Overview and Introduction
Statistics is a tricky business. The casual reader doesn't understand statistics in any great depth, while the experienced reader often knows a lot about the subject. Balancing between these two extremes is often difficult, and far from natural. The following resource is meant as a guide to writing statistics.
This guide is not meant to teach you statistics, but rather how to use statistics more effectively in your writing. This guide is designed to help you understand both how to write using other people's statistics, and how to write using your own statistics. If you want to learn how to interpret statistics, then take a course taught by a professional. For an excellent beginner's textbook, see Introduction to the Practice of Statistics by David S. Moore and George P. McCabe.
What is a Statistic?
In the casual sense, a statistic is any number that describes a group of objects. There are two main categories of statistics, descriptive and inferential.
- Descriptive: Statistics that merely describe the group they belong to.
- Inferential: Statistics that are used to draw conclusions about a larger group of people.
Examples of Descriptive Statistics
Many times, however this group of objects is a smaller subset of a larger group. By examining the smaller subset, it is often thought that information can be inferred upon the larger population. This is the basis of inferential statistics.
Examples of Inferential Statistics
In these last two examples, the researchers have not studied all people, they have studied a small group of people, and are generalizing the results to lots of people. This is known as inferential statistics, because you are inferring properties about a large group from a smaller group. As a statistician or a researcher, it is your hope that this smaller group is representative of the larger group, and that the two groups behave the same way. If they do not, then your inference may not be correct.
If you merely want to describe the data that you have for one single group, then you are using descriptive statistics. If you want to say something about a larger group, or you want your reader to infer something about a larger group, then you need to use inferential statistics. It is important to understand the difference between these two because how you use a statistic depends on what type of statistic it is.
This handout contains the following information:
- Overview and Introduction: A brief introduction to statistics and how to write with them.
- Quick Tips: An overall list of dos and don'ts about writing statistics.
- Descriptive Statistics: Statistics that are used to describe data.
- Writing Descriptive Statistics: Tips on writing descriptive statistics.
- Inferential Statistics: Statistics that are used to infer patterns in a population. This section includes some definitions, basic theory, and applications.
- Writing Inferential Statistics: Tips on writing inferential statistics. This section includes p-values and their interpretation.
- Using Visuals and Statistics: How to use visuals to represent statistics.
- Terms: Formal definitions of useful terms.
Quick Tips On Writing with Statistics
1. Never calculate or use a statistical procedure you don't fully understand. If you need a statistical procedure, and you don't understand it, then you need to consult someone or learn how to do it properly.
2. Never attempt to interpret the results of a statistical procedure you don't fully understand. If you need to interpret a particular statistic, talk with a professional statistician and make sure you understand the proper interpretation. Unlike descriptive statistics, inferential statistics is anything but black and white, there may be several valid interpretations of a given statistic, and you need to be aware of which ones are better under which circumstances.
3. If you are using statistics in a paper, consider your audience. Will they understand the statistics you are using? If not, you may need to explain the procedure that you are using in detail. This is not inappropriate! It is better to include too much information than too little. Depending on your field, this may be done using an appendix, footnotes, or directly in the text.
4. Present as much information as needed so that your reader can make their own interpretation of your data. Certainly, your job is to help the reader interpret your data, but most statistics are used to support a persuasive argument. You need to give your reader enough information that they can reconstruct your argument from your statistics. If you don't give them enough information, people will think that you are being deceptive, which can damage your credibility. You can't convince someone of anything if they are convinced that you are misleading them!
5. Use graphics and tables. Statistics can contain a lot of information, and using visuals can display a lot of information in a manner that can be quickly understood. See the section on visuals and statistics for more information.
6. If it's applicable, and you can calculate it, do include some measure of variability; typically this is a standard deviation. Even if you aren't doing any inferential statistics, this statistic provides excellent information about your data set.
7. Be wary of using statistics from other places that are not peer-reviewed. Popular magazines are notorious for including bad statistics. Oftentimes their 'sample' is a section of people who choose to respond to some online query. Their sample often includes mostly women or mostly men (depending on the magazine) but rarely do they have a good representation from all genders, and many times the magazines imply that the results generalize to the entire population. While some statistics might be generalizable, many are not. If it's not from a reliable source, then don't use it.
8. Speaking of sources, if you used a statistic, you need to provide your audience with additional information including where the statistic came from. You should be wary of statistics that seem to appear out of nowhere.
- A poor example: The ten largest cities in the U.S. comprised 54% of the total U.S. population.
- A good example: According to the United States Census Bureau, in 2000, the ten largest cities in the U.S. comprised 54% of the total U.S. population.
In the second example, your audience knows exactly where the statistic comes from (if they don't believe your statistic, they can go and check themselves) and it comes from a reputable source (the U.S. Census Bureau).
9. If you calculated a statistic, how did you calculate it? In some fields, you don't need to tell your readers how you calculated some statistics. For example, in psychology, you don't need to explain how you did an ANOVA or a t-test, but in other areas you might need to explain this in more detail.
10. Be clear as to what population(s) your statistic is meant to generalize to. If your sample included only male college students, you should be very careful if you want to generalize your results to female lawyers. Don't imply that your sample generalizes to everyone if your statistic was calculated from a more specific population.
11. If you are using inferential statistics, try to speak as plainly as possible, and put the statistics at the end of the sentence. See the Writing Inferential Statistics section for more information.
Descriptive Statistics
The mean, the mode, the median, the range, and the standard deviation are all examples of descriptive statistics. Descriptive statistics are used because in most cases, it isn't possible to present all of your data in any form that your reader will be able to quickly interpret.
Generally, when writing descriptive statistics, you want to present at least one form of central tendency (or average), that is, either the mean, median, or mode. In addition, you should present one form of variability, usually the standard deviation.
Measures of Central Tendency and Other Commonly Used Descriptive Statistics
The mean, median, and the mode are all measures of central tendency. They attempt to describe what the typical data point might look like. In essence, they are all different forms of 'the average.' When writing statistics, you never want to say 'average' because it is difficult, if not impossible, for your reader to understand if you are referring to the mean, the median, or the mode.
The Mean
The mean is the most common form of central tendency, and is what most people usually are referring to when the say average. It is simply the total sum of all the numbers in a data set, divided by the total number of data points. For example, the following data set has a mean of 4: {-1, 0, 1, 16}. That is, 16 divided by 4 is 4. If there isn't a good reason to use one of the other forms of central tendency, then you should use the mean to describe the central tendency.
The Median
The median is simply the middle value of a data set. In order to calculate the median, all values in the data set need to be ordered, from either highest to lowest, or vice versa. If there are an odd number of values in a data set, then the median is easy to calculate. If there is an even number of values in a data set, then the calculation becomes more difficult. Statisticians still debate how to properly calculate a median when there is an even number of values, but for most purposes, it is appropriate to simply take the mean of the two middle values. The median is useful when describing data sets that are skewed or have extreme values. Incomes of baseballs players, for example, are commonly reported using a median because a small minority of baseball players makes a lot of money, while most players make more modest amounts. The median is less influenced by extreme scores than the mean.
The Mode
The mode is the most commonly occurring number in the data set. The mode is best used when you want to indicate the most common response or item in a data set. For example, if you wanted to predict the score of the next football game, you may want to know what the most common score is for the visiting team, but having an average score of 15.3 won't help you if it is impossible to score 15.3 points. Likewise, a median score may not be very informative either, if you are interested in what score is most likely.
Standard Deviation
The standard deviation is a measure of variability (it is not a measure of central tendency). Conceptually it is best viewed as the 'average distance that individual data points are from the mean.' Data sets that are highly clustered around the mean have lower standard deviations than data sets that are spread out.
For example, the first data set would have a higher standard deviation than the second data set:
Notice that both groups have the same mean (5) and median (also 5), but the two groups contain different numbers and are organized much differently. This organization of a data set is often referred to as a distribution. Because the two data sets above have the same mean and median, but different standard deviation, we know that they also have different distributions. Understanding the distribution of a data set helps us understand how the data behave.
Writing with Descriptive Statistics
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.
Basic Inferential Statistics: Theory and Application
The heart of statistics is inferential statistics. Descriptive statistics are typically straightforward and easy to interpret. Unlike descriptive statistics, inferential statistics are often complex and may have several different interpretations.
The goal of inferential statistics is to discover some property or general pattern about a large group by studying a smaller group of people in the hopes that the results will generalize to the larger group. For example, we may ask residents of New York City their opinion about their mayor. We would probably poll a few thousand individuals in New York City in an attempt to find out how the city as a whole views their mayor. The following section examines how this is done.
Basic Inferential Statistics: Theory and Application
A population is the entire group of people you would like to know something about. In our previous example of New York City, the population is all of the people living in New York City. It should not include people from England, visitors in New York, or even people who know a lot about New York City.
A sample is a subset of the population. Just like you may sample different types of ice cream at the grocery store, a sample of a population should be just a smaller version of the population.
It is extremely important to understand how the sample being studied was drawn from the population. The sample should be as representative of the population as possible. There are several valid ways of creating a sample from a population, but inferential statistics works best when the sample is drawn at random from the population. Given a large enough sample, drawing at random ensures a fair and representative sample of a population.
Comparing two or more groups
Much of statistics, especially in medicine and psychology, is used to compare two or more groups and attempts to figure out if the two groups are different from one another.
Example: Drug X
Let us say that a drug company has developed a pill, which they think increases the recovery time from the common cold. How would they actually find out if the pill works or not? What they might do is get two groups of people from the same population (say, people from a small town in Indiana who had just caught a cold) and administer the pill to one group, and give the other group a placebo. They could then measure how many days each group took to recover (typically, one would calculate the mean of each group). Let's say that the mean recovery time for the group with the new drug was 5.4 days, and the mean recovery time for the group with the placebo was 5.8 days.
The question becomes, is this difference due to random chance, or does taking the pill actually help you recover from the cold faster? The means of the two groups alone does not help us determine the answer to this question. We need additional information.
Sample Size
If our example study only consisted of two people (one from the drug group and one from the placebo group) there would be so few participants that we would not have much confidence that there is a difference between the two groups. That is to say, there is a high probability that chance explains our results (any number of explanations might account for this, for example, one person might be younger, and thus have a better immune system). However, if our sample consisted of 1,000 people in each group, then the results become much more robust (while it might be easy to say that one person is younger than another, it is hard to say that 1,000 random people are younger than another 1,000 random people). If the sample is drawn at random from the population, then these 'random' variations in participants should be approximately equal in the two groups, given that the two groups are large. This is why inferential statistics works best when there are lots of people involved.
Be wary of statistics that have small sample sizes, unless they are in a peer-reviewed journal. Professional statisticians can interpret results correctly from small sample sizes, and often do, but not everyone is a professional, and novice statisticians often incorrectly interpret results. Also, if your author has an agenda, they may knowingly misinterpret results. If your author does not give a sample size, then he or she is probably not a professional, and you should be wary of the results. Sample sizes are required information in almost all peer-reviewed journals, and therefore, should be included in anything you write as well.
Variability
Even if we have a large enough sample size, we still need more information to reach a conclusion. What we need is some measure of variability. We know that the typical person takes about 5-6 days to recover from a cold, but does everyone recover around 5-6 days, or do some people recover in 1 day, and others recover in 10 days? Understanding the spread of the data will tell us how effective the pill is. If everyone in the placebo group takes exactly 5.8 days to recover, then it is clear that the pill has a positive effect, but if people have a wide variability in their length of recovery (and they probably do) then the picture becomes a little fuzzy. Only when the mean, sample size, and variability have been calculated can a proper conclusion be made. In our case, if the sample size is large, and the variability is small, then we would receive a small p-value (probability-value). Small p-values are good, and this term is prominent enough to warrant further discussion.
P-values
In classic inferential statistics, we make two hypotheses before we start our study, the null hypothesis, and the alternative hypothesis.
Null Hypothesis: States that the two groups we are studying are the same.
Alternative Hypothesis: States that the two groups we are studying are different.
The goal in classic inferential statistics is to prove the null hypothesis wrong. The logic says that if the two groups aren't the same, then they must be different. A low p-value indicates a low probability that the null hypothesis is correct (thus, providing evidence for the alternative hypothesis).
Remember: It's good to have low p-values.
Writing with Inferential Statistics
Writing Statistics Plainly
In general, you should always 'translate' your statistics into some understandable form for your reader.
The example above is complicated and hard to read. It's better to say something plainly first, then provide the statistical evidence afterwards:
In the second example, your reader understands the relationship, it's not filled with jargon, but all of the same information is presented. Note that different fields have their own way of writing with statistics—please refer to your field's style guide for specific guidelines.
When using a complicated inferential procedure that your readers would be unfamiliar with, explain it. It may be necessary to go over it in detail. You may want to cite who used it first, and why they used it, and explain how it is applicable to your situation. A footnote or an appendix is a fine place for such an explanation.
If you include statistics that many of your readers would not understand, consider adding the statistics in a footnote or appendix if you can, especially if it is not central to your argument.
Writing Statistics Accurately
If you aren't sure how to calculate a particular statistic, either find out how, or don't use it. Along the same lines, never plug in numbers into a computer program, such as SPSS, and think that the output is "correct." Computer programs don't think for us; they simply allow for fast calculations. They cannot and do not interpret results. You should never interpret the results of a statistic that you don't fully understand. This is extremely important.
When in doubt, keep it simple. If the only thing you can say for certain is that the mean of one group is higher than the mean of another group, then that is fine. This is evidence, albeit it's not as strong as other types of evidence.
Remember that inferential statistics can never "prove" anything. You should think of statistics as a body of evidence (much like a fingerprint at a crime scene) that provides support for your argument. Sometimes it can be used as primary evidence or sometimes it is used in a more supporting role.
Focusing on Statistics
How you frame the use of your statistics is extremely important. In a more scientific field, you'll probably want your statistics as a focal point, but in other fields (say politics, for instance) you may use statistics to support a stance or policy, but it may be only one of many reasons for that policy. Knowing how your audience will react to statistics should affect how you use it. If your audience doesn't use a lot of statistics, you probably shouldn't make statistics the focal point of your argument, or if you do, you need to be very good about explaining the logic behind your statistics.
Statistics and Visuals
Tables
Don't be afraid to use graphics. Statistics can contain a lot of information. Visuals can display a lot of information in a manner that can be quickly understood. The same thing applies to tables. For example:
It' s hard to read! Imagine trying to make sense of this. Instead, provide your data in a table for easy reading:
Group A | Group B | Group C | Group D | |
Mean | 10.5 | 12.3 | 15.9 | 21.3 |
S.D. | 2.1 | 1.2 | 1.8 | 2.5 |
A table is much easier to read than blocks of text. It can help sort the information for both you and your readers. It also makes group comparisons easy. For example, suppose you want to point out to the reader the difference between group A and group D (perhaps this was a new weight training program comparing the number of 80 lbs. dumbbell reps).
Group A | Group B | Group C | Group D | |
Mean | 10.5 | 12.3 | 15.9 | 21.3 |
S.D. | 2.1 | 1.2 | 1.8 | 2.5 |
Or, you could do this:
Group A | Group B | Group C | Group D | |
Mean | 10.5 * | 12.3 | 15.9 | 21.3 * |
S.D. | 2.1 | 1.2 | 1.8 | 2.5 |
Don't be afraid to bold, use asterisks, or otherwise highlight important groups or comparisons.
Graphs
Graphs are an excellent alternative to tables, and they are used by virtually everyone in every field. Papers and articles are like faces. Graphics are like makeup. Makeup is always good in small doses, but don't over apply, or you will end up looking worse than if you didn't use any make up at all. Use visuals, but be careful not to over use them. This is a good example of a visual using the data from the previous table:
Consider distributions of information for a moment. Imagine that we are teaching a class and displaying the students' first homework grades to the students for their benefit. This is one of the ways we could display their homework grades.
In this graph, each of these bars represents a student (each student gets a different color). This is an example of using too much make-up. While the graph does convey a lot of information, it is hard to read. The following graph is much better, and it actually gives you some useful information regarding the class:
Now we can clearly see that one person did really poorly, but that most people were clustered between 70-90%. In the first graph of student scores, we can't really 'see' the distribution, but in this second graph we have a much clearer image of the distribution of scores.
Key Terms
Data Point: A data point is one particular number or item from a data set.
Data Set: A data set is simply a group of numbers. In formal mathematics, data sets are distinguished from each other by using brackets. A more formal mathematical definition allows a data set to contain other things besides numbers (such as letters, items, or even concepts and ideas). The following data set contains only the numbers 2, 5, and 7.
Distribution: A distribution is simply how the data points are clustered. Are they spread apart evenly, or do most of them cluster in the middle and fall off towards the edge like a bell-shaped curve? Two data sets may have the same mean or median, but having different distributions gives them radically different properties.
Mean: The mean (or arithmetic mean) is what most people are referring to when the say average. It is simply the total sum of all the numbers in a data set, divided by the number of different data points.
Median: The middle data point in a data set.
Mode: The most common data point in a data set. This is the value that occurs with greatest frequency.
Population: A population is all of the members contained within a group. In statistics, the population is the group you want your results to generalize about. For example, if you are studying a particular species of fish. such as a Yellow Fin Tuna, then your population is all Yellow Fin Tuna. Your population would not be all fish, nor would your population be all the different species of tuna.
Sample: A sample is all of the units or members that you have studied, drawn from a larger population. In our tuna example, researchers may have found 50 particular yellow fin tuna to study. The sample therefore would consist of 50 yellow fin tuna. As a researcher, you hope that your sample is as representative of your population as possible. The closer the sample represents the population, the stronger and more accurate an inference drawn from the sample will be. This is why you want a large sample to study from.
T-test: A t-test is a common statistical test used to compare two groups, typically two groups' means (the difference of two means divided by a measure of variability). A t-test takes into account the number of units in the sample.