aids so that your readers can draw their own conclusions from the facts and verify your assertions. If what you have cut out could affect the reader’s interpretation of your data, then you might consider keeping it.

4. Give bases of all percentages

Because percentages are always derived from a specific base, they are meaningless until associated with a base. So even if I tell you that after this reading this handout, you will be 23% more persuasive as a writer, that is not a very meaningful assertion because you have no idea what it is based on—23% more persuasive than what?

Let’s look at crime rates to see how this works. Suppose we have two cities, Springfield and Shelbyville. In Springfield, the murder rate has gone up 75%, while in Shelbyville, the rate has only increased by 10%. Which city is having a bigger murder problem? Well, that’s obvious, right? It has to be Springfield. After all, 75% is bigger than 10%.

Hold on a second, because this is actually much less clear than it looks. In order to really know which city has a worse problem, we have to look at the actual numbers. If I told you that Springfield had 4 murders last year and 7 this year, and Shelbyville had 30 murders last year and 33 murders this year, would you change your answer? Maybe, since 33 murders are significantly more than 7. One would certainly feel safer in Springfield, right?

Not so fast, because we still do not have all the facts. We have to make the comparison between the two based on equivalent standards. To do that, we have to look at the per capita rate (often given in rates per 100,000 people per year). If Springfield has 700 residents while Shelbyville has 3.3 million, then Springfield has a murder rate of 1,000 per 100,000 people, and Shelbyville’s rate is merely 1 per 100,000. Gadzooks! The residents of Springfield are dropping like flies. I think I’ll stick with nice, safe Shelbyville, thank you very much.

Percentages are really no different from any other form of statistics: they gain their meaning only through their context. Consequently, percentages should be presented in context so that readers can draw their own conclusions as you emphasize facts important to your argument. Remember, if your statistics really do support your point, then you should have no fear of revealing the larger context that frames them.

Important questions to ask (and answer) about statistics

• Is the question being asked relevant?
• Do the data come from reliable sources?
• Margin of error/confidence interval—when is a change really a change?
• Are all data reported, or just the best/worst?
• Are the data presented in context?
• Have the data been interpreted correctly?
• Does the author confuse correlation with causation?

Now that you have learned the lessons of statistics, you have two options. Use this knowledge to manipulate your numbers to your advantage, or use this knowledge to better understand and use statistics to make accurate and fair arguments. The choice is yours. Nine out of ten writers, however, prefer the latter, and the other one later regrets his or her decision.

You may reproduce it for non-commercial use if you use the entire handout and attribute the source: The Writing Center, University of North Carolina at Chapel Hill

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How To Write a Statistical Analysis Essay

Home » Videos » How To Write a Statistical Analysis Essay

Statistical analysis is a powerful tool used to draw meaningful insights from data. It can be applied to almost any field and has been used in everything from natural sciences, economics, and sociology to sports analytics and business decisions. Writing a statistical analysis essay requires an understanding of the concepts behind it as well as proficiency with data manipulation techniques.

In this guide, we’ll look at the steps involved in writing a statistical analysis essay. Experts in research paper writing from https://domypaper.me/write-my-research-paper/ give detailed instructions on how to properly conduct a statistical analysis and make valid conclusions.

Overview of statistical analysis essays

A statistical analysis essay is an academic paper that involves analyzing quantitative data and interpreting the results. It is often used in social sciences, economics and business to draw meaningful conclusions from the data. The objective of a statistical analysis essay is to analyze a specific dataset or multiple datasets in order to answer a question or prove or disprove a hypothesis. To achieve this effectively, the information must be analyzed using appropriate statistical techniques such as descriptive statistics, inferential statistics, regression analysis and correlation analysis.

Researching the subject matter

Before writing your statistical analysis essay it is important to research the subject matter thoroughly so that you have an understanding of what you are dealing with. This can include collecting and organizing any relevant data sets as well as researching different types of statistical techniques available for analyzing them. Furthermore, it is important to become familiar with the terminology associated with statistical analysis such as mean, median and mode.

Structuring your statistical analysis essay

The structure of your essay will depend on the type of data you are analyzing and the research question or hypothesis that you are attempting to answer. Generally speaking, it should include an introduction which introduces the research question or hypothesis; a body section which includes an overview of relevant literature; a description of how the data was collected and analyzed and any conclusions drawn from it; and finally a conclusion which summarizes all findings.

Analyzing data and drawing conclusions from it

After collecting and organizing your data, you must analyze it in order to draw meaningful conclusions from it. This involves using appropriate statistical techniques such as descriptive statistics, inferential statistics, regression analysis and correlation analysis. It is important to understand the assumptions made when using each technique in order to analyze the data correctly and draw accurate conclusions from it. When choosing a statistical technique for your research, it is important to consult with an expert https://typemyessay.me/service/research-paper-writing-service who can guide you on the most appropriate approach for your study.

Interpreting results and writing a conclusion

Once you have analyzed the data successfully, you must interpret the results carefully in order to answer your research question or prove/disprove your hypothesis. This involves making sure that any conclusions drawn are soundly based on the evidence presented. After interpreting the results, you should write a conclusion which summarizes all of your findings.

Using sources in your analysis

In order to back up your claims and provide support for your arguments, it is important to use credible sources within your analysis. This could include peer-reviewed articles, journals and books which can provide evidence to support your conclusion. It is also important to cite all sources used in order to avoid plagiarism.

Proofreading and finalizing your work

Once you have written your essay it is important to proofread it carefully before submitting it. This involves checking for grammar, spelling and punctuation errors as well as ensuring that the flow of the essay makes sense. Additionally, make sure that any references cited are correct and up-to-date. If you find it hard to complete your research statistical paper, you may want to consider buying a research paper for sale . This service can save you time and money, allowing you to focus on other important tasks.

Tips for writing a successful statistical analysis essay

Here are some tips for writing a successful statistical analysis essay:

• Research your subject matter thoroughly before writing your essay.
• Structure your paper according to the type of data you are analyzing.
• Analyze your data using appropriate statistical techniques.
• Interpret and draw meaningful conclusions from your results.
• Use credible sources to back up any claims or arguments made.
• Proofread and finalize your work before submitting it.

These tips will help ensure that your essay is well researched, structured correctly and contains accurate information. Following these tips will help you write a successful statistical analysis essay which can be used to answer research questions or prove/disprove hypotheses.

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Understanding and Using Statistical Methods

Statistics is a set of tools used to organize and analyze data. Data must either be numeric in origin or transformed by researchers into numbers. For instance, statistics could be used to analyze percentage scores English students receive on a grammar test: the percentage scores ranging from 0 to 100 are already in numeric form. Statistics could also be used to analyze grades on an essay by assigning numeric values to the letter grades, e.g., A=4, B=3, C=2, D=1, and F=0.

Employing statistics serves two purposes, (1) description and (2) prediction. Statistics are used to describe the characteristics of groups. These characteristics are referred to as variables . Data is gathered and recorded for each variable. Descriptive statistics can then be used to reveal the distribution of the data in each variable.

Statistics is also frequently used for purposes of prediction. Prediction is based on the concept of generalizability : if enough data is compiled about a particular context (e.g., students studying writing in a specific set of classrooms), the patterns revealed through analysis of the data collected about that context can be generalized (or predicted to occur in) similar contexts. The prediction of what will happen in a similar context is probabilistic . That is, the researcher is not certain that the same things will happen in other contexts; instead, the researcher can only reasonably expect that the same things will happen.

Prediction is a method employed by individuals throughout daily life. For instance, if writing students begin class every day for the first half of the semester with a five-minute freewriting exercise, then they will likely come to class the first day of the second half of the semester prepared to again freewrite for the first five minutes of class. The students will have made a prediction about the class content based on their previous experiences in the class: Because they began all previous class sessions with freewriting, it would be probable that their next class session will begin the same way. Statistics is used to perform the same function; the difference is that precise probabilities are determined in terms of the percentage chance that an outcome will occur, complete with a range of error. Prediction is a primary goal of inferential statistics.

Revealing Patterns Using Descriptive Statistics

Descriptive statistics, not surprisingly, "describe" data that have been collected. Commonly used descriptive statistics include frequency counts, ranges (high and low scores or values), means, modes, median scores, and standard deviations. Two concepts are essential to understanding descriptive statistics: variables and distributions .

Statistics are used to explore numerical data (Levin, 1991). Numerical data are observations which are recorded in the form of numbers (Runyon, 1976). Numbers are variable in nature, which means that quantities vary according to certain factors. For examples, when analyzing the grades on student essays, scores will vary for reasons such as the writing ability of the student, the students' knowledge of the subject, and so on. In statistics, these reasons are called variables. Variables are divided into three basic categories:

Nominal Variables

Nominal variables classify data into categories. This process involves labeling categories and then counting frequencies of occurrence (Runyon, 1991). A researcher might wish to compare essay grades between male and female students. Tabulations would be compiled using the categories "male" and "female." Sex would be a nominal variable. Note that the categories themselves are not quantified. Maleness or femaleness are not numerical in nature, rather the frequencies of each category results in data that is quantified -- 11 males and 9 females.

Ordinal Variables

Ordinal variables order (or rank) data in terms of degree. Ordinal variables do not establish the numeric difference between data points. They indicate only that one data point is ranked higher or lower than another (Runyon, 1991). For instance, a researcher might want to analyze the letter grades given on student essays. An A would be ranked higher than a B, and a B higher than a C. However, the difference between these data points, the precise distance between an A and a B, is not defined. Letter grades are an example of an ordinal variable.

Interval Variables

Interval variables score data. Thus the order of data is known as well as the precise numeric distance between data points (Runyon, 1991). A researcher might analyze the actual percentage scores of the essays, assuming that percentage scores are given by the instructor. A score of 98 (A) ranks higher than a score of 87 (B), which ranks higher than a score of 72 (C). Not only is the order of these three data points known, but so is the exact distance between them -- 11 percentage points between the first two, 15 percentage points between the second two and 26 percentage points between the first and last data points.

Distributions

A distribution is a graphic representation of data. The line formed by connecting data points is called a frequency distribution. This line may take many shapes. The single most important shape is that of the bell-shaped curve, which characterizes the distribution as "normal." A perfectly normal distribution is only a theoretical ideal. This ideal, however, is an essential ingredient in statistical decision-making (Levin, 1991). A perfectly normal distribution is a mathematical construct which carries with it certain mathematical properties helpful in describing the attributes of the distribution. Although frequency distribution based on actual data points seldom, if ever, completely matches a perfectly normal distribution, a frequency distribution often can approach such a normal curve.

The closer a frequency distribution resembles a normal curve, the more probable that the distribution maintains those same mathematical properties as the normal curve. This is an important factor in describing the characteristics of a frequency distribution. As a frequency distribution approaches a normal curve, generalizations about the data set from which the distribution was derived can be made with greater certainty. And it is this notion of generalizability upon which statistics is founded. It is important to remember that not all frequency distributions approach a normal curve. Some are skewed. When a frequency distribution is skewed, the characteristics inherent to a normal curve no longer apply.

Making Predictions Using Inferential Statistics

Inferential statistics are used to draw conclusions and make predictions based on the descriptions of data. In this section, we explore inferential statistics by using an extended example of experimental studies. Key concepts used in our discussion are probability, populations, and sampling.

Experiments

A typical experimental study involves collecting data on the behaviors, attitudes, or actions of two or more groups and attempting to answer a research question (often called a hypothesis). Based on the analysis of the data, a researcher might then attempt to develop a causal model that can be generalized to populations.

A question that might be addressed through experimental research might be "Does grammar-based writing instruction produce better writers than process-based writing instruction?" Because it would be impossible and impractical to observe, interview, survey, etc. all first-year writing students and instructors in classes using one or the other of these instructional approaches, a researcher would study a sample – or a subset – of a population. Sampling – or the creation of this subset of a population – is used by many researchers who desire to make sense of some phenomenon.

To analyze differences in the ability of student writers who are taught in each type of classroom, the researcher would compare the writing performance of the two groups of students.

Dependent Variables

In an experimental study, a variable whose score depends on (or is determined or caused by) another variable is called a dependent variable. For instance, an experiment might explore the extent to which the writing quality of final drafts of student papers is affected by the kind of instruction they received. In this case, the dependent variable would be writing quality of final drafts.

Independent Variables

In an experimental study, a variable that determines (or causes) the score of a dependent variable is called an independent variable. For instance, an experiment might explore the extent to which the writing quality of final drafts of student papers is affected by the kind of instruction they received. In this case, the independent variable would be the kind of instruction students received.

Probability

Beginning researchers most often use the word probability to express a subjective judgment about the likelihood, or degree of certainty, that a particular event will occur. People say such things as: "It will probably rain tomorrow." "It is unlikely that we will win the ball game." It is possible to assign a number to the event being predicted, a number between 0 and 1, which represents degree of confidence that the event will occur. For example, a student might say that the likelihood an instructor will give an exam next week is about 90 percent, or .9. Where 100 percent, or 1.00, represents certainty, .9 would mean the student is almost certain the instructor will give an exam. If the student assigned the number .6, the likelihood of an exam would be just slightly greater than the likelihood of no exam. A rating of 0 would indicate complete certainty that no exam would be given(Shoeninger, 1971).

The probability of a particular outcome or set of outcomes is called a p-value . In our discussion, a p-value will be symbolized by a p followed by parentheses enclosing a symbol of the outcome or set of outcomes. For example, p(X) should be read, "the probability of a given X score" (Shoeninger). Thus p(exam) should be read, "the probability an instructor will give an exam next week."

A population is a group which is studied. In educational research, the population is usually a group of people. Researchers seldom are able to study every member of a population. Usually, they instead study a representative sample – or subset – of a population. Researchers then generalize their findings about the sample to the population as a whole.

Sampling is performed so that a population under study can be reduced to a manageable size. This can be accomplished via random sampling, discussed below, or via matching.

Random sampling is a procedure used by researchers in which all samples of a particular size have an equal chance to be chosen for an observation, experiment, etc (Runyon and Haber, 1976). There is no predetermination as to which members are chosen for the sample. This type of sampling is done in order to minimize scientific biases and offers the greatest likelihood that a sample will indeed be representative of the larger population. The aim here is to make the sample as representative of the population as possible. Note that the closer a sample distribution approximates the population distribution, the more generalizable the results of the sample study are to the population. Notions of probability apply here. Random sampling provides the greatest probability that the distribution of scores in a sample will closely approximate the distribution of scores in the overall population.

Matching is a method used by researchers to gain accurate and precise results of a study so that they may be applicable to a larger population. After a population has been examined and a sample has been chosen, a researcher must then consider variables, or extrinsic factors, that might affect the study. Matching methods apply when researchers are aware of extrinsic variables before conducting a study. Two methods used to match groups are:

Precision Matching

In precision matching , there is an experimental group that is matched with a control group. Both groups, in essence, have the same characteristics. Thus, the proposed causal relationship/model being examined allows for the probabilistic assumption that the result is generalizable.

Frequency Distribution

Frequency distribution is more manageable and efficient than precision matching. Instead of one-to-one matching that must be administered in precision matching, frequency distribution allows the comparison of an experimental and control group through relevant variables. If three Communications majors and four English majors are chosen for the control group, then an equal proportion of three Communications major and four English majors should be allotted to the experiment group. Of course, beyond their majors, the characteristics of the matched sets of participants may in fact be vastly different.

Although, in theory, matching tends to produce valid conclusions, a rather obvious difficulty arises in finding subjects which are compatible. Researchers may even believe that experimental and control groups are identical when, in fact, a number of variables have been overlooked. For these reasons, researchers tend to reject matching methods in favor of random sampling.

Statistics can be used to analyze individual variables, relationships among variables, and differences between groups. In this section, we explore a range of statistical methods for conducting these analyses.

Statistics can be used to analyze individual variables, relationships among variables, and differences between groups.

Analyzing Individual Variables

The statistical procedures used to analyze a single variable describing a group (such as a population or representative sample) involve measures of central tendency and measures of variation . To explore these measures, a researcher first needs to consider the distribution , or range of values of a particular variable in a population or sample. Normal distribution occurs if the distribution of a population is completely normal. When graphed, this type of distribution will look like a bell curve; it is symmetrical and most of the scores cluster toward the middle. Skewed Distribution simply means the distribution of a population is not normal. The scores might cluster toward the right or the left side of the curve, for instance. Or there might be two or more clusters of scores, so that the distribution looks like a series of hills.

Once frequency distributions have been determined, researchers can calculate measures of central tendency and measures of variation. Measures of central tendency indicate averages of the distribution, and measures of variation indicate the spread, or range, of the distribution (Hinkle, Wiersma and Jurs 1988).

Measures of Central Tendency

Central tendency is measured in three ways: mean , median and mode . The mean is simply the average score of a distribution. The median is the center, or middle score within a distribution. The mode is the most frequent score within a distribution. In a normal distribution, the mean, median and mode are identical.

Measures of Variation

Measures of variation determine the range of the distribution, relative to the measures of central tendency. Where the measures of central tendency are specific data points, measures of variation are lengths between various points within the distribution. Variation is measured in terms of range, mean deviation, variance, and standard deviation (Hinkle, Wiersma and Jurs 1988).

The range is the distance between the lowest data point and the highest data point. Deviation scores are the distances between each data point and the mean.

Mean deviation is the average of the absolute values of the deviation scores; that is, mean deviation is the average distance between the mean and the data points. Closely related to the measure of mean deviation is the measure of variance .

Variance also indicates a relationship between the mean of a distribution and the data points; it is determined by averaging the sum of the squared deviations. Squaring the differences instead of taking the absolute values allows for greater flexibility in calculating further algebraic manipulations of the data. Another measure of variation is the standard deviation .

Standard deviation is the square root of the variance. This calculation is useful because it allows for the same flexibility as variance regarding further calculations and yet also expresses variation in the same units as the original measurements (Hinkle, Wiersma and Jurs 1988).

Analyzing Differences Between Groups

Statistical tests can be used to analyze differences in the scores of two or more groups. The following statistical tests are commonly used to analyze differences between groups:

A t-test is used to determine if the scores of two groups differ on a single variable. A t-test is designed to test for the differences in mean scores. For instance, you could use a t-test to determine whether writing ability differs among students in two classrooms.

Note: A t-test is appropriate only when looking at paired data. It is useful in analyzing scores of two groups of participants on a particular variable or in analyzing scores of a single group of participants on two variables.

Matched Pairs T-Test

This type of t-test could be used to determine if the scores of the same participants in a study differ under different conditions. For instance, this sort of t-test could be used to determine if people write better essays after taking a writing class than they did before taking the writing class.

Analysis of Variance (ANOVA)

The ANOVA (analysis of variance) is a statistical test which makes a single, overall decision as to whether a significant difference is present among three or more sample means (Levin 484). An ANOVA is similar to a t-test. However, the ANOVA can also test multiple groups to see if they differ on one or more variables. The ANOVA can be used to test between-groups and within-groups differences. There are two types of ANOVAs:

One-Way ANOVA: This tests a group or groups to determine if there are differences on a single set of scores. For instance, a one-way ANOVA could determine whether freshmen, sophomores, juniors, and seniors differed in their reading ability.

Multiple ANOVA (MANOVA): This tests a group or groups to determine if there are differences on two or more variables. For instance, a MANOVA could determine whether freshmen, sophomores, juniors, and seniors differed in reading ability and whether those differences were reflected by gender. In this case, a researcher could determine (1) whether reading ability differed across class levels, (2) whether reading ability differed across gender, and (3) whether there was an interaction between class level and gender.

Analyzing Relationships Among Variables

Statistical relationships between variables rely on notions of correlation and regression. These two concepts aim to describe the ways in which variables relate to one another:

Correlation

Correlation tests are used to determine how strongly the scores of two variables are associated or correlated with each other. A researcher might want to know, for instance, whether a correlation exists between students' writing placement examination scores and their scores on a standardized test such as the ACT or SAT. Correlation is measured using values between +1.0 and -1.0. Correlations close to 0 indicate little or no relationship between two variables, while correlations close to +1.0 (or -1.0) indicate strong positive (or negative) relationships (Hayes et al. 554).

Correlation denotes positive or negative association between variables in a study. Two variables are positively associated when larger values of one tend to be accompanied by larger values of the other. The variables are negatively associated when larger values of one tend to be accompanied by smaller values of the other (Moore 208).

An example of a strong positive correlation would be the correlation between age and job experience. Typically, the longer people are alive, the more job experience they might have.

An example of a strong negative relationship might occur between the strength of people's party affiliations and their willingness to vote for a candidate from different parties. In many elections, Democrats are unlikely to vote for Republicans, and vice versa.

Regression analysis attempts to determine the best "fit" between two or more variables. The independent variable in a regression analysis is a continuous variable, and thus allows you to determine how one or more independent variables predict the values of a dependent variable.

Simple Linear Regression is the simplest form of regression. Like a correlation, it determines the extent to which one independent variables predicts a dependent variable. You can think of a simple linear regression as a correlation line. Regression analysis provides you with more information than correlation does, however. It tells you how well the line "fits" the data. That is, it tells you how closely the line comes to all of your data points. The line in the figure indicates the regression line drawn to find the best fit among a set of data points. Each dot represents a person and the axes indicate the amount of job experience and the age of that person. The dotted lines indicate the distance from the regression line. A smaller total distance indicates a better fit. Some of the information provided in a regression analysis, as a result, indicates the slope of the regression line, the R value (or correlation), and the strength of the fit (an indication of the extent to which the line can account for variations among the data points).

Multiple Linear Regression allows one to determine how well multiple independent variables predict the value of a dependent variable. A researcher might examine, for instance, how well age and experience predict a person's salary. The interesting thing here is that one would no longer be dealing with a regression "line." Instead, since the study deals with three dimensions (age, experience, and salary), it would be dealing with a plane, that is, with a two-dimensional figure. If a fourth variable was added to the equations, one would be dealing with a three-dimensional figure, and so on.

Misuses of Statistics

Statistics consists of tests used to analyze data. These tests provide an analytic framework within which researchers can pursue their research questions. This framework provides one way of working with observable information. Like other analytic frameworks, statistical tests can be misused, resulting in potential misinterpretation and misrepresentation. Researchers decide which research questions to ask, which groups to study, how those groups should be divided, which variables to focus upon, and how best to categorize and measure such variables. The point is that researchers retain the ability to manipulate any study even as they decide what to study and how to study it.

Potential Misuses:

• Manipulating scale to change the appearance of the distribution of data
• Eliminating high/low scores for more coherent presentation
• Inappropriately focusing on certain variables to the exclusion of other variables
• Presenting correlation as causation

Measures Against Potential Misuses:

• Testing for reliability and validity
• Testing for statistical significance
• Critically reading statistics

Annotated Bibliography

Dear, K. (1997, August 28). SurfStat australia . Available: http://surfstat.newcastle.edu.au/surfstat/main/surfstat-main.html

A comprehensive site contain an online textbook, links together statistics sites, exercises, and a hotlist for Java applets.

de Leeuw, J. (1997, May 13 ). Statistics: The study of stability in variation . Available: http://www.stat.ucla.edu/textbook/ [1997, December 8].

An online textbook providing discussions specifically regarding variability.

Ewen, R.B. (1988). The workbook for introductory statistics for the behavioral sciences. Orlando, FL: Harcourt Brace Jovanovich.

A workbook providing sample problems typical of the statistical applications in social sciences.

Glass, G. (1996, August 26). COE 502: Introduction to quantitative methods . Available: http://seamonkey.ed.asu.edu/~gene/502/home.html

Outline of a basic statistics course in the college of education at Arizona State University, including a list of statistic resources on the Internet and access to online programs using forms and PERL to analyze data.

Hartwig, F., Dearing, B.E. (1979). Exploratory data analysis . Newberry Park, CA: Sage Publications, Inc.

Hayes, J. R., Young, R.E., Matchett, M.L., McCaffrey, M., Cochran, C., and Hajduk, T., eds. (1992). Reading empirical research studies: The rhetoric of research . Hillsdale, NJ: Lawrence Erlbaum Associates.

A text focusing on the language of research. Topics vary from "Communicating with Low-Literate Adults" to "Reporting on Journalists."

Hinkle, Dennis E., Wiersma, W. and Jurs, S.G. (1988). Applied statistics for the behavioral sciences . Boston: Houghton.

This is an introductory text book on statistics. Each of 22 chapters includes a summary, sample exercises and highlighted main points. The book also includes an index by subject.

Kleinbaum, David G., Kupper, L.L. and Muller K.E. Applied regression analysis and other multivariable methods 2nd ed . Boston: PWS-KENT Publishing Company.

An introductory text with emphasis on statistical analyses. Chapters contain exercises.

Kolstoe, R.H. (1969). Introduction to statistics for the behavioral sciences . Homewood, ILL: Dorsey.

Though more than 25-years-old, this textbook uses concise chapters to explain many essential statistical concepts. Information is organized in a simple and straightforward manner.

Levin, J., and James, A.F. (1991). Elementary statistics in social research, 5th ed . New York: HarperCollins.

This textbook presents statistics in three major sections: Description, From Description to Decision Making and Decision Making. The first chapter underlies reasons for using statistics in social research. Subsequent chapters detail the process of conducting and presenting statistics.

Liebetrau, A.M. (1983). Measures of association . Newberry Park, CA: Sage Publications, Inc.

Mendenhall, W.(1975). Introduction to probability and statistics, 4th ed. North Scltuate, MA: Duxbury Press.

An introductory textbook. A good overview of statistics. Includes clear definitions and exercises.

Moore, David S. (1979). Statistics: Concepts and controversies , 2nd ed . New York: W. H. Freeman and Company.

Introductory text. Basic overview of statistical concepts. Includes discussions of concrete applications such as opinion polls and Consumer Price Index.

Mosier, C.T. (1997). MG284 Statistics I - notes. Available: http://phoenix.som.clarkson.edu/~cmosier/statistics/main/outline/index.html

Explanations of fundamental statistical concepts.

Newton, H.J., Carrol, J.H., Wang, N., & Whiting, D.(1996, Fall). Statistics 30X class notes. Available: http://stat.tamu.edu/stat30x/trydouble2.html [1997, December 10].

This site contains a hyperlinked list of very comprehensive course notes from and introductory statistics class. A large variety of statistical concepts are covered.

Runyon, R.P., and Haber, A. (1976). Fundamentals of behavioral statistics , 3rd ed . Reading, MA: Addison-Wesley Publishing Company.

This is a textbook that divides statistics into categories of descriptive statistics and inferential statistics. It presents statistical procedures primarily through examples. This book includes sectional reviews, reviews of basic mathematics and also a glossary of symbols common to statistics.

Schoeninger, D.W. and Insko, C.A. (1971). Introductory statistics for the behavioral sciences . Boston: Allyn and Bacon, Inc.

An introductory text including discussions of correlation, probability, distribution, and variance. Includes statistical tables in the appendices.

Stevens, J. (1986). Applied multivariate statistics for the social sciences . Hillsdale, NJ: Lawrence Erlbaum Associates.

Stockberger, D. W. (1996). Introductory statistics: Concepts, models and applications . Available: http://www.psychstat.smsu.edu/ [1997, December 8].

Describes various statistical analyses. Includes statistical tables in the appendix.

Local Resources

If you are a member of the Colorado State University community and seek more in-depth help with analyzing data from your research (e.g., from an undergraduate or graduate research project), please contact CSU's Graybill Statistical Laboratory for statistical consulting assistance at http://www.stat.colostate.edu/statlab.html .

Jackson, Shawna, Karen Marcus, Cara McDonald, Timothy Wehner, & Mike Palmquist. (2005). Statistics: An Introduction. Writing@CSU . Colorado State University. https://writing.colostate.edu/guides/guide.cfm?guideid=67

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Writing with Descriptive Statistics

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Usually there is no good way to write a statistic. It rarely sounds good, and often interrupts the structure or flow of your writing. Oftentimes the best way to write descriptive statistics is to be direct. If you are citing several statistics about the same topic, it may be best to include them all in the same paragraph or section.

The mean of exam two is 77.7. The median is 75, and the mode is 79. Exam two had a standard deviation of 11.6.

Overall the company had another excellent year. We shipped 14.3 tons of fertilizer for the year, and averaged 1.7 tons of fertilizer during the summer months. This is an increase over last year, where we shipped only 13.1 tons of fertilizer, and averaged only 1.4 tons during the summer months. (Standard deviations were as followed: this summer .3 tons, last summer .4 tons).

Some fields prefer to put means and standard deviations in parentheses like this:

If you have lots of statistics to report, you should strongly consider presenting them in tables or some other visual form. You would then highlight statistics of interest in your text, but would not report all of the statistics. See the section on statistics and visuals for more details.

If you have a data set that you are using (such as all the scores from an exam) it would be unusual to include all of the scores in a paper or article. One of the reasons to use statistics is to condense large amounts of information into more manageable chunks; presenting your entire data set defeats this purpose.

At the bare minimum, if you are presenting statistics on a data set, it should include the mean and probably the standard deviation. This is the minimum information needed to get an idea of what the distribution of your data set might look like. How much additional information you include is entirely up to you. In general, don't include information if it is irrelevant to your argument or purpose. If you include statistics that many of your readers would not understand, consider adding the statistics in a footnote or appendix that explains it in more detail.

How to present data and statistics in your paper

In most academic fields, presenting stats and data is key. words like 'values', 'equations', 'numbers', and 'tests' are common in theses and papers. but how do you use these words; what other words do they usually combine with in this analysis, we explore what phrases authors use most often when they present data and statistics..

Our analysis

We built a data set of 300 million sentences from published papers. From these sentences, we extracted all three-word combinations following the pattern subject + verb + object (for example, 'data shows difference').

We then collected the 100 most frequent combinations and their frequency, and visualized these (see image below). The 3 most-used triples were 'equation have solution', 'data provide evidence', and 'test show difference'.

Note that all phrases are lemmatized: they reflect the total counts of all forms. For example, the phrase 'test show difference' includes 'tests showing differences', 'tests showed differences', and others. The combined words were also not necessarily adjacent in the original sentence; for instance, an occurrence of 'test show difference' might have been 'test A showed a small difference' in the original paper.

The image below shows the most frequently used word combinations. The subject is shown in bold, the verb in regular script, and the object in italics. The figure uses hierarchical clustering, with the phrases first being grouped by subject and then by verb.

Not surprisingly, ‘data’ is the most frequent subject. It is often combined with the verbs 'provide', 'show', and 'support'. For example, data 'provide' 'evidence', 'information', or 'insights'; data 'show' 'differences', 'increases', and 'correlations'; and data 'support' 'hypotheses', 'notions', and 'ideas'. The subject 'test' is also frequent, and most often followed by 'reveal', 'indicate', or 'show' '(a) difference'.

Next time you’re writing your methods or results section and you’re stuck for words, see if this image helps you! It might give you the words you’re looking for.

About the author

Hilde is Chief Applied Linguist at Writefull.

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Introductory essay

Written by the educators who created Visualizing Data, a brief look at the key facts, tough questions and big ideas in their field. Begin this TED Study with a fascinating read that gives context and clarity to the material.

The reality of today

All of us now are being blasted by information design. It's being poured into our eyes through the Web, and we're all visualizers now; we're all demanding a visual aspect to our information...And if you're navigating a dense information jungle, coming across a beautiful graphic or a lovely data visualization, it's a relief, it's like coming across a clearing in the jungle. David McCandless

In today's complex 'information jungle,' David McCandless observes that "Data is the new soil." McCandless, a data journalist and information designer, celebrates data as a ubiquitous resource providing a fertile and creative medium from which new ideas and understanding can grow. McCandless's inspiration, statistician Hans Rosling, builds on this idea in his own TEDTalk with his compelling image of flowers growing out of data/soil. These 'flowers' represent the many insights that can be gleaned from effective visualization of data.

We're just learning how to till this soil and make sense of the mountains of data constantly being generated. As Gary King, Director of Harvard's Institute for Quantitative Social Science says in his New York Times article "The Age of Big Data":

It's a revolution. We're really just getting under way. But the march of quantification, made possible by enormous new sources of data, will sweep through academia, business and government. There is no area that is going to be untouched.

How do we deal with all this data without getting information overload? How do we use data to gain real insight into the world? Finding ways to pull interesting information out of data can be very rewarding, both personally and professionally. The managing editor of Financial Times observed on CNN's Your Money : "The people who are able to in a sophisticated and practical way analyze that data are going to have terrific jobs." Those who learn how to present data in effective ways will be valuable in every field.

Many people, when they think of data, think of tables filled with numbers. But this long-held notion is eroding. Today, we're generating streams of data that are often too complex to be presented in a simple "table." In his TEDTalk, Blaise Aguera y Arcas explores images as data, while Deb Roy uses audio, video, and the text messages in social media as data.

Some may also think that only a few specialized professionals can draw insights from data. When we look at data in the right way, however, the results can be fun, insightful, even whimsical — and accessible to everyone! Who knew, for example, that there are more relationship break-ups on Monday than on any other day of the week, or that the most break-ups (at least those discussed on Facebook) occur in mid-December? David McCandless discovered this by analyzing thousands of Facebook status updates.

Data, data, everywhere

There is more data available to us now than we can possibly process. Every minute , Internet users add the following to the big data pool (i):

• 204,166,667 email messages sent
• More than 2,000,000 Google searches
• 684,478 pieces of content added on Facebook
• $272,070 spent by consumers via online shopping
• More than 100,000 tweets on Twitter
• 47,000 app downloads from Apple
• 34,722 "likes" on Facebook for different brands and organizations
• 27,778 new posts on Tumblr blogs
• 3,600 new photos on Instagram
• 3,125 new photos on Flickr
• 2,083 check-ins on Foursquare
• 571 new websites created
• 347 new blog posts published on Wordpress
• 217 new mobile web users
• 48 hours of new video on YouTube

These numbers are almost certainly higher now, as you read this. And this just describes a small piece of the data being generated and stored by humanity. We're all leaving data trails — not just on the Internet, but in everything we do. This includes reams of financial data (from credit cards, businesses, and Wall Street), demographic data on the world's populations, meteorological data on weather and the environment, retail sales data that records everything we buy, nutritional data on food and restaurants, sports data of all types, and so on.

Governments are using data to search for terrorist plots, retailers are using it to maximize marketing strategies, and health organizations are using it to track outbreaks of the flu. But did you ever think of collecting data on every minute of your child's life? That's precisely what Deb Roy did. He recorded 90,000 hours of video and 140,000 hours of audio during his son's first years. That's a lot of data! He and his colleagues are using the data to understand how children learn language, and they're now extending this work to analyze publicly available conversations on social media, allowing them to take "the real-time pulse of a nation."

Data can provide us with new and deeper insight into our world. It can help break stereotypes and build understanding. But the sheer quantity of data, even in just any one small area of interest, is overwhelming. How can we make sense of some of this data in an insightful way?

The power of visualizing data

Visualization can help transform these mountains of data into meaningful information. In his TEDTalk, David McCandless comments that the sense of sight has by far the fastest and biggest bandwidth of any of the five senses. Indeed, about 80% of the information we take in is by eye. Data that seems impenetrable can come alive if presented well in a picture, graph, or even a movie. Hans Rosling tells us that "Students get very excited — and policy-makers and the corporate sector — when they can see the data."

It makes sense that, if we can effectively display data visually, we can make it accessible and understandable to more people. Should we worry, however, that by condensing data into a graph, we are simplifying too much and losing some of the important features of the data? Let's look at a fascinating study conducted by researchers Emre Soyer and Robin Hogarth . The study was conducted on economists, who are certainly no strangers to statistical analysis. Three groups of economists were asked the same question concerning a dataset:

• One group was given the data and a standard statistical analysis of the data; 72% of these economists got the answer wrong.
• Another group was given the data, the statistical analysis, and a graph; still 61% of these economists got the answer wrong.
• A third group was given only the graph, and only 3% got the answer wrong.

Visualizing data can sometimes be less misleading than using the raw numbers and statistics!

What about all the rest of us, who may not be professional economists or statisticians? Nathalie Miebach finds that making art out of data allows people an alternative entry into science. She transforms mountains of weather data into tactile physical structures and musical scores, adding both touch and hearing to the sense of sight to build even greater understanding of data.

Another artist, Chris Jordan, is concerned about our ability to comprehend big numbers. As citizens of an ever-more connected global world, we have an increased need to get useable information from big data — big in terms of the volume of numbers as well as their size. Jordan's art is designed to help us process such numbers, especially numbers that relate to issues of addiction and waste. For example, Jordan notes that the United States has the largest percentage of its population in prison of any country on earth: 2.3 million people in prison in the United States in 2005 and the number continues to rise. Jordan uses art, in this case a super-sized image of 2.3 million prison jumpsuits, to help us see that number and to help us begin to process the societal implications of that single data value. Because our brains can't truly process such a large number, his artwork makes it real.

The role of technology in visualizing data

The TEDTalks in this collection depend to varying degrees on sophisticated technology to gather, store, process, and display data. Handling massive amounts of data (e.g., David McCandless tracking 10,000 changes in Facebook status, Blaise Aguera y Arcas synching thousands of online images of the Notre Dame Cathedral, or Deb Roy searching for individual words in 90,000 hours of video tape) requires cutting-edge computing tools that have been developed specifically to address the challenges of big data. The ability to manipulate color, size, location, motion, and sound to discover and display important features of data in a way that makes it readily accessible to ordinary humans is a challenging task that depends heavily on increasingly sophisticated technology.

The importance of good visualization

There are good ways and bad ways of presenting data. Many examples of outstanding presentations of data are shown in the TEDTalks. However, sometimes visualizations of data can be ineffective or downright misleading. For example, an inappropriate scale might make a relatively small difference look much more substantial than it should be, or an overly complicated display might obfuscate the main relationships in the data. Statistician Kaiser Fung's blog Junk Charts offers many examples of poor representations of data (and some good ones) with descriptions to help the reader understand what makes a graph effective or ineffective. For more examples of both good and bad representations of data, see data visualization architect Andy Kirk's blog at visualisingdata.com . Both consistently have very current examples from up-to-date sources and events.

Creativity, even artistic ability, helps us see data in new ways. Magic happens when interesting data meets effective design: when statistician meets designer (sometimes within the same person). We are fortunate to live in a time when interactive and animated graphs are becoming commonplace, and these tools can be incredibly powerful. Other times, simpler graphs might be more effective. The key is to present data in a way that is visually appealing while allowing the data to speak for itself.

Changing perceptions through data

While graphs and charts can lead to misunderstandings, there is ultimately "truth in numbers." As Steven Levitt and Stephen Dubner say in Freakonomics , "[T]eachers and criminals and real-estate agents may lie, and politicians, and even C.I.A. analysts. But numbers don't." Indeed, consideration of data can often be the easiest way to glean objective insights. Again from Freakonomics : "There is nothing like the sheer power of numbers to scrub away layers of confusion and contradiction."

Data can help us understand the world as it is, not as we believe it to be. As Hans Rosling demonstrates, it's often not ignorance but our preconceived ideas that get in the way of understanding the world as it is. Publicly-available statistics can reshape our world view: Rosling encourages us to "let the dataset change your mindset."

Chris Jordan's powerful images of waste and addiction make us face, rather than deny, the facts. It's easy to hear and then ignore that we use and discard 1 million plastic cups every 6 hours on airline flights alone. When we're confronted with his powerful image, we engage with that fact on an entirely different level (and may never see airline plastic cups in the same way again).

The ability to see data expands our perceptions of the world in ways that we're just beginning to understand. Computer simulations allow us to see how diseases spread, how forest fires might be contained, how terror networks communicate. We gain understanding of these things in ways that were unimaginable only a few decades ago. When Blaise Aguera y Arcas demonstrates Photosynth, we feel as if we're looking at the future. By linking together user-contributed digital images culled from all over the Internet, he creates navigable "immensely rich virtual models of every interesting part of the earth" created from the collective memory of all of us. Deb Roy does somewhat the same thing with language, pulling in publicly available social media feeds to analyze national and global conversation trends.

Roy sums it up with these powerful words: "What's emerging is an ability to see new social structures and dynamics that have previously not been seen. ...The implications here are profound, whether it's for science, for commerce, for government, or perhaps most of all, for us as individuals."

Let's begin with the TEDTalk from David McCandless, a self-described "data detective" who describes how to highlight hidden patterns in data through its artful representation.

David McCandless

The beauty of data visualization.

i. Data obtained June 2012 from “How Much Data Is Created Every Minute?” on http://mashable.com/2012/06/22/data-created-every-minute/ .

Relevant talks

Hans Rosling

The magic washing machine.

Nathalie Miebach

Art made of storms.

Chris Jordan

Turning powerful stats into art.

Blaise Agüera y Arcas

How photosynth can connect the world's images.

The birth of a word

Writing With Statistics: Mistakes to Watch Out For

Numbers are power. Adding relevant statistics to your content can strengthen any argument. But if not used carefully, numbers create more problems than they solve.

I wrote a book for content writers called From Reads To Leads . You should check it out to learn what rules you need to follow to write content that converts readers into leads. One of these rules is about using statistics in writing. Go to my home page to get the book or read the first chapter.

Many writers pick up numbers off the street to make their messages more compelling. They aren’t looking to support their arguments or to make their stories more accurate. They aren’t looking for truth. 

Let's look at this example:

If 36 percent of Americans use food delivery services, does this mean that the popularity of these services is growing? To show growth in popularity, we would need to compare the percentage of Americans who used food delivery services in March 2019, for example, with the percentage who used them in March 2021. Growth can only be demonstrated over time. Since there’s nothing to which readers can compare this 36 percent, they might doubt whether the popularity of food delivery services is actually growing.

Let's look at another example:

Any percentage is meaningless to your readers unless you compare it against some base percentage. The stats in the example I've just mentioned look reasonable. The author compares Black Friday sales completed using mobile devices last year with sales completed using mobile devices a few years ago. But let’s think about it for a second. Don’t these statistics raise any questions? Firstly of all, they come from two different sources. They may have been collected using wildly different methodologies and by surveying entirely different demographics. There’s no way for the reader to know whether these percentages can reasonably be compared.

Sometimes a percentage might look high, but without context, it might not be telling the whole story. You need to dig deeper to uncover the truth:

The author did solid research to help her readers arrive at the conclusion that despite a seemingly large number of women-owned businesses, there’s still gender inequality in entrepreneurship.

How to write with statistics

As you use statistics in your writing, here are a few things you need to pay attention to:

1 . Numbers can be just as ambiguous as words and need just as much explanation.

For example:

Is 57 percent good or bad? This statement requires an explanation:

Now it’s clear that we’re making progress. 

2. Don’t just throw numbers everywhere you can because it’s considered a good SEO practice. Your statistics need to help you make your point . They can make your arguments believable. 

For example, let’s say our key message is “eating fat keeps you healthy.” One of the arguments we can use to support this message is that polyunsaturated fats can lower cholesterol levels, which, as a result, can lower the risk of heart disease and stroke. We can use statistics to make this argument believable:

‍ We didn’t use these stats to demonstrate expertise . We added them to support our key message.

3. When trying to prove your argument with statistics, you need to be sure that what you’re saying is true. I you have doubts, you need to look first at the numbers to help you shape your message and ideas. Don’t do it the other way round. 

If you’ve already shaped your message about something, you’ll be tempted to look for data to prove that you’re right. But the truth is, you might be wrong. You can find seemingly legitimate evidence to support any claim, but your argument won’t be convincing if it’s built on a shaky foundation. 

The next time you find yourself thinking that what you want to say might not be accurate, don’t head to Google to prove you’re right. Instead, look to answer the question with data. Good writing is about telling the truth, not trying to dupe the reader.

4. Numbers without context or specific details are just that—numbers. They don’t help readers arrive at conclusions.

Now let’s see how to author, uses details to communicate the gravity of the situation:

The statement “the cost of college increased by more than 25% in the last 10 years” doesn’t give readers a clear idea of the growing cost of higher education. But comparing what students paid (on average) for a college education in 1978, 2008, and today helps the reader realize that the costs of college are increasing at a breakneck pace and that something has to be done about it.

You need statistics to prove your arguments. And when citing statistics, you must obey the same rules of clarity you obey with words. Learn more about these rules in my book From Reads To Leads.

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When it comes to attracting leads to your business, there’s no substitute for content that teaches your audience something valuable. Read on to discover how you can share valuable knowledge through different types of content.

Do you know for sure who exactly will be reading your content? You don't. Read on to learn how to write content that serves and resonates with as many people as possible.

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1.2: Importance of Statistics

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Learning Objectives

• Give examples of statistics encountered in everyday life
• Give examples of how statistics can lend credibility to an argument

Like most people, you probably feel that it is important to "take control of your life." But what does this mean? Partly, it means being able to properly evaluate the data and claims that bombard you every day. If you cannot distinguish good from faulty reasoning, then you are vulnerable to manipulation and to decisions that are not in your best interest. Statistics provides tools that you need in order to react intelligently to information you hear or read. In this sense, statistics is one of the most important things that you can study.

To be more specific, here are some claims that we have heard on several occasions. (We are not saying that each one of these claims is true!)

• \(4\) out of \(5\) dentists recommend Dentine.
• Almost \(85\%\) of lung cancers in men and \(45\%\) in women are tobacco-related.
• Condoms are effective \(94\%\) of the time.
• Native Americans are significantly more likely to be hit crossing the street than are people of other ethnicities.
• People tend to be more persuasive when they look others directly in the eye and speak loudly and quickly.
• Women make \(75\) cents to every dollar a man makes when they work the same job.
• A surprising new study shows that eating egg whites can increase one's life span.
• People predict that it is very unlikely there will ever be another baseball player with a batting average over \(400\).
• There is an \(80\%\) chance that in a room full of \(30\) people that at least two people will share the same birthday.
• \(79.48\%\) of all statistics are made up on the spot.

All of these claims are statistical in character. We suspect that some of them sound familiar; if not, we bet that you have heard other claims like them. Notice how diverse the examples are. They come from psychology, health, law, sports, business, etc. Indeed, data and data interpretation show up in discourse from virtually every facet of contemporary life.

Statistics are often presented in an effort to add credibility to an argument or advice. You can see this by paying attention to television advertisements. Many of the numbers thrown about in this way do not represent careful statistical analysis. They can be misleading and push you into decisions that you might find cause to regret. For these reasons, learning about statistics is a long step towards taking control of your life. (It is not, of course, the only step needed for this purpose.) The present textbook is designed to help you learn statistical essentials. It will make you into an intelligent consumer of statistical claims.

You can take the first step right away. To be an intelligent consumer of statistics, your first reflex must be to question the statistics that you encounter. The British Prime Minister Benjamin Disraeli is quoted by Mark Twain as having said, "There are three kinds of lies -- lies, damned lies, and statistics." This quote reminds us why it is so important to understand statistics. So let us invite you to reform your statistical habits from now on. No longer will you blindly accept numbers or findings. Instead, you will begin to think about the numbers, their sources, and most importantly, the procedures used to generate them.

We have put the emphasis on defending ourselves against fraudulent claims wrapped up as statistics. We close this section on a more positive note. Just as important as detecting the deceptive use of statistics is the appreciation of the proper use of statistics. You must also learn to recognize statistical evidence that supports a stated conclusion. Statistics are all around you, sometimes used well, sometimes not. We must learn how to distinguish the two cases.

Quant Analysis 101: Descriptive Statistics

Everything You Need To Get Started (With Examples)

By: Derek Jansen (MBA) | Reviewers: Kerryn Warren (PhD) | October 2023

If you’re new to quantitative data analysis , one of the first terms you’re likely to hear being thrown around is descriptive statistics. In this post, we’ll unpack the basics of descriptive statistics, using straightforward language and loads of examples . So grab a cup of coffee and let’s crunch some numbers!

Overview: Descriptive Statistics

What are descriptive statistics.

• Descriptive vs inferential statistics
• Why the descriptives matter
• The “ Big 7 ” descriptive statistics
• Key takeaways

At the simplest level, descriptive statistics summarise and describe relatively basic but essential features of a quantitative dataset – for example, a set of survey responses. They provide a snapshot of the characteristics of your dataset and allow you to better understand, roughly, how the data are “shaped” (more on this later). For example, a descriptive statistic could include the proportion of males and females within a sample or the percentages of different age groups within a population.

Another common descriptive statistic is the humble average (which in statistics-talk is called the mean ). For example, if you undertook a survey and asked people to rate their satisfaction with a particular product on a scale of 1 to 10, you could then calculate the average rating. This is a very basic statistic, but as you can see, it gives you some idea of how this data point is shaped .

What about inferential statistics?

Now, you may have also heard the term inferential statistics being thrown around, and you’re probably wondering how that’s different from descriptive statistics. Simply put, descriptive statistics describe and summarise the sample itself , while inferential statistics use the data from a sample to make inferences or predictions about a population .

Put another way, descriptive statistics help you understand your dataset , while inferential statistics help you make broader statements about the population , based on what you observe within the sample. If you’re keen to learn more, we cover inferential stats in another post , or you can check out the explainer video below.

Why do descriptive statistics matter?

While descriptive statistics are relatively simple from a mathematical perspective, they play a very important role in any research project . All too often, students skim over the descriptives and run ahead to the seemingly more exciting inferential statistics, but this can be a costly mistake.

The reason for this is that descriptive statistics help you, as the researcher, comprehend the key characteristics of your sample without getting lost in vast amounts of raw data. In doing so, they provide a foundation for your quantitative analysis . Additionally, they enable you to quickly identify potential issues within your dataset – for example, suspicious outliers, missing responses and so on. Just as importantly, descriptive statistics inform the decision-making process when it comes to choosing which inferential statistics you’ll run, as each inferential test has specific requirements regarding the shape of the data.

Long story short, it’s essential that you take the time to dig into your descriptive statistics before looking at more “advanced” inferentials. It’s also worth noting that, depending on your research aims and questions, descriptive stats may be all that you need in any case . So, don’t discount the descriptives! 

The “Big 7” descriptive statistics

With the what and why out of the way, let’s take a look at the most common descriptive statistics. Beyond the counts, proportions and percentages we mentioned earlier, we have what we call the “Big 7” descriptives. These can be divided into two categories – measures of central tendency and measures of dispersion.

Measures of central tendency

True to the name, measures of central tendency describe the centre or “middle section” of a dataset. In other words, they provide some indication of what a “typical” data point looks like within a given dataset. The three most common measures are:

The mean , which is the mathematical average of a set of numbers – in other words, the sum of all numbers divided by the count of all numbers. 

The median , which is the middlemost number in a set of numbers, when those numbers are ordered from lowest to highest.

The mode , which is the most frequently occurring number in a set of numbers (in any order). Naturally, a dataset can have one mode, no mode (no number occurs more than once) or multiple modes.

To make this a little more tangible, let’s look at a sample dataset, along with the corresponding mean, median and mode. This dataset reflects the service ratings (on a scale of 1 – 10) from 15 customers.

As you can see, the mean of 5.8 is the average rating across all 15 customers. Meanwhile, 6 is the median . In other words, if you were to list all the responses in order from low to high, Customer 8 would be in the middle (with their service rating being 6). Lastly, the number 5 is the most frequent rating (appearing 3 times), making it the mode.

Together, these three descriptive statistics give us a quick overview of how these customers feel about the service levels at this business. In other words, most customers feel rather lukewarm and there’s certainly room for improvement. From a more statistical perspective, this also means that the data tend to cluster around the 5-6 mark , since the mean and the median are fairly close to each other.

To take this a step further, let’s look at the frequency distribution of the responses . In other words, let’s count how many times each rating was received, and then plot these counts onto a bar chart.

As you can see, the responses tend to cluster toward the centre of the chart , creating something of a bell-shaped curve. In statistical terms, this is called a normal distribution .

As you delve into quantitative data analysis, you’ll find that normal distributions are very common , but they’re certainly not the only type of distribution. In some cases, the data can lean toward the left or the right of the chart (i.e., toward the low end or high end). This lean is reflected by a measure called skewness , and it’s important to pay attention to this when you’re analysing your data, as this will have an impact on what types of inferential statistics you can use on your dataset.

Measures of dispersion

While the measures of central tendency provide insight into how “centred” the dataset is, it’s also important to understand how dispersed that dataset is . In other words, to what extent the data cluster toward the centre – specifically, the mean. In some cases, the majority of the data points will sit very close to the centre, while in other cases, they’ll be scattered all over the place. Enter the measures of dispersion, of which there are three:

Range , which measures the difference between the largest and smallest number in the dataset. In other words, it indicates how spread out the dataset really is.

Variance , which measures how much each number in a dataset varies from the mean (average). More technically, it calculates the average of the squared differences between each number and the mean. A higher variance indicates that the data points are more spread out , while a lower variance suggests that the data points are closer to the mean.

Standard deviation , which is the square root of the variance . It serves the same purposes as the variance, but is a bit easier to interpret as it presents a figure that is in the same unit as the original data . You’ll typically present this statistic alongside the means when describing the data in your research.

Again, let’s look at our sample dataset to make this all a little more tangible.

As you can see, the range of 8 reflects the difference between the highest rating (10) and the lowest rating (2). The standard deviation of 2.18 tells us that on average, results within the dataset are 2.18 away from the mean (of 5.8), reflecting a relatively dispersed set of data .

For the sake of comparison, let’s look at another much more tightly grouped (less dispersed) dataset.

As you can see, all the ratings lay between 5 and 8 in this dataset, resulting in a much smaller range, variance and standard deviation . You might also notice that the data are clustered toward the right side of the graph – in other words, the data are skewed. If we calculate the skewness for this dataset, we get a result of -0.12, confirming this right lean.

In summary, range, variance and standard deviation all provide an indication of how dispersed the data are . These measures are important because they help you interpret the measures of central tendency within context . In other words, if your measures of dispersion are all fairly high numbers, you need to interpret your measures of central tendency with some caution , as the results are not particularly centred. Conversely, if the data are all tightly grouped around the mean (i.e., low dispersion), the mean becomes a much more “meaningful” statistic).

Key Takeaways

We’ve covered quite a bit of ground in this post. Here are the key takeaways:

• Descriptive statistics, although relatively simple, are a critically important part of any quantitative data analysis.
• Measures of central tendency include the mean (average), median and mode.
• Skewness indicates whether a dataset leans to one side or another
• Measures of dispersion include the range, variance and standard deviation

If you’d like hands-on help with your descriptive statistics (or any other aspect of your research project), check out our private coaching service , where we hold your hand through each step of the research journey. 

Psst… there’s more!

This post is an extract from our bestselling short course, Methodology Bootcamp . If you want to work smart, you don't want to miss this .

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Importance of Statistics in Daily Life Essay

In order to explore the use of statistics in everyday life, the essay should start with a debatable statement. On a daily basis, people collect and analyze a lot of information presented in numbers, and it is closely associated with different aspects of their lives. Thus, it is typical to apply the elementary statistical approaches to examining the learned material about everyday activities. It can help to get the average results in relation to actual events or phenomena.

However, many people do not guess that they use such principles as the base for their knowledge. So, to reveal the importance of statistics in daily life, this essay will provide different real-life examples and explain the application of various data analysis methods.

Moreover, when persons have to present the solution to this or that question or decide how to act in the definite situation they also use the statistical data on the issue as one of the main arguments which can influence the further development of the case. That is why statistics can be defined as the science which deals with the data’s collection and its interpretation according to the certain task, and the results of the research can be effectively used in many spheres. From this point, the relative value of statistics for the everyday life is in the fact that people have the opportunity to plan their actions according to the statistical data with references to those results which can satisfy or not their expectations.

People are usually interested in the average temperature and the weather forecasts, in the amount of people who prefer this or that product which they usually purchase. These persons listen to the economical news in which the data of statistics on the state’s development are presented and pay attention to the risks of the transport incidents before going out the house. The statistical data influence all the aspects of the people’s life during the whole day.

When an individual wants to learn about the latest news he concentrates on the information which is interesting for him personally, and these facts are often given in the form of numbers. The average results in different fields and areas from the average level of incomes in the country and the average level of attendance the local library till the average data on the consumers’ preference of brands and services can provide the basics for the people’s choices and usual decisions which are made as a part of the daily rituals and routines.

One more effective advantage of statistics is the possibility to offer the prognoses of the development of definite situations and processes. People are inclined to use the statistical prognoses when they plan such significant changes in their life as the search of the new job, new investments in companies, travelling, and long-term projects. Statistics as the science is based on the strict mathematical calculations and formulas (Bluman, 2009). That is why its methods can be discussed as the effective ways of interpreting the collected quantitative information on any aspect of the life.

It is possible to analyze the tendencies of the world’s development with references to the statistical approach and use this approach as the means to organize the everyday life according to these trends. Furthermore, many people focus on the results of the statistical researches not only at the elementary level in their daily life but also as the part of their work. Thus, accounting, economics, logistics, and many other spheres of the knowledge use statistics (Black, 2009). Moreover, working with their computers, people often refer to the statistical analysis of the data in order to receive the average result or form the picture of the process’s development (Mann, 2010).

Today, it is not necessary for people to examine and test a lot of material to get the information about its appropriateness for the people’s everyday activity or about tendencies of the phenomenon’s progress because all these data can be taken in the form of the statistical graphs or percentages. There are many daily questions the answers to which are hidden in the statistical data.

Black, K. (2009). Business statistics: Contemporary decision making . USA: Wiley.

Bluman, A. G. (2009). Elementary statistics: A step by step approach. New York, NY: McGraw-Hill.

Mann, P. S. (2010). Introductory statistics . Hoboken, NJ: Wiley.

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IvyPanda. (2023, October 30). Importance of Statistics in Daily Life Essay. https://ivypanda.com/essays/statistics-in-the-everyday-life/

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• Guidance hub

Writing about statistics

Policy details, table of contents, things to be aware of.

Your department may have standards that you have to work to when it comes to writing about statistics, particularly when it involves writing content for webpages.

This guidance is a general set of advice and guidelines for any and all statistical commentary. In the future, producers from across the Government Statistical Service (GSS) may produce different products for different users. If this happens, each different product will have its own unique approach to statistical commentary to fit the intended user that it is aimed at.

This guidance replaces the following:

• Standards for statistical reports
• Writing about statistics: guidance for the Government Statistical Service on preparing first releases
• National statistician’s guidance: presentation and publication of official statistics

Introduction

Statistical commentary is required to bring numbers to life..

Commentary should do much more than just describe the statistics in words. It should help the user to understand the meaning of patterns, trends and limitations, and build on any factual and public information already known about the subject matter.

Clear, insightful and professionally sound commentary supports informed decision-making and democratic debate.

Statistics and data should be presented clearly, explained meaningfully and provide authoritative insights that serve the public good.

Code of Practice for Statistics, United Kingdom (UK) Statistics Authority, 2018

What we mean by good commentary

Good commentary draws attention to important findings, puts them in context and provides a clear take away message for users. It supports and enables the appropriate use of statistics. It clearly explains issues of quality and reliability, how these impact on the use of the statistics and the conclusions that can be drawn from them.

Good commentary opens up the statistics for re-use. It ensures users fully understand the nature of the statistics and the top-line results they should be able to reproduce if undertaking further analysis.

Who this guidance is for

This guidance has been developed for producers of commentary about official statistics. The guidance may also be helpful for others who produce and report on statistics.

The guidance has been developed by the Government Statistical Service Good Practice Team.

 Th e aim of this guidance

The aim of this guidance is to help producers to write statistical commentary that provides insight, and is impartial, helpful and accessible to a range of audiences.

What this guidance covers

This guidance is not a set of standards, but rather provides a common approach for writing about statistics, drawing on recognised good practice.

We look at how to present a full picture of the subject, and how factors like structure and language can impact upon the messages that readers take away. We discuss the importance of considering the users of the statistics when writing commentary. We also explore how to convey to users what the statistics mean in practice, whilst keeping the commentary objective and impartial.

Top tips for writing about statistics

You should:

• understand the users and uses of your statistics — find out who uses your statistics and what the statistics are used for
• use plain language — balance the need for technically exact but complex terminology and clarity
• put the statistics into context — provide neutral and impartial information about users and uses, strengths and limitations, other relevant statistics, long-term trends and changes, geographical comparisons, and why the statistics have been collected
• present main messages clearly and concisely — concentrate on the main points of interest and do not try to summarise all of the findings in the publication
• tell users about quality and methods — be upfront and specific about caveats that are vital to interpretation and provide links to more detailed information that some users may want
• explore patterns, relationships, causes, and effects
• help users find the information they need — a contents page can be helpful for longer releases
• write clear and informative titles
• use structure to tell the statistical story
• consider the online experience — think about how users access information online
• think beyond statistical bulletins

Understand the users and uses of your statistics

A sound understanding of the users and uses of your statistics is essential to delivering effective commentary. Effective commentary caters well for different audiences.

Find out who uses your statistics and what the statistics are used for

The audience for statistics is diverse. Your commentary will be most useful if you have a clear understanding of your users and how they draw value from the statistics. What decisions are taken and arguments made? Are the statistics re-used in further analysis or publications? How do different results and levels of quality affect users’ actions?

Research might be required to identify the wide range of users and uses of the statistics. There is guidance for working with users on the GSS website.

Ensure commentary is accessible to all

Government bodies have a legal obligation to make publications accessible to all.  Avoid barriers to accessibility such as small fonts, colour contrasts that are hard to distinguish and complex walls of text.

Think about the requirements of different audiences, including people with disabilities. The Government Digital Service blog post “ Writing content for everyone ” is very helpful.

Engage with users online

StatsUserNet is an interactive website for communication between users and producers of official statistics, hosted by the Office for National Statistics. With over 3,000 individual members, it is a well-established forum for online user engagement. Make an effort to monitor the site regularly and engage with users’ posts and discussions.

Get feedback from users

When writing about statistics, it is easy to become too close to the process and unable to judge whether content is accessible, understandable and valuable. A second opinion is usually helpful.

Possible approaches:

Ask a colleague or non-specialist in your department to peer review your writing, placing themselves as a lay reader without your expert knowledge.

Consider inviting wider peer review, either through a group inside your department, or initiatives like the Government Statistical Service’s ‘scrum’ programme. Getting a perspective from outside your department can be very valuable.

Ask your users for feedback. Do they find the commentary easy to understand? Are the main messages clear?

Think about what users are trying to achieve

A ‘jobs to be done’ approach can provide a useful, simple and quick way of gaining user insight. Think about what users are trying to achieve with your statistics, not simply their inherent characteristics.

The Office for National Statistics (ONS) developed a set of user personas, based on research done with users of the ONS website.  Personas can help us think about how to present and tailor commentary for different types of user. The ONS research identified five user personas :

• Expert analysts
• Information foragers
• Inquiring citizens
• Technical users
• Policy influencers

This NHS digital publication: “ Health and Care of People with Learning Disabilities ” has a summary report, an easy read version and data files to address these different user personas.

Put the statistics into context

Help users understand the statistics in the context of the wider world: the economy, society, or the environment.

Be clear on what your statistics can and cannot be used for

Depending on the type of publication you are working on it might be useful to discuss the users and uses of the statistics. You might also describe the types of decisions that can be made using the statistics. This information demonstrates the relevance and public value of the statistics.

Mention both the known and likely uses. It is acceptable to make assumptions about what the statistics might be used for.

“The Summary Hospital-level Mortality Indicator (SHMI) is not a measure of quality of care. A higher than expected number of deaths should not immediately be interpreted as indicating poor performance and instead should be viewed as a ‘smoke alarm’ which requires further investigation. Similarly, an ‘as expected’ or ‘lower than expected’ SHMI should not immediately be interpreted as indicating satisfactory or good performance.” NHS Digital, March 2018, Summary Hospital-level Mortality Indicator (SHMI)

Discuss the findings in the context of long-term trends and changes

Don’t focus solely on the latest numbers, or on point-to-point comparisons in isolation. Instead, give the overall picture, drawing attention to individual movements only where they add value to the story.

Do not report on changes without discussing the context

For example, if you report a 2% rise, help the user understand whether this is typical or unusual in comparison to previous statistics, to other countries, or in respect to policy targets.

“The Creative Industries accounted for 9.1% of all UK services imports in 2016, the highest proportion contributed since 2010. The proportion of the UK total contributed by the Creative Industries has generally been growing since 2010, with the change in the contribution between 2015 and 2016 (up 2.0 percentage points) being higher than usual.” Department for Digital, Culture, Media and Sport, June 2018, DCMS Sectors Economic Estimates 2016: Trade

“In 2018 the UK farmland bird index was 45% of its 1970 value. The majority of this decline occurred between the late 1970s and the 1980s largely due to the negative impact of rapid changes in farmland management during this period. The decline has continued at a slower rate more recently; the smoothed index significantly decreased by 6% between 2012 and 2017.” Department for Environment, Food and Rural Affairs, November 2017, Wild Bird Population in the UK, 1970 to 2016

Make, or enable users to make, geographical comparisons

Comparisons may be made between regions, countries in the UK or internationally. Establish where equivalent data and publications are held. Comment on these and include links to the relevant websites, where appropriate. If there are differences in methods or definitions, provide appropriate caveats to avoid misleading comparisons.

Explain the strengths and limitations of the statistics in relation to likely uses

If there are key issues that affect how the statistics should be used or interpreted, mention them up front to support appropriate use.

Don’t bury important limitations in the supporting information. Avoid any implication that the statistics are free from error. Include descriptions of the main likely errors, their potential impact on the statistics, and the implications for use.

Further information about quality and methods can be found in other sections of this guidance and in our Communicating quality, uncertainty and change guidance.

Pages 4, 5 and 6 of the Welsh Index of Multiple Deprivation 2019 report outline what the index can and can’t be used for.

Be neutral and impartial

Describe policies and targets in factual terms. Don’t endorse or comment on the effectiveness of current or past policies, or comment on the appropriateness of targets.

Departmental logos are helpful for orientation but be cautious before using the logo or branding of a government programme to which the statistics relate. This can carry the risk of perceived endorsement.

Explain why the statistics have been collected

Include relevant, factual information about the policy and operational context. If the statistics are used to measure policies or targets, list or provide links to them.

• what is measured
• what the statistics show in relation to the policies or targets
• any relevant frameworks or indicators
• any relevant previous targets
• why the policy is being monitored

“Indicators are useful tools for summarising and communicating broad trends. They are not intended to incorporate all the relevant information available in the UK. They are best seen, as their name suggests, as indicative of wider changes. The UK biodiversity indicators formed a major part of the UK’s 5th National Report to the CBD in 2014, supplemented with other information relating to UK biodiversity and implementation of the Strategic Plan for Biodiversity 2011-2020. It is expected that the indicators will be amongst the information used to produce the 6th National Report to the CBD (due to be submitted in December 2018). In 2015, JNCC produced an updated mapping of the indicators against both global and European biodiversity targets.”

Department for Environment and Rural Affairs, August 2017, UK Biodiversity Indicators 2017

But, remember the message!

Remember to think about the type of context that adds the most to the messages you are trying to communicate.  Don’t give users the statistics in every type of context.

Provide interpretation for the statistics

Good commentary should help users to understand and interpret the messages from the statistics, and should be insightful and objective.

Explore relationships, causes and effects

Explore relationships, causes and effects to the extent that they can be supported by evidence. Include possible reasons, appropriately justified, to explain what the statistics show.

It can be challenging to provide insightful commentary without straying into opinion and conjecture, but you have an obligation to explain how any contextual information has been used to validate your statistics.

Explore potential reasons for the patterns that you see

Do research and keep up to date with the latest developments in your subject area. Sound knowledge of your topic and its theoretical context will help you to interpret the statistics and add value through your commentary.

“The decline in the number of certificates in Functional Skills is likely due to the changes in funding rules by the Education and Skills Funding Agency and revised guidance from DfE that post 16 students who have a grade D/grade 3 in English or maths must now be entered for GCSE resits rather than Functional Skills. In addition, colleges are also incentivised to enter students with grade E for GCSE as they gain more credit for distance travelled by improving a GCSE grade than for Functional Skills attainment.”

Office of Qualifications and Examinations Regulation (Ofqual), June 2018, ‘ Vocational and other qualifications quarterly: January to March 2018 ‘

“It is reductions from the energy production and manufacturing sectors that have been the strongest drivers for the long term trend of decreasing emissions, by switching fuel use from coal to gas and the fitting of flue gas desulphurisation in the remaining coal fired plants in the power sector. The decrease in SO2 emissions in recent years, with UK emissions falling by 61 per cent between 2012 and 2016, was largely due to the closure of a number of coal-fired power stations that had reached the end of their working lifetime. These closures, together with the conversion of a few other coal-fired units to burn biomass instead, have significantly reduced the overall coal-burning capacity.”

Department for Environment and Rural Affairs, February 2018, “ Emissions of air pollutants in the UK, 1970 to 2016 “

“The total Net Ingredient Cost (NIC) for items prescribed for alcohol dependence in 2017 was £4.42 million. This is 9% lower than in 2016 and breaks the recent trend of successive year on year increases. The decrease in cost has been mainly driven by reduced prescriptions items for Disulfiram.”

NHS Digital, May 2018, ‘ Statistics on Alcohol: England 2018 ‘

“Likewise, the decrease in passenger journeys on some systems (for example, Docklands Light Railway and Sheffield Supertram) are likely to be a result of planned work closure.” Department for Transport, June 2018, “ Light Rail and Tram Statistics England , 2017/18”

Provide insights into any trends

Mention relevant special events or circumstances that may have affected the statistics. Don’t start time series at a point that could be perceived as not being impartial. Similarly, avoid comparisons of two points that could be perceived as not being impartial.

Avoid ‘elevator’ commentary that describes every rise and fall in the numbers. Graphing the series and pointing out important features will help when examining trends.

Consult with policy teams and other specialists

Establish if there have been policy, societal or economic changes or new initiatives that may have caused the results observed and reflect this information in the commentary. Providing the analysis is evidence-based and impartial, this can legitimately be done in compliance with the Code of Practice as part of the quality assurance process.

Be mindful that what is relevant or important may change between releases

Don’t just update the numbers into the narrative of a previous release. Provide a relevant and insightful story behind the latest figures, particularly for topics that become of high national interest or feature in political debate.

Figure 2.2 in the Home Office report on Hate Crime in England and Wales 2016/17 has annotation outlining high profile incidents that aid interpretation of the statistics.

Describe the extent of the uncertainty in the statistics

Good commentary will help the reader to understand the extent of uncertainty in the statistics. It should draw attention to and make clear the nature and implications of the uncertainty associated with the statistics.

See the “Communicating Uncertainty and Change Guidance” on the GSS Policy Store for more information [20].

Present main messages clearly and concisely

Conveying the main messages from the statistics is essential to maximise public value. Focus on the most important, useful and relevant messages and present these up front.

“Statistics should be accompanied by a clear description of the main statistical messages that explains the relevance and meaning of the statistics in a way that is not materially misleading.”

Code of Practice for Statistics, UK Statistics Authority, 2018

Focus on the main points of interest

Take into account your users’ requirements and the current context. If your statistics say something important about a current debate, try to incorporate this information to add public value.

Write accessible and easy to understand main messages

Try to write the main messages so that any user can understand them. Peer review can really help here.

Update the messages as well as the numbers

Are the messages from the last reporting period still the most relevant and newsworthy, or should you revisit them? Remember that the biggest change may not be the most important one. Take account of the current context.

Don’t try to summarise all of the findings in the publication

Main points should include up to six bullet points each no longer than one sentence.

Ensure messages can stand alone

Journalists and press offices often use main messages verbatim. Well drafted messaging increases the chance of the media identifying and re-presenting appropriately. Consider whether the messages can stand alone in a newspaper article without additional explanation. If not, they may be taken out of context.

Use structure to tell the statistical story

The structure of the publication should help users understand the story behind the statistics.

Summarise the main messages at the start of the release

These should be the points that are most relevant or interesting to your users and for public debate. This ensures users come away with the main messages even if they don’t read the whole publication.

Use the inverted pyramid structure

The inverted pyramid structure is used by journalists and differs from the traditional style of academic reporting. The most important information is presented first. Further detail and less critical information can be provided afterwards.

The inverted pyramid structure breaks content up into three parts:

• Nice to know

Don’t start with lengthy background information or technical definitions

Include a short paragraph explaining what the publication is about. More detailed information can be placed in a separate section.

Descriptive subheadings help users

Active headings outline the main message making them more memorable for users.

Only include information which adds to the statistical story

Consider each sentence and whether it adds to the story. If not, the information can be presented elsehwhere without disrupting the commentary.

Write clear and informative titles

Users need clear and informative titles to help them to identify whether the statistics are of interest and relevant to them. They also help users scan online content.

Titles should stand alone

Titles should include:

• a concise description of the statistics
• the time period covered
• the geographical coverage
• how often the statistics are released
• whether the statistics are provisional or final, if applicable

Avoid ‘producer-focused’ titles

Some titles betray the author’s understandable desire to publicise the work they have done on a data collection. This may also be a legacy title used for many years, but changing titles can (and has been) done.

Aim to convey a user’s perspective of the output. The data source can be included in a subtitle.

Don’t overload the title with too much information

If necessary, provide a short paragraph of additional detail on the front page.

Use plain language

The language used in commentary should be simple, clear and appropriate for all audiences.

Use plain English

Avoid technical language, jargon and acronyms.

“Gross weekly pay in the bottom income decile was below £276 for full-time employees” is much easier to understand when written like this:

“One in ten full-time employees earned less than £276 per week”.

The plain English campaign gives more information on this area.

The Government Digital Service have put together plain English guidance and a list of words to avoid.

Be impartial and objective

Avoid sensationalism. Do not use terms that reflect a value judgement such as ‘relatively strong rate’, ‘very few’, and ‘only’.

Avoid suggestions of partiality to government, e.g. by referring to government as ‘our’ or ‘we’.

Balance the need for technically exact but complex terminology and clarity

Users will understand that even with a well understood term like ‘unemployment’ there are detailed decisions taken about classifications. These do not need to be spelt out in the main messages.

There will be times where more detail is necessary to avoid risk of confusion between related concepts.

If technical terms and definitions are unavoidable, explain them on first use

Some well-known abbreviations and acronyms may need no explanation, but it is best to be cautious and to explain any terms that may be unfamiliar to most readers.

Embedding complex definitions into the main story makes the language complex and hard to follow. Hyperlinks can take users to definitions they need.

Include a glossary of specialist terms

Signpost users to a glossary, but don’t force them to rely on one. Do not place glossaries at the start of a release.

Be cautious if using words with specific meanings in the context of statistics

Take care not to misuse words like “significant”.

In some cases, there may be plausible but uncertain explanations for patterns in the statistics. It is important to apply sound professional judgement. With careful wording, less certain explanations can also be included.

Words which suggest causality: affect, cause, consequence, effect, impact.

Words which suggest relationship but not causality: association, correlation, corresponding, equivalent, parallel.

Words which suggest a more provisional explanation: expect, believe, think, predict, envisage, forecast.

Be consistent

Use the same terms, abbreviations and units throughout to help the reader understand and draw comparisons. For example, don’t switch between “0.3 million” and “300 thousand”.

Round numbers appropriately

Make sure that the level of numerical detail is appropriate given the precision of the numbers you are reporting. Figures with lots of detail give an impression of high accuracy that may be unwarranted.

Users find it difficult to process long, complex numbers. For example, use “3.5 million” instead of “3,546,882”. Use commas to separate out thousands when writing numbers.

Write short sentences and paragraphs. Aim for 15 to 20 words per sentence and one concept per paragraph. Don’t overload sentences with lots of numbers.

Use tools to improve readability

Most word processors include tools to check readability. For example, Microsoft Word can give you a “Flesch-Kincaid Grade Level”. How to find and understand your readability score on Word.

Alternatively, if your content does not contain any sensitive unpublished material, paste it into the online Hemingway App . This will also give you a reading grade and it can help to improve your content by identifying complex sentence structures, phrasing and words.

Help users find the information they need

Help users quickly and easily navigate the publication and identify points of relevance.

A contents page can be helpful for longer releases

A contents list should to be used to give a broad overview of the structure of the publication. A long, detailed list of tables and figures is off-putting and detracts from the main messages.

Clearly state whether the statistics are accredited official statistics or official statistics

Include clear labelling in the release and supporting documents.

Where the statistics are designated as such, always use the National Statistics logo. Never use the logo on outputs that are not accredited official statistics. OSR are currently consulting on new badges following the introduction of new label of ‘accredited official statistics’. You can  find out more about the National Statistics designation refresh project on the OSR website .

Also include:

• timing of the next release
• copyright terms
• contact details for the producer

Provide or direct users to relevant supporting information

Supporting information helps users to understand and use the statistics correctly. Information should be readily available from the website landing page of a release.

Provide the underlying and any related data to enable further analysis

Where possible, include links to supplementary tables and datasets (e.g. lower geographies, time series) in a convenient format to allow for the reuse of the data.

Consider providing the data in machine readable open data formats. The Connected Open Government Statistics project has more information on this.

Outline the disclosure controls in place. Consider providing links to any related datasets.

Using a standard template can ensure a consistent structure

Standard templates can be useful for regular users who will be able to locate the information they need quickly and easily. Templates also demonstrate to users that publications are from a group of similar or related statistics.

Consider the online experience

It is best practice to publish statistical releases in a HTML format (i.e. text on a webpage), as opposed to a PDF. This means thinking about the online experience is very important as it is a different user experience to reading a printed document.

Why HTML is better

Reasons for moving away from publishing content in documents (e.g. PDF, Word, Excel):

• they are not best practice in terms of accessibility
• search engines cannot look inside document formats meaning content is harder to find
• we cannot use analytical tools to assess how users read documents
• internet search engines sometimes take users directly to documents, but when this happens the user will often not be able to find out where on a website that document lives, this means it is difficult for users to tell if a document is out of date.

Read more about  why website content should be published in HTML and not PDF .

Reading speed

Reading tends to be slower online, but people expect quicker results and spend little time on a page. Clear identification of the main messages and being able to easily scan content is even more important.

Essential information and key words should be on the top left

Our eyes move across web pages from left to right, top to bottom, in an F-pattern. This places most attention on the top left of the page. Use the right and lower part of the page for supporting information that is not essential for the main story.

Style.ONS has further guidance on how we read on the web. 

Use web analytics to gain user insight

Publishing content in HTML allows producers to use website data to gain insight into how users access and navigate in publications.

Analysis of visitors to the ONS website found that only 20% of users scroll a quarter of the way down a bulletin, and 53% of people who land on a bulletin page leave the site immediately. Bulletins take 9.5 times longer to read than people actually spend on the page.

Tell users about quality and methods

Commentary should be supported by information that describes the quality of the statistics and the methods used to derive them.

“The quality of the statistics and data, including their accuracy and reliability, coherence and comparability, and timeliness and punctuality, should be monitored and reported regularly.”

Be upfront about any important caveats

Any caveats that arise because of the quality of the statistics or the methods used should be presented early on in the publication. However, ensure that these details do not dilute or obscure the main messages.

Use progressive disclosure

Adopt a tiered approach with different levels of information available for different users.

Think about user personas, In general, nontechnical users will not need to know the detailed methods involved to use the numbers with confidence.

Detailed quality and methods information should be provided and users should be signposted to it.

Be specific

Avoid general statements about the quality of the statistics. Instead, focus on how quality and methods impact on use.

Explain complex concepts

When discussing confidence intervals and other quality measures use a plain English explanation. Make sure that such explanations are as easy as possible to understand and not overly detailed. Ask a colleague or non-expert to peer review the explanation.

Explanation of confidence intervals from Department for Work and Pensions on page 2 of ‘ Fraud and error in the benefit system: final 2016 to 2017 estimates first release (PDF, 665KB) ‘

Explain the statistics are initial estimates if normally subject to later revision

“Scheduled revisions, or unscheduled corrections that result from errors, should be explained alongside the statistics, being clear on the scale, nature, cause and impact.”

Include a revisions statement which outlines:

• when the statistics are likely to be revised
• the extent and direction of any likely revision (take care to avoid conjecture)
• a link to a published Revisions Policy relating to the statistics

Smaller revisions are a measure of reliability. However, small revisions do not necessarily mean that the statistics are accurate.

To prevent confusion or the use of incorrect figures, ensure only the latest version of a revised dataset is available. Explain the nature and extent of revisions, and how these revisions affect the interpretation of the statistics.

Report quality against the European Statistical System’s quality dimensions

Relevance is the degree to which a statistical product meets user needs in terms of content and coverage.

Accuracy and reliability is how close the estimated value in the initial and final outputs are to the true result.

Timeliness and punctuality describes the time between the date of publication and the date to which the data refers, and the time between the actual publication and the planned publication of a statistic.

Accessibility and clarity is the quality and sufficiency of metadata, illustrations and accompanying advice.

Coherence and comparability is the degree to which data derived from different sources or methods, but that refers to the same topic, is similar, and the degree to which data can be compared over time and domain, for example, geographic level.

Think beyond bulletins

All of the statistical outputs made available to users should include appropriate and accessible commentary.

Adapt commentary for different users and outputs

Commentary in statistical bulletins may not be appropriate for all types of users. Good commentary can be adapted from bulletins and used elsewhere.

Use main messages from your commentary for policy colleagues, social media outputs, infographics and board reports—but adapt to suit these different users.

Use social media alongside statistical releases

Social media can help reach a wide audience and convey headline messages quickly. The Government Digital Service’s ‘Social Media Playbook ’ provides comprehensive guidance on using social media in government.

• If you have any feedback on the content or design of this page, please feel free to leave your comments: If you would like us to get in touch with you then please leave your contact details or email [email protected] directly.
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Challenging assumption-based design: red and green flag statements

Stakeholder mapping, understanding the difference between official statistics and published management information, data sharing for national crisis response, accessibility legislation: what you need to know.

Essay on Statistics: Meaning and Definition of Statistics

“Statistics”, that a word is often used, has been derived from the Latin word ‘Status’ that means a group of numbers or figures; those represent some information of our human interest.

We find statistics in everyday life, such as in books or other information papers or TV or newspapers.

Although, in the beginning it was used by Kings only for collecting information about states and other information which was needed about their people, their number, revenue of the state etc.

This was known as the science of the state because it was used only by the Kings. So it got its development as ‘Kings’ subject or ‘Science of Kings’ or we may call it as “Political Arithmetic’s”. It was for the first time, perhaps in Egypt to conduct census of population in 3050 B.C. because the king needed money to erect pyramids. But in India, it is thought, that, it started dating back to Chandra Gupta Maurya’s kingdom under Chankya to collect the data of births and deaths. TM has also been stated in Chankya’s Arthshastra.

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But now-a-days due to its pervading nature, its scope has increased and widened. It is now used in almost in all the fields of human knowledge and skills like Business, Commerce, Economics, Social Sciences, Politics, Planning, Medicine and other sciences, Physical as well as Natural.

Definition :

The term ‘Statistics’ has been defined in two senses, i.e. in Singular and in Plural sense.

“Statistics has two meanings, as in plural sense and in singular sense”.

—Oxford Dictionary

In plural sense, it means a systematic collection of numerical facts and in singular sense; it is the science of collecting, classifying and using statistics.

A. In the Plural Sense :

“Statistics are numerical statements of facts in any department of enquiry placed in relation to each other.” —A.L. Bowley

“The classified facts respecting the condition of the people in a state—especially those facts which can be stated in numbers or in tables of numbers or in any tabular or classified arrangement.” —Webster

These definitions given above give a narrow meaning to the statistics as they do not indicate its various aspects as are witnessed in its practical applications. From the this point of view the definition given by Prof. Horace Sacrist appears to be the most comprehensive and meaningful:

“By statistics we mean aggregates of facts affected to a marked extent by multiplicity of causes, numerically expressed, enumerated or estimated according to reasonable standard of accuracy, collected in a systematic manner for a predetermined purpose, and placed in relation to each other.”—Horace Sacrist

B. In the Singular Sense :

“Statistics refers to the body of technique or methodology, which has been developed for the collection, presentation and analysis of quantitative data and for the use of such data in decision making.” —Ncttor and Washerman

“Statistics may rightly be called the science of averages.” —Bowleg

“Statistics may be defined as the collection, presentation, analysis, and interpretation of numerical data.” —Croxton and Cowden

Stages of Investigations :

1. Collection of Data:

It is the first stage of investigation and is regarding collection of data. It is determined that which method of collection is needed in this problem and then data are collected.

2. Organisation of Data:

It is second stage. The data are simplified and made comparative and are classified according to time and place.

3. Presentation of Data:

In this third stage, organised data are made simple and attractive. These are presented in the form of tables diagrams and graphs.

4. Analysis of Data:

Forth stage of investigation is analysis. To get correct results, analysis is necessary. It is often undertaken using Measures of central tendencies, Measures of dispersion, correlation, regression and interpolation etc.

5. Interpretation of Data:

In this last stage, conclusions are enacted. Use of comparisons is made. On this basis, forecasting is made.

Some Modern Definitions :

From the above two senses of statistics, modem definitions have emerged as given below:

“Statistics is a body of methods for making wise decisions on the face of uncertainty.” —Wallis and Roberts

“Statistics is a body of methods for obtaining and analyzing numerical data in order to make better decisions in an uncertain world.” —Edward N. Dubois

So, from above definitions we find that science of statistics also includes the methods of collecting, organising, presenting, analysing and interpreting numerical facts and decisions are taken on their basis.

The most proper definition of statistics can be given as following after analysing the various definitions of statistics.

“Statistics in the plural sense are numerical statements of facts capable of some meaningful analysis and interpretation, and in singular sense, it relates to the collection, classification, presentation and interpretation of numerical data.”

Related Articles:

• 7 Main Characteristics of Statistics – Explained!
• Use of Statistics in Economics: Origin, Meaning and Other Details
• Nature and Subject Matter of Statistics
• Relation of Statistics with other Sciences

Stanford's Rachel Heck pens first-person essay to explain why she won't go pro

T his spring, after Rachel Heck completes her senior year at Stanford, she’ll put her golf clubs away and take on an internship in private equity. She’ll also be pinned as a Lieutenant of the United States Air Force. Heck explained her reasons for not turning professional in a first-person essay on nolayingup.com.

“I was strongly considering attributing my decision to my injuries,” wrote Heck, who has grappled with several in recent years. “It is true that even if I wanted to, I do not know if my body would hold up on tour. But frankly, after a couple of years of painful deliberation, I have come to realize that I do not want to play professional golf.

"I do not want a life on the road and in the public eye. I no longer dream of the U.S. Open trophies and the Hall of Fame. And I realize now that these dreams were never what my dad intended when he first put a club in my hand.”

https://twitter.com/rachelheck2020/status/1772274878592266370

Heck qualified for the U.S. Women’s Open at age 15 and, as a hotshot junior, suffered a back injury that left her sidelined from the game. Without golf, she felt lost, and during a period of darkness, decided that she wanted to pursue the Air Force ROTC to find something more. Heck’s parents told her she was crazy, but she persisted.

As a freshman at Stanford, with dreams of playing on the LPGA and serving in the Air Force in full throttle , Heck set an NCAA scoring record (69.72) en route to sweeping the postseason.

Heck won six times in nine starts in 2021, including her last five events. She became the third player in NCAA history to sweep the postseason, winning the Pac-12 Championship, NCAA regionals and nationals. She posted 15 of 25 rounds in the 60s, including 12 consecutive.

But, as her college career progressed, more injuries followed. While Heck intends to pass on the professional life, she does plan to continue to play amateur golf, following a similar path set by Wake Forest grad Emilia Migliaccio.

“I have grappled with anger, hope, depression, joy, and everything in between,” Heck wrote, “but amid each trial in which I so desperately sought the clarity of a deeper meaning, God always showed me the next step. Right now, the next step is not professional golf.”

This article originally appeared on Golfweek: Stanford's Rachel Heck pens first-person essay to explain why she won't go pro

‘Dimbulb’ Trump Torched After Rambling Attempt To Explain Gettysburg Goes Wrong

Overnight Editor, HuffPost

Donald Trump’s attempt to explain the Battle of Gettysburg took some strange verbal detours ― and his critics were quick to call him out over it.

“Gettysburg, what an unbelievable battle that was. The Battle of Gettysburg,” the former president said at a rally in Pennsylvania on Saturday. “What an unbelievable ― I mean, it was so much and so interesting, and so vicious and horrible, and so beautiful in so many different ways.”

Trump continued:

“Gettysburg. Wow. I go to Gettysburg, Pennsylvania, to look and to watch. And the statement of Robert E. Lee ― who’s no longer in favor, did you ever notice that? No longer in favor ― ‘Never fight uphill, me boys, never fight uphill.’ They were fighting uphill. He said, ‘Wow, that was a big mistake.’ He lost his great general, and they were fighting. ‘Never fight uphill, me boys!’ But it was too late.”

The ramble was made even more surreal when someone just over Trump’s left shoulder began making odd faces midway through:

Trump goes on a weird rant about the battle of Gettysburg and then notes of Robert E Lee that "he's no longer in favor. Did you ever notice that?" pic.twitter.com/hs9GjmCh6K — Aaron Rupar (@atrupar) April 14, 2024

Trump has been prone to verbal gaffes , stumbles and miscues , especially lately .

But even critics on X, formerly Twitter, were left perplexed by his attempt to describe one of the most famous battles in American history:

This is the part where Trump does a mash-up of the Civil War and Pirates of the Caribbean. https://t.co/H8aszJvzwF — Heather Thomas (@HeatherThomasAF) April 14, 2024

Trump: "Gettysburg! Wow!" What a dimbulb. — Stephen King (@StephenKing) April 14, 2024

So @realDonaldTrump Gettysburg was "Beautiful" and "it represented such a big portion of the success of this country." Really? Oh and "Robert E. Lee is no longer in favor"! Do you know why he is no longer in favor? Because he was a damn insurrectionist! On June 7, 1865, Robert E.… https://t.co/VUxRNPfLjR — Michael Steele (@MichaelSteele) April 14, 2024

His utter stupidity has always amazed me more than his psychopathy. https://t.co/X3pU976ln6 — George Conway (@gtconway3d) April 14, 2024

Trump: Gettysburg, what an amazing, horrible, just incredible, classy, terrible thing, really beautiful. I kinda went there, but had the wrong address. Robert E Lee a war hero that wasn’t captured, loser on the hill, but we miss him, really a great guy, believe me https://t.co/CZ0ABdwk1s — Jared Moskowitz (@JaredEMoskowitz) April 14, 2024

Trump, in his bizarro history lesson, has Robert E. Lee saying to his troops "Never fight uphill, me boys," as if he was the Lucky Charms leprechaun. https://t.co/vpkYzPTUsI — James Surowiecki (@JamesSurowiecki) April 15, 2024

“I go to Gettysburg, Pennsylvania, to look and to watch” I wish more people would note how frequently Trump says simple sentences with simple words that are still so utterly crazy that no one in history has ever said them before https://t.co/5bqRXHuFhY — Roger Sollenberger (@SollenbergerRC) April 15, 2024

Trump just gave his own Gettysburg Address. It was incoherent. https://t.co/3G04VaIF3b — davidrlurie (@davidrlurie) April 14, 2024

The man's brain is mush. Imagine thinking this demented buffoon should become the president of the United States. https://t.co/y6FvYJdr4v — Republicans against Trump (@RpsAgainstTrump) April 14, 2024

If we asked Trump which General led Pickett’s Charge at Gettysburg, is there any chance he would give the right answer? — Ron Filipkowski (@RonFilipkowski) April 14, 2024

Civil War historian here anyone who is a fan of the Confederate traitor and enslaver Robert E Lee should not ever be the President of the United States. https://t.co/SkcA8MoL2H — Manisha Sinha (@ProfMSinha) April 14, 2024

Guarantee this is the totality of what Trump knows about Gettysburg https://t.co/p6D2M48QWW — Greg Pinelo (@gregpinelo) April 14, 2024

“Never fight uphill, me boys.” Was Robert E Lee an Irish pirate? https://t.co/OVCxH0IuSD — Shannon Watts (@shannonrwatts) April 14, 2024

Trump is one of those idiots who asks Gettysburg park rangers if the monuments were there during the battle. — Bob Cesca (@bobcesca_go) April 14, 2024

Donald Trump always talks about history (or, well, *anything*) like a fourth-grader doing a book report on a book he didn’t read. — Mrs. Betty Bowers (@BettyBowers) April 14, 2024

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The Beginner's Guide to Statistical Analysis | 5 Steps & Examples

Statistical analysis means investigating trends, patterns, and relationships using quantitative data . It is an important research tool used by scientists, governments, businesses, and other organisations.

To draw valid conclusions, statistical analysis requires careful planning from the very start of the research process . You need to specify your hypotheses and make decisions about your research design, sample size, and sampling procedure.

After collecting data from your sample, you can organise and summarise the data using descriptive statistics . Then, you can use inferential statistics to formally test hypotheses and make estimates about the population. Finally, you can interpret and generalise your findings.

This article is a practical introduction to statistical analysis for students and researchers. We’ll walk you through the steps using two research examples. The first investigates a potential cause-and-effect relationship, while the second investigates a potential correlation between variables.

Table of contents

Step 1: write your hypotheses and plan your research design, step 2: collect data from a sample, step 3: summarise your data with descriptive statistics, step 4: test hypotheses or make estimates with inferential statistics, step 5: interpret your results, frequently asked questions about statistics.

To collect valid data for statistical analysis, you first need to specify your hypotheses and plan out your research design.

Writing statistical hypotheses

The goal of research is often to investigate a relationship between variables within a population . You start with a prediction, and use statistical analysis to test that prediction.

A statistical hypothesis is a formal way of writing a prediction about a population. Every research prediction is rephrased into null and alternative hypotheses that can be tested using sample data.

While the null hypothesis always predicts no effect or no relationship between variables, the alternative hypothesis states your research prediction of an effect or relationship.

• Null hypothesis: A 5-minute meditation exercise will have no effect on math test scores in teenagers.
• Alternative hypothesis: A 5-minute meditation exercise will improve math test scores in teenagers.
• Null hypothesis: Parental income and GPA have no relationship with each other in college students.
• Alternative hypothesis: Parental income and GPA are positively correlated in college students.

Planning your research design

A research design is your overall strategy for data collection and analysis. It determines the statistical tests you can use to test your hypothesis later on.

First, decide whether your research will use a descriptive, correlational, or experimental design. Experiments directly influence variables, whereas descriptive and correlational studies only measure variables.

• In an experimental design , you can assess a cause-and-effect relationship (e.g., the effect of meditation on test scores) using statistical tests of comparison or regression.
• In a correlational design , you can explore relationships between variables (e.g., parental income and GPA) without any assumption of causality using correlation coefficients and significance tests.
• In a descriptive design , you can study the characteristics of a population or phenomenon (e.g., the prevalence of anxiety in U.S. college students) using statistical tests to draw inferences from sample data.

Your research design also concerns whether you’ll compare participants at the group level or individual level, or both.

• In a between-subjects design , you compare the group-level outcomes of participants who have been exposed to different treatments (e.g., those who performed a meditation exercise vs those who didn’t).
• In a within-subjects design , you compare repeated measures from participants who have participated in all treatments of a study (e.g., scores from before and after performing a meditation exercise).
• In a mixed (factorial) design , one variable is altered between subjects and another is altered within subjects (e.g., pretest and posttest scores from participants who either did or didn’t do a meditation exercise).
• Experimental
• Correlational

First, you’ll take baseline test scores from participants. Then, your participants will undergo a 5-minute meditation exercise. Finally, you’ll record participants’ scores from a second math test.

In this experiment, the independent variable is the 5-minute meditation exercise, and the dependent variable is the math test score from before and after the intervention. Example: Correlational research design In a correlational study, you test whether there is a relationship between parental income and GPA in graduating college students. To collect your data, you will ask participants to fill in a survey and self-report their parents’ incomes and their own GPA.

Measuring variables

When planning a research design, you should operationalise your variables and decide exactly how you will measure them.

For statistical analysis, it’s important to consider the level of measurement of your variables, which tells you what kind of data they contain:

• Categorical data represents groupings. These may be nominal (e.g., gender) or ordinal (e.g. level of language ability).
• Quantitative data represents amounts. These may be on an interval scale (e.g. test score) or a ratio scale (e.g. age).

Many variables can be measured at different levels of precision. For example, age data can be quantitative (8 years old) or categorical (young). If a variable is coded numerically (e.g., level of agreement from 1–5), it doesn’t automatically mean that it’s quantitative instead of categorical.

Identifying the measurement level is important for choosing appropriate statistics and hypothesis tests. For example, you can calculate a mean score with quantitative data, but not with categorical data.

In a research study, along with measures of your variables of interest, you’ll often collect data on relevant participant characteristics.

In most cases, it’s too difficult or expensive to collect data from every member of the population you’re interested in studying. Instead, you’ll collect data from a sample.

Statistical analysis allows you to apply your findings beyond your own sample as long as you use appropriate sampling procedures . You should aim for a sample that is representative of the population.

Sampling for statistical analysis

There are two main approaches to selecting a sample.

• Probability sampling: every member of the population has a chance of being selected for the study through random selection.
• Non-probability sampling: some members of the population are more likely than others to be selected for the study because of criteria such as convenience or voluntary self-selection.

In theory, for highly generalisable findings, you should use a probability sampling method. Random selection reduces sampling bias and ensures that data from your sample is actually typical of the population. Parametric tests can be used to make strong statistical inferences when data are collected using probability sampling.

But in practice, it’s rarely possible to gather the ideal sample. While non-probability samples are more likely to be biased, they are much easier to recruit and collect data from. Non-parametric tests are more appropriate for non-probability samples, but they result in weaker inferences about the population.

If you want to use parametric tests for non-probability samples, you have to make the case that:

• your sample is representative of the population you’re generalising your findings to.
• your sample lacks systematic bias.

Keep in mind that external validity means that you can only generalise your conclusions to others who share the characteristics of your sample. For instance, results from Western, Educated, Industrialised, Rich and Democratic samples (e.g., college students in the US) aren’t automatically applicable to all non-WEIRD populations.

If you apply parametric tests to data from non-probability samples, be sure to elaborate on the limitations of how far your results can be generalised in your discussion section .

Create an appropriate sampling procedure

Based on the resources available for your research, decide on how you’ll recruit participants.

• Will you have resources to advertise your study widely, including outside of your university setting?
• Will you have the means to recruit a diverse sample that represents a broad population?
• Do you have time to contact and follow up with members of hard-to-reach groups?

Your participants are self-selected by their schools. Although you’re using a non-probability sample, you aim for a diverse and representative sample. Example: Sampling (correlational study) Your main population of interest is male college students in the US. Using social media advertising, you recruit senior-year male college students from a smaller subpopulation: seven universities in the Boston area.

Calculate sufficient sample size

Before recruiting participants, decide on your sample size either by looking at other studies in your field or using statistics. A sample that’s too small may be unrepresentative of the sample, while a sample that’s too large will be more costly than necessary.

There are many sample size calculators online. Different formulas are used depending on whether you have subgroups or how rigorous your study should be (e.g., in clinical research). As a rule of thumb, a minimum of 30 units or more per subgroup is necessary.

To use these calculators, you have to understand and input these key components:

• Significance level (alpha): the risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
• Statistical power : the probability of your study detecting an effect of a certain size if there is one, usually 80% or higher.
• Expected effect size : a standardised indication of how large the expected result of your study will be, usually based on other similar studies.
• Population standard deviation: an estimate of the population parameter based on a previous study or a pilot study of your own.

Once you’ve collected all of your data, you can inspect them and calculate descriptive statistics that summarise them.

Inspect your data

There are various ways to inspect your data, including the following:

• Organising data from each variable in frequency distribution tables .
• Displaying data from a key variable in a bar chart to view the distribution of responses.
• Visualising the relationship between two variables using a scatter plot .

By visualising your data in tables and graphs, you can assess whether your data follow a skewed or normal distribution and whether there are any outliers or missing data.

A normal distribution means that your data are symmetrically distributed around a center where most values lie, with the values tapering off at the tail ends.

In contrast, a skewed distribution is asymmetric and has more values on one end than the other. The shape of the distribution is important to keep in mind because only some descriptive statistics should be used with skewed distributions.

Extreme outliers can also produce misleading statistics, so you may need a systematic approach to dealing with these values.

Calculate measures of central tendency

Measures of central tendency describe where most of the values in a data set lie. Three main measures of central tendency are often reported:

• Mode : the most popular response or value in the data set.
• Median : the value in the exact middle of the data set when ordered from low to high.
• Mean : the sum of all values divided by the number of values.

However, depending on the shape of the distribution and level of measurement, only one or two of these measures may be appropriate. For example, many demographic characteristics can only be described using the mode or proportions, while a variable like reaction time may not have a mode at all.

Calculate measures of variability

Measures of variability tell you how spread out the values in a data set are. Four main measures of variability are often reported:

• Range : the highest value minus the lowest value of the data set.
• Interquartile range : the range of the middle half of the data set.
• Standard deviation : the average distance between each value in your data set and the mean.
• Variance : the square of the standard deviation.

Once again, the shape of the distribution and level of measurement should guide your choice of variability statistics. The interquartile range is the best measure for skewed distributions, while standard deviation and variance provide the best information for normal distributions.

Using your table, you should check whether the units of the descriptive statistics are comparable for pretest and posttest scores. For example, are the variance levels similar across the groups? Are there any extreme values? If there are, you may need to identify and remove extreme outliers in your data set or transform your data before performing a statistical test.

From this table, we can see that the mean score increased after the meditation exercise, and the variances of the two scores are comparable. Next, we can perform a statistical test to find out if this improvement in test scores is statistically significant in the population. Example: Descriptive statistics (correlational study) After collecting data from 653 students, you tabulate descriptive statistics for annual parental income and GPA.

It’s important to check whether you have a broad range of data points. If you don’t, your data may be skewed towards some groups more than others (e.g., high academic achievers), and only limited inferences can be made about a relationship.

A number that describes a sample is called a statistic , while a number describing a population is called a parameter . Using inferential statistics , you can make conclusions about population parameters based on sample statistics.

Researchers often use two main methods (simultaneously) to make inferences in statistics.

• Estimation: calculating population parameters based on sample statistics.
• Hypothesis testing: a formal process for testing research predictions about the population using samples.

You can make two types of estimates of population parameters from sample statistics:

• A point estimate : a value that represents your best guess of the exact parameter.
• An interval estimate : a range of values that represent your best guess of where the parameter lies.

If your aim is to infer and report population characteristics from sample data, it’s best to use both point and interval estimates in your paper.

You can consider a sample statistic a point estimate for the population parameter when you have a representative sample (e.g., in a wide public opinion poll, the proportion of a sample that supports the current government is taken as the population proportion of government supporters).

There’s always error involved in estimation, so you should also provide a confidence interval as an interval estimate to show the variability around a point estimate.

A confidence interval uses the standard error and the z score from the standard normal distribution to convey where you’d generally expect to find the population parameter most of the time.

Hypothesis testing

Using data from a sample, you can test hypotheses about relationships between variables in the population. Hypothesis testing starts with the assumption that the null hypothesis is true in the population, and you use statistical tests to assess whether the null hypothesis can be rejected or not.

Statistical tests determine where your sample data would lie on an expected distribution of sample data if the null hypothesis were true. These tests give two main outputs:

• A test statistic tells you how much your data differs from the null hypothesis of the test.
• A p value tells you the likelihood of obtaining your results if the null hypothesis is actually true in the population.

Statistical tests come in three main varieties:

• Comparison tests assess group differences in outcomes.
• Regression tests assess cause-and-effect relationships between variables.
• Correlation tests assess relationships between variables without assuming causation.

Your choice of statistical test depends on your research questions, research design, sampling method, and data characteristics.

Parametric tests

Parametric tests make powerful inferences about the population based on sample data. But to use them, some assumptions must be met, and only some types of variables can be used. If your data violate these assumptions, you can perform appropriate data transformations or use alternative non-parametric tests instead.

A regression models the extent to which changes in a predictor variable results in changes in outcome variable(s).

• A simple linear regression includes one predictor variable and one outcome variable.
• A multiple linear regression includes two or more predictor variables and one outcome variable.

Comparison tests usually compare the means of groups. These may be the means of different groups within a sample (e.g., a treatment and control group), the means of one sample group taken at different times (e.g., pretest and posttest scores), or a sample mean and a population mean.

• A t test is for exactly 1 or 2 groups when the sample is small (30 or less).
• A z test is for exactly 1 or 2 groups when the sample is large.
• An ANOVA is for 3 or more groups.

The z and t tests have subtypes based on the number and types of samples and the hypotheses:

• If you have only one sample that you want to compare to a population mean, use a one-sample test .
• If you have paired measurements (within-subjects design), use a dependent (paired) samples test .
• If you have completely separate measurements from two unmatched groups (between-subjects design), use an independent (unpaired) samples test .
• If you expect a difference between groups in a specific direction, use a one-tailed test .
• If you don’t have any expectations for the direction of a difference between groups, use a two-tailed test .

The only parametric correlation test is Pearson’s r . The correlation coefficient ( r ) tells you the strength of a linear relationship between two quantitative variables.

However, to test whether the correlation in the sample is strong enough to be important in the population, you also need to perform a significance test of the correlation coefficient, usually a t test, to obtain a p value. This test uses your sample size to calculate how much the correlation coefficient differs from zero in the population.

You use a dependent-samples, one-tailed t test to assess whether the meditation exercise significantly improved math test scores. The test gives you:

• a t value (test statistic) of 3.00
• a p value of 0.0028

Although Pearson’s r is a test statistic, it doesn’t tell you anything about how significant the correlation is in the population. You also need to test whether this sample correlation coefficient is large enough to demonstrate a correlation in the population.

A t test can also determine how significantly a correlation coefficient differs from zero based on sample size. Since you expect a positive correlation between parental income and GPA, you use a one-sample, one-tailed t test. The t test gives you:

• a t value of 3.08
• a p value of 0.001

The final step of statistical analysis is interpreting your results.

Statistical significance

In hypothesis testing, statistical significance is the main criterion for forming conclusions. You compare your p value to a set significance level (usually 0.05) to decide whether your results are statistically significant or non-significant.

Statistically significant results are considered unlikely to have arisen solely due to chance. There is only a very low chance of such a result occurring if the null hypothesis is true in the population.

This means that you believe the meditation intervention, rather than random factors, directly caused the increase in test scores. Example: Interpret your results (correlational study) You compare your p value of 0.001 to your significance threshold of 0.05. With a p value under this threshold, you can reject the null hypothesis. This indicates a statistically significant correlation between parental income and GPA in male college students.

Note that correlation doesn’t always mean causation, because there are often many underlying factors contributing to a complex variable like GPA. Even if one variable is related to another, this may be because of a third variable influencing both of them, or indirect links between the two variables.

Effect size

A statistically significant result doesn’t necessarily mean that there are important real life applications or clinical outcomes for a finding.

In contrast, the effect size indicates the practical significance of your results. It’s important to report effect sizes along with your inferential statistics for a complete picture of your results. You should also report interval estimates of effect sizes if you’re writing an APA style paper .

With a Cohen’s d of 0.72, there’s medium to high practical significance to your finding that the meditation exercise improved test scores. Example: Effect size (correlational study) To determine the effect size of the correlation coefficient, you compare your Pearson’s r value to Cohen’s effect size criteria.

Decision errors

Type I and Type II errors are mistakes made in research conclusions. A Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s false.

You can aim to minimise the risk of these errors by selecting an optimal significance level and ensuring high power . However, there’s a trade-off between the two errors, so a fine balance is necessary.

Frequentist versus Bayesian statistics

Traditionally, frequentist statistics emphasises null hypothesis significance testing and always starts with the assumption of a true null hypothesis.

However, Bayesian statistics has grown in popularity as an alternative approach in the last few decades. In this approach, you use previous research to continually update your hypotheses based on your expectations and observations.

Bayes factor compares the relative strength of evidence for the null versus the alternative hypothesis rather than making a conclusion about rejecting the null hypothesis or not.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

The research methods you use depend on the type of data you need to answer your research question .

• If you want to measure something or test a hypothesis , use quantitative methods . If you want to explore ideas, thoughts, and meanings, use qualitative methods .
• If you want to analyse a large amount of readily available data, use secondary data. If you want data specific to your purposes with control over how they are generated, collect primary data.
• If you want to establish cause-and-effect relationships between variables , use experimental methods. If you want to understand the characteristics of a research subject, use descriptive methods.

Statistical analysis is the main method for analyzing quantitative research data . It uses probabilities and models to test predictions about a population from sample data.

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• Central Limit Theorem | Formula, Definition & Examples
• Central Tendency | Understanding the Mean, Median & Mode
• Correlation Coefficient | Types, Formulas & Examples
• Descriptive Statistics | Definitions, Types, Examples
• How to Calculate Standard Deviation (Guide) | Calculator & Examples
• How to Calculate Variance | Calculator, Analysis & Examples
• How to Find Degrees of Freedom | Definition & Formula
• How to Find Interquartile Range (IQR) | Calculator & Examples
• How to Find Outliers | Meaning, Formula & Examples
• How to Find the Geometric Mean | Calculator & Formula
• How to Find the Mean | Definition, Examples & Calculator
• How to Find the Median | Definition, Examples & Calculator
• How to Find the Range of a Data Set | Calculator & Formula
• Inferential Statistics | An Easy Introduction & Examples
• Levels of measurement: Nominal, ordinal, interval, ratio
• Missing Data | Types, Explanation, & Imputation
• Normal Distribution | Examples, Formulas, & Uses
• Null and Alternative Hypotheses | Definitions & Examples
• Poisson Distributions | Definition, Formula & Examples
• Skewness | Definition, Examples & Formula
• T-Distribution | What It Is and How To Use It (With Examples)
• The Standard Normal Distribution | Calculator, Examples & Uses
• Type I & Type II Errors | Differences, Examples, Visualizations
• Understanding Confidence Intervals | Easy Examples & Formulas
• Variability | Calculating Range, IQR, Variance, Standard Deviation
• What is Effect Size and Why Does It Matter? (Examples)
• What Is Interval Data? | Examples & Definition
• What Is Nominal Data? | Examples & Definition
• What Is Ordinal Data? | Examples & Definition
• What Is Ratio Data? | Examples & Definition
• What Is the Mode in Statistics? | Definition, Examples & Calculator

Mentor Public Library hosts talk on how to care…

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Mentor public library hosts talk on how to care for precious papers, artifacts.

Those interested can learn how to protect their heirlooms with a curator from the National Park Service at noon, May 8, at Mentor Public Library’s Main Branch, 8215 Mentor Ave.

Kelsey Voit from the James A. Garfield National Historic Site will explain how to properly care for precious papers and artifacts. This program is part of the Leaders and Legacies of the Civil War lecture series — a partnership between the Garfield site and the library.

It’s free to attend, but due to limited space, registration is required. Sign up at www.mentorpl.org or call the library at 440-255-8811 ext. 1.

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Is Mike Trout elite again? Two things Angels star has changed in 2024 and one thing that remains the same

Trout is off to a fantastic start, and we think we can explain why.

There used to be a rule of thumb among baseball writers that the season wasn't "real" until Los Angeles Angels star Mike Trout led the majors in Wins Above Replacement. It never took him long to climb the leaderboard, but once he did -- and he always did --  you could feel more confident in analyzing individual performances. 

It's been a while since that guidepost was helpful. Trout's rash of injuries, plus former teammate Shohei Ohtani's historic brilliance, has made the latter more befitting of being the standard bearer. Despite what you might be expecting us to write next, Trout is not actually atop the leaderboard right now. He is pretty close though, sitting sixth on FanGraphs and tied for eighth on Baseball Reference . That's a result of him batting .290/.362/.710 (202 OPS+) with seven home runs and 10 RBI over his first 16 games.

Mike Trout with a two-run homer to left and the #Angels take a 2-1 lead in the eighth inning. It's his seventh of the year. pic.twitter.com/C0tVNjfMPw — Rhett Bollinger (@RhettBollinger) April 16, 2024

With that in mind, we decided this would be the opportune time to dig in on Trout. What we found is that two notable things have changed about his game while one has remained the same. Scroll slowly with us while we reveal all, won't you?

1. Improved contact rate

Last August, we observed that Trout had made a philosophical shift to his game by prioritizing slugging over contact . Not only was he whiffing more frequently, he was pulling the ball more often. It wasn't too surprising of a development: the all-or-nothing approach to hitting has become MLB's house style.

If Trout's season to date is any indication, that might be changing.

Trout has significantly improved his contact rate, connecting so far on more than 83% of his swings. That would represent a new career-best figure if he's able to maintain it over the rest of the season. Additionally, he's nearly halved his overall whiff rate from last year. He's also improved his contact rate on two-strike pitches, lifting it from 65.5% to 80.7%. Trout, in turn, has seen his strikeout rate dip from 28.7% to 18.8%.

Observant readers might be thinking to themselves that this all sounds a bit like what Cody Bellinger did last year, when he authored his finest season in ages. That comparison works in another respect, too.

2. Different exit velocity

Part of what made Bellinger such a polarizing free agent was the effect his contact-heavy approach had on his exit velocity. Simply put, he wasn't hitting the ball with the force you would expect from someone who once launched 47 home runs.

Trout is following a similar pattern. His average exit velocity is down more than 2 mph in 2024 and his hard-hit percentage (that is, the share of batted balls with a 95 mph exit velocity or greater) has declined from 51.9% to 36.7%. That's the difference between being in the top 5% of the league or being around the league-average mark.

deep in his bag like a grandma with a peppermint 👝 #RepTheHalo pic.twitter.com/P6s9D33hwj — Los Angeles Angels (@Angels) April 10, 2024

Intuitively, it tracks that a heightened emphasis on making contact would result in a lower average exit velocity; you're probably employing your B-hack more often, resulting in more squibblers, flares, and dumps. 

The difference between Trout and Bellinger is that Trout is still showing top-end power. He's already clobbered two balls over 110 mph, including one that clocked in over 113 mph; Bellinger's peak last season was 109.2 mph. 

That's part of why, if we had to guess, we suspect Trout will see his aforementioned hard-hit percentage improve over the coming weeks. There's simply too much strength in his swing for him to be hitting the ball hard at a league-average rate.

As for that part of Trout's game that hasn't changed ....

3. Disciplined approach

A few years ago, some coach told a reporter that he was showing heat maps of Trout's swing rate to a young hitter to educate them on making better decisions. We would love to share the specifics of that story, but Google was unhelpful in uncovering it. As such, you'll have to take our word for it. Besides, we don't know that the finer details matter here: we suspect many coaches have used Trout as the model.

Trout, as a rule, doesn't swing very often. When he does, he swings at strikes. And, when he's in an early count -- say fewer than two strikes -- he'll look for one pitch in one area. If he gets it, he'll unleash; if he doesn't, he'll bide his time. 

If you think about Trout's approach too much you'll become convinced hitting is that simple. Right, and Beethovan just tapped some keys.

Anyhow, Trout remains incredibly disciplined. He's sporting his lowest swing rate (42.3%) and chase rate (18.2%) since the 2021 season. His swing rate heat map in zero-strike counts hasn't achieved the same orderly look of those earlier in his career, when there was one neat little glowing area, but it still informs that he has a few favorite spots he likes to hunt and he otherwise doesn't bother:

Take Trout's vision (and we mean that in multiple senses) and combine it with a renewed focus on making contact (while maintaining displays of immense raw strength). What do you get? A white-hot start to the season that has us remembering that Trout still has it.

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Descriptive Statistics | Definitions, Types, Examples

Published on July 9, 2020 by Pritha Bhandari . Revised on June 21, 2023.

Descriptive statistics summarize and organize characteristics of a data set. A data set is a collection of responses or observations from a sample or entire population.

In quantitative research , after collecting data, the first step of statistical analysis is to describe characteristics of the responses, such as the average of one variable (e.g., age), or the relation between two variables (e.g., age and creativity).

The next step is inferential statistics , which help you decide whether your data confirms or refutes your hypothesis and whether it is generalizable to a larger population.

Table of contents

Types of descriptive statistics, frequency distribution, measures of central tendency, measures of variability, univariate descriptive statistics, bivariate descriptive statistics, other interesting articles, frequently asked questions about descriptive statistics.

There are 3 main types of descriptive statistics:

• The distribution concerns the frequency of each value.
• The central tendency concerns the averages of the values.
• The variability or dispersion concerns how spread out the values are.

You can apply these to assess only one variable at a time, in univariate analysis, or to compare two or more, in bivariate and multivariate analysis.

• Go to a library
• Watch a movie at a theater
• Visit a national park

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A data set is made up of a distribution of values, or scores. In tables or graphs, you can summarize the frequency of every possible value of a variable in numbers or percentages. This is called a frequency distribution .

• Simple frequency distribution table
• Grouped frequency distribution table

From this table, you can see that more women than men or people with another gender identity took part in the study. In a grouped frequency distribution, you can group numerical response values and add up the number of responses for each group. You can also convert each of these numbers to percentages.

Measures of central tendency estimate the center, or average, of a data set. The mean, median and mode are 3 ways of finding the average.

Here we will demonstrate how to calculate the mean, median, and mode using the first 6 responses of our survey.

The mean , or M , is the most commonly used method for finding the average.

To find the mean, simply add up all response values and divide the sum by the total number of responses. The total number of responses or observations is called N .

The median is the value that’s exactly in the middle of a data set.

To find the median, order each response value from the smallest to the biggest. Then , the median is the number in the middle. If there are two numbers in the middle, find their mean.

The mode is the simply the most popular or most frequent response value. A data set can have no mode, one mode, or more than one mode.

To find the mode, order your data set from lowest to highest and find the response that occurs most frequently.

Measures of variability give you a sense of how spread out the response values are. The range, standard deviation and variance each reflect different aspects of spread.

The range gives you an idea of how far apart the most extreme response scores are. To find the range , simply subtract the lowest value from the highest value.

Standard deviation

The standard deviation ( s or SD ) is the average amount of variability in your dataset. It tells you, on average, how far each score lies from the mean. The larger the standard deviation, the more variable the data set is.

There are six steps for finding the standard deviation:

• List each score and find their mean.
• Subtract the mean from each score to get the deviation from the mean.
• Square each of these deviations.
• Add up all of the squared deviations.
• Divide the sum of the squared deviations by N – 1.
• Find the square root of the number you found.

Step 5: 421.5/5 = 84.3

Step 6: √84.3 = 9.18

The variance is the average of squared deviations from the mean. Variance reflects the degree of spread in the data set. The more spread the data, the larger the variance is in relation to the mean.

To find the variance, simply square the standard deviation. The symbol for variance is s 2 .

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Univariate descriptive statistics focus on only one variable at a time. It’s important to examine data from each variable separately using multiple measures of distribution, central tendency and spread. Programs like SPSS and Excel can be used to easily calculate these.

If you were to only consider the mean as a measure of central tendency, your impression of the “middle” of the data set can be skewed by outliers, unlike the median or mode.

Likewise, while the range is sensitive to outliers , you should also consider the standard deviation and variance to get easily comparable measures of spread.

If you’ve collected data on more than one variable, you can use bivariate or multivariate descriptive statistics to explore whether there are relationships between them.

In bivariate analysis, you simultaneously study the frequency and variability of two variables to see if they vary together. You can also compare the central tendency of the two variables before performing further statistical tests .

Multivariate analysis is the same as bivariate analysis but with more than two variables.

Contingency table

In a contingency table, each cell represents the intersection of two variables. Usually, an independent variable (e.g., gender) appears along the vertical axis and a dependent one appears along the horizontal axis (e.g., activities). You read “across” the table to see how the independent and dependent variables relate to each other.

Interpreting a contingency table is easier when the raw data is converted to percentages. Percentages make each row comparable to the other by making it seem as if each group had only 100 observations or participants. When creating a percentage-based contingency table, you add the N for each independent variable on the end.

From this table, it is more clear that similar proportions of children and adults go to the library over 17 times a year. Additionally, children most commonly went to the library between 5 and 8 times, while for adults, this number was between 13 and 16.

Scatter plots

A scatter plot is a chart that shows you the relationship between two or three variables . It’s a visual representation of the strength of a relationship.

In a scatter plot, you plot one variable along the x-axis and another one along the y-axis. Each data point is represented by a point in the chart.

From your scatter plot, you see that as the number of movies seen at movie theaters increases, the number of visits to the library decreases. Based on your visual assessment of a possible linear relationship, you perform further tests of correlation and regression.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Statistical power
• Pearson correlation
• Degrees of freedom
• Statistical significance

Methodology

• Cluster sampling
• Stratified sampling
• Focus group
• Systematic review
• Ethnography
• Double-Barreled Question

Research bias

• Implicit bias
• Publication bias
• Cognitive bias
• Placebo effect
• Pygmalion effect
• Hindsight bias
• Overconfidence bias

Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset.

• Distribution refers to the frequencies of different responses.
• Measures of central tendency give you the average for each response.
• Measures of variability show you the spread or dispersion of your dataset.
• Univariate statistics summarize only one variable  at a time.
• Bivariate statistics compare two variables .
• Multivariate statistics compare more than two variables .

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AF&PA Releases March 2024 Packaging Papers Monthly Report

WASHINGTON – The American Forest & Paper Association (AF&PA) released its March 2024 Packaging Papers Monthly report.

Total packaging papers & specialty packaging shipments in March decreased 5% compared to March 2023. They were down 2% when compared to the same 3 months of 2023.

• The operating rate for bleached packaging papers was 82.5%, up 5.7 points from March 2023 and up 9.5 points year-to-date.
• Shipments of the biggest subgrade in unbleached packaging papers -- bag & sack -- were 99,700 short tons for the month of March, down 0.3% from the same month last year but up 3.5% year-to-date.

The complete report with detailed tables, charts and historical data can be purchased by contacting Kory Bockman at [email protected] or 202-463-4716.

The American Forest & Paper Association (AF&PA) serves to advance U.S. paper and wood products manufacturers through fact-based public policy and marketplace advocacy. The forest products industry is circular by nature. AF&PA member companies make essential products from renewable and recyclable resources, generate renewable bioenergy and are committed to continuous improvement through the industry’s sustainability initiative — Better Practices, Better Planet 2030: Sustainable Products for a Sustainable Future . The forest products industry accounts for approximately 5% of the total U.S. manufacturing GDP, manufactures about $350 billion in products annually and employs about 925,000 people. The industry meets a payroll of about $65 billion annually and is among the top 10 manufacturing sector employers in 43 states. Visit AF&PA online at afandpa.org  or follow us on Twitter @ForestandPaper . 

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In March 2024, paper shipments, U.S. purchases, and inventory levels dropped by 1% compared to last year or February 2024.

AF&PA Releases February 2024 Printing-Writing Monthly Report

According to the report, total printing-writing paper shipments decreased 1% in February compared to February 2023. U.S. purchases of total printing-writing papers remained essentially flat (-0.1%) in February compared to the same month last year. Total printing-writing paper inventory levels decreased 1% when compared to January 2024.

AF&PA Releases February 2024 Packaging Papers Monthly Report

Total packaging papers & specialty packaging shipments in February increased 5% compared to February 2023. They were essentially flat when compared to the same 2 months of 2023.