54 Expressing Your Results
Learning Objectives
- Write out simple descriptive statistics in American Psychological Association (APA) style.
- Interpret and create simple APA-style figures—including bar graphs, line graphs, and scatterplots.
- Interpret and create simple APA-style tables—including tables of group or condition means and correlation matrices.
Once you have conducted your descriptive statistical analyses, you will need to present them to others. In this section, we focus on presenting descriptive statistical results in writing, in figures, and in tables—following American Psychological Association (APA) guidelines for written research reports. These principles can be adapted easily to other presentation formats such as posters and slide show presentations.
Presenting Descriptive Statistics in Writing
Recall that APA style includes several rules for presenting numerical results in the text (see 4.31–4.34 in the APA Publication Manual) . These include using words only for numbers less than 10 that do not represent precise statistical results and using numerals for numbers 10 and higher. However, statistical results are always presented in the form of numerals rather than words and are usually rounded to two decimal places (e.g., “2.00” rather than “two” or “2”). They can be presented either in the narrative description of the results or parenthetically—much like reference citations. When you have a small number of results to report, it is often most efficient to write them out. Here are some examples:
The mean age of the participants was 22.43 years with a standard deviation of 2.34.
Among the participants with low self-esteem, those in a negative mood expressed stronger intentions to have unprotected sex (M = 4.05, SD = 2.32) than those in a positive mood (M = 2.15, SD = 2.27).
The treatment group had a mean of 23.40 (SD = 9.33), while the control group had a mean of 20.87 (SD = 8.45).
The test-retest correlation was .96.
There was a moderate negative correlation between the alphabetical position of respondents’ last names and their response time (r = −.27).
Notice that when presented in the narrative, the terms mean and standard deviation are written out, but when presented parenthetically, the symbols M and SD are used instead. Notice also that it is especially important to use parallel construction to express similar or comparable results in similar ways. The third example is much better than the following nonparallel alternative:
The treatment group had a mean of 23.40 (SD = 9.33), while 20.87 was the mean of the control group, which had a standard deviation of 8.45.
Presenting Descriptive Statistics in Figures
When you have a large number of results to report, you can often do it more clearly and efficiently with a graphical depiction of the data, such as pie charts, bar graphs, or scatterplots. In an APA style research report, these graphs are presented as figures. When you prepare figures for an APA-style research report, there are some general guidelines that you should keep in mind. First, the figure should always add important information rather than repeat information that already appears in the text or in a table (if a figure presents information more clearly or efficiently, then you should keep the figure and eliminate the text or table.) Second, figures should be as simple as possible. For example, the Publication Manual discourages the use of color unless it is absolutely necessary (although color can still be an effective element in posters, slide show presentations, or textbooks.) Third, figures should be interpretable on their own. A reader should be able to understand the basic result based only on the figure and its caption and should not have to refer to the text for an explanation.
There are also several more technical guidelines for presentation of figures that include the following (see the APA Publication Manual section 5.20 through 5.30):
- Layout of graphs
- In general, scatterplots, bar graphs, and line graphs should be slightly wider than they are tall.
- The independent variable should be plotted on the x-axis and the dependent variable on the y-axis.
- Values should increase from left to right on the x-axis and from bottom to top on the y-axis.
- The x-axis and y-axis should begin with the value zero.
- Axis Labels and Legends
- Axis labels should be clear and concise and include the units of measurement if they do not appear in the caption.
- Axis labels should be parallel to the axis.
- Legends should appear within the figure.
- Text should be in the same simple font throughout and no smaller than 8 point and no larger than 14 point.
- Captions
- Captions are titled with the word “Figure”, followed by the figure number in the order in which it appears in the text, and terminated with a period. This title is italicized.
- After the title is a brief description of the figure terminated with a period (e.g., “Reaction times of the control versus experimental group.”)
- Following the description, include any information needed to interpret the figure, such as any abbreviations, units of measurement (if not in the axis label), units of error bars, etc.
Bar Graphs
As we have seen throughout this book, bar graphs are generally used to present and compare the mean scores for two or more groups or conditions. The bar graph in Figure 12.11 is an APA-style version of Figure 12.4. Notice that it conforms to all the guidelines listed. A new element in Figure 12.11 is the smaller vertical bars that extend both upward and downward from the top of each main bar. These are error bars, and they represent the variability in each group or condition. Although they sometimes extend one standard deviation in each direction, they are more likely to extend one standard error in each direction (as in Figure 12.11). The standard error is the standard deviation of the group divided by the square root of the sample size of the group. The standard error is used because, in general, a difference between group means that is greater than two standard errors is statistically significant. Thus one can “see” whether a difference is statistically significant based on a bar graph with error bars.
Line Graphs
Line graphs are used when the independent variable is measured in a more continuous manner (e.g., time) or to present correlations between quantitative variables when the independent variable has, or is organized into, a relatively small number of distinct levels. Each point in a line graph represents the mean score on the dependent variable for participants at one level of the independent variable. Figure 12.12 is an APA-style version of the results of Carlson and Conard. Notice that it includes error bars representing the standard error and conforms to all the stated guidelines.
In most cases, the information in a line graph could just as easily be presented in a bar graph. In Figure 12.12, for example, one could replace each point with a bar that reaches up to the same level and leave the error bars right where they are. This emphasizes the fundamental similarity of the two types of statistical relationship. Both are differences in the average score on one variable across levels of another. The convention followed by most researchers, however, is to use a bar graph when the variable plotted on the x-axis is categorical and a line graph when it is quantitative.
Scatterplots
Scatterplots are used to present correlations and relationships between quantitative variables when the variable on the x-axis (typically the independent variable) has a large number of levels. Each point in a scatterplot represents an individual rather than the mean for a group of individuals, and there are no lines connecting the points. The graph in Figure 12.13 is an APA-style version of Figure 12.7, which illustrates a few additional points. First, when the variables on the x-axis and y-axis are conceptually similar and measured on the same scale—as here, where they are measures of the same variable on two different occasions—this can be emphasized by making the axes the same length. Second, when two or more individuals fall at exactly the same point on the graph, one way this can be indicated is by offsetting the points slightly along the x-axis. Other ways are by displaying the number of individuals in parentheses next to the point or by making the point larger or darker in proportion to the number of individuals. Finally, the straight line that best fits the points in the scatterplot, which is called the regression line, can also be included.
Expressing Descriptive Statistics in Tables
Like graphs, tables can be used to present large amounts of information clearly and efficiently. The same general principles apply to tables as apply to graphs. They should add important information to the presentation of your results, be as simple as possible, and be interpretable on their own. Again, we focus here on tables for an APA-style manuscript.
The most common use of tables is to present several means and standard deviations—usually for complex research designs with multiple independent and dependent variables. Figure 12.14, for example, shows the results of a hypothetical study similar to the one by MacDonald and Martineau (2002)[1] (The means in Figure 12.14 are the means reported by MacDonald and Martineau, but the standard errors are not). Recall that these researchers categorized participants as having low or high self-esteem, put them into a negative or positive mood, and measured their intentions to have unprotected sex. They also measured participants’ attitudes toward unprotected sex. Notice that the table includes horizontal lines spanning the entire table at the top and bottom, and just beneath the column headings. Furthermore, every column has a heading—including the leftmost column—and there are additional headings that span two or more columns that help to organize the information and present it more efficiently. Finally, notice that APA-style tables are numbered consecutively starting at 1 (Table 1, Table 2, and so on) and given a brief but clear and descriptive title.
Another common use of tables is to present correlations—usually measured by Pearson’s r—among several variables. This kind of table is called a correlation matrix. Figure 12.15 is a correlation matrix based on a study by David McCabe and colleagues (McCabe, Roediger, McDaniel, Balota, & Hambrick, 2010)[2]. They were interested in the relationships between working memory and several other variables. We can see from the table that the correlation between working memory and executive function, for example, was an extremely strong .96, that the correlation between working memory and vocabulary was a medium .27, and that all the measures except vocabulary tend to decline with age. Notice here that only half the table is filled in because the other half would have identical values. For example, the Pearson’s r value in the upper right corner (working memory and age) would be the same as the one in the lower left corner (age and working memory). The correlation of a variable with itself is always 1.00, so these values are replaced by dashes to make the table easier to read.
As with graphs, precise statistical results that appear in a table do not need to be repeated in the text. Instead, the writer can note major trends and alert the reader to details (e.g., specific correlations) that are of particular interest.
- MacDonald, T. K., & Martineau, A. M. (2002). Self-esteem, mood, and intentions to use condoms: When does low self-esteem lead to risky health behaviors? Journal of Experimental Social Psychology, 38, 299–306. ↵
- McCabe, D. P., Roediger, H. L., McDaniel, M. A., Balota, D. A., & Hambrick, D. Z. (2010). The relationship between working memory capacity and executive functioning. Neuropsychology, 24(2), 222–243. doi:10.1037/a0017619 ↵
Differences between the groups may reflect the generation that people come from rather than a direct effect of age.
Research that is non-experimental because it focuses on recording systemic observations of behavior in a natural or laboratory setting without manipulating anything.
A method in which an equal number of participants complete each possible order of conditions.
A research that lacks the manipulation of an independent variable.
Is a small-scale study conducted to make sure that a new procedure works as planned.
Learning Objectives
- Identify the key components of empirical journal articles
- Define the basic elements of the results section in a journal article
- Describe statistical significance and confidence intervals
Reading scholarly articles can be a more challenging than reading a book, magazine, news article—or even some textbooks. Theoretical and practical articles are, generally speaking, easier to understand. Empirical articles, because they add new knowledge, must go through great detail to demonstrate that the information they offer is based on solid science. Empirical articles can be challenging to read, and this section is designed to make that process easier for you.
Nearly all articles will have an abstract, the short paragraph at the beginning of an article that summarizes the author’s research question, methods used to answer the question, and key findings. The abstract may also give you some idea about the theoretical perspective of the author. In effect, the abstract provides you with a framework to understand the rest of the article and the article's punch line: what the author(s) found, and whether the article is relevant to your area of inquiry. For this reason, I suggest skimming abstracts as part of the literature search process.
As you will recall from Chapter 2, theoretical articles have no set structure and will look similar to reading a chapter of a book. Empirical articles contain the following sections (although exact section names vary): introduction, methods, results, and discussion. The introduction contains the literature review for the article and is an excellent source of information as you build your own literature review. The methods section reviews how the author gathered their sample, how they measured their variables, and how the data were analyzed. The results section provides an in-depth discussion of the findings of the study. The discussion section reviews the main findings and addresses how those findings fit in with the existing literature. At the end, there will be a list of references (which you should read!) and there may be a few tables, figures, or appendices if applicable.
While you should get into the habit of familiarizing yourself with each part of the articles you wish to cite, there are strategic ways to read journal articles that can make them a little easier to digest. Once you have read the abstract for an article and determined it is one you’d like to read in full, read through the introduction and discussion sections next. The introduction section will showcase other articles and findings that are significant in your topic area, so reading through it will be beneficial for your own information-gathering process for your literature review. Reading an article’s discussion section helps you understand what the author views as their study’s major findings and how the author perceives those findings to relate to other research.
As you progress through your research methods course, you will pick up additional research elements that are important to understand. You will learn how to identify qualitative and quantitative methods, as well as exploratory, explanatory, and descriptive research methods. You will also learn the criteria for establishing causality and the different types of causality. Subsequent chapters of this textbook will address other elements of journal articles, including choices about measurement, sampling, and design. As you learn about these additional items, you will find that the methods and results sections begin to make more sense and you will understand how the authors reached their conclusions.
As you read a research report, there are several questions you can ask yourself about each section, from abstract to conclusion. Those questions are summarized in Table 3.1. Keep in mind that the questions covered here are designed to help you, the reader, to think critically about the research you come across and to get a general understanding of the strengths, weaknesses, and key takeaways from a given study. I hope that by considering how you might respond to the following questions while reading research reports, you’ll gain confidence in describing the report to others and discussing its meaning and impact with them.
| Report section | Questions worth asking |
| Abstract | What are the key findings? How were those findings reached? What framework does the researcher employ? |
| Acknowledgments | Who are this study’s major stakeholders? Who provided feedback? Who provided support in the form of funding or other resources? |
| Problem statement (introduction) | How does the author frame their research focus? What other possible ways of framing the problem exist? Why might the author have chosen this particular way of framing the problem? |
| Literature review (introduction) |
How selective does the researcher appear to have been in identifying relevant literature to discuss? Does the review of literature appear appropriately extensive? Does the researcher provide a critical review? |
| Sample (methods) | Where was the data collected? Did the researcher collect their own data or use someone else's data? What population is the study trying to make claims about, and does the sample represent that population well? What are the sample’s major strengths and major weaknesses? |
| Data collection (methods) | How were the data collected? What do you know about the relative strengths and weaknesses of the method employed? What other methods of data collection might have been employed, and why was this particular method employed? What do you know about the data collection strategy and instruments (e.g., questions asked, locations observed)? What don’t you know about the data collection strategy and instruments? |
| Data analysis (methods) | How were the data analyzed? Is there enough information provided for you to feel confident that the proper analytic procedures were employed accurately? |
| Results | What are the study’s major findings? Are findings linked back to previously described research questions, objectives, hypotheses, and literature? Are sufficient amounts of data (e.g., quotes and observations in qualitative work, statistics in quantitative work) provided in order to support conclusions drawn? Are tables readable? |
| Discussion/conclusion | Does the author generalize to some population beyond her/his/their sample? How are these claims presented? Are claims made supported by data provided in the results section (e.g., supporting quotes, statistical significance)? Have limitations of the study been fully disclosed and adequately addressed? Are implications sufficiently explored? |
Understanding the results section
As mentioned previously in this chapter, reading the abstract that appears in most reports of scholarly research will provide you with an excellent, easily digestible review of a study’s major findings and of the framework the author is using to position their findings. Abstracts typically contain just a few hundred words, so reading them is a nice way to quickly familiarize yourself with a study. If the study seems relevant to your paper, it’s probably worth reading more. If it’s not, then you have only spent a minute or so reading the abstract. Another way to get a snapshot of the article is to scan the headings, tables, and figures throughout the report (Green & Simon, 2012). [1]
At this point, I have read hundreds of literature reviews written by students. One of the challenges I have noted is that students will report the summarized results from the abstract, rather than the detailed findings in the results section of the article. This is a problem when you are writing a literature review because you need to provide specific and clear facts that support your reading of the literature. The abstract may say something like: “we found that poverty is associated with mental health status.” For your literature review, you want the details, not the summary. In the results section of the article, you may find a sentence that states: “for households in poverty, children are three times more likely to have a mental health diagnosis.” This more detailed information provides a stronger basis on which to build a literature review.
Using the summarized results in an abstract is an understandable mistake to make. The results section often contains terminology, diagrams, and symbols that may be hard to understand without having completed advanced coursework on statistical or qualitative analysis. To that end, the purpose of this section is to improve reading comprehension by providing an introduction to the basic components of a results section.
Journal articles often contain tables, and scanning them is a good way to begin reading an article. A table provides a quick, condensed summary of the report’s key findings. The use of tables is not limited to one form or type of data, though they are used most commonly in quantitative research. Tables are a concise way to report large amounts of data. Some tables present descriptive information about a researcher’s sample, which is often the first table in a results section. These tables will likely contain frequencies (N) and percentages (%). For example, if gender happened to be an important variable for the researcher’s analysis, a descriptive table would show how many and what percent of all study participants are women, men, or other genders. Frequencies or counts will probably be listed as N, while the percent symbol (%) might be used to indicate percentages.
In a table presenting a causal relationship, two sets of variables are represented. The independent variable, or cause, and the dependent variable, the effect. We’ll go into more detail on variables in Chapter 6. The independent variable attributes are typically presented in the table’s columns, while dependent variable attributes are presented in rows. This allows the reader to scan across a table’s rows to see how values on the dependent variable attributes change as the independent variable attribute values change. Tables displaying results of quantitative analysis will also likely include some information about the strength and statistical significance of the relationships presented in the table. These details tell the reader how likely it is that the relationships presented will have occurred simply by chance.
Let’s look at a specific example. Table 3.2 shows data from a study of older adults that was conducted by Dr. Blackstone, an original author of this textbook. It presents the causal relationship between gender and the experience of harassing behaviors in the workplace. In this example, gender is the independent variable (the cause) and the harassing behaviors listed are the dependent variables (the effects). [2] Therefore, we place gender in the table’s columns and harassing behaviors in the table’s rows.
Reading across the table’s top row, we see that 2.9% of women in the sample reported experiencing subtle or obvious threats to their safety at work, while 4.7% of men in the sample reported the same. We can read across each of the rows of the table in this way. Reading across the bottom row, we see that 9.4% of women in the sample reported experiencing staring or invasion of their personal space at work while just 2.3% of men in the sample reported having the same experience. We’ll discuss p value later in this section.
| Behavior Experienced at work | Women | Men | p value |
| Subtle or obvious threats to your safety | 2.9% | 4.7% | 0.623 |
| Being hit, pushed, or grabbed | 2.2% | 4.7% | 0.480 |
| Comments or behaviors that demean your gender | 6.5% | 2.3% | 0.184 |
| Comments or behaviors that demean your age | 13.8% | 9.3% | 0.407 |
| Staring or invasion of your personal space | 9.4% | 2.3% | 0.039 |
| Note: Sample size was 138 for women and 43 for men. | |||
These statistics represent what the researchers found in their sample, and they are using their sample to make conclusions about the true population of all employees in the real world. Because the methods we use in social science are never perfect, there is some amount of error in that value. The researchers in this study estimated the true value we would get if we asked every employee in the world the same questions on our survey. Researchers will often provide a confidence interval, or a range of values in which the true value is likely to be, to provide a more accurate description of their data. For example, at the time I’m writing this, my wife and I are expecting our first child next month. The doctor told us our due date was August 15th. But the doctor also told us that August 15th was only their best estimate. They were actually 95% sure our baby might be born any time between August 1st and September 1st. Confidence intervals are often listed with a percentage, like 90% or 95%, and a range of values, such as between August 1st and September 1st. You can read that as: we are 95% sure your baby will be born between August 1st and September 1st. So, while we get a due date of August 15th, the uncertainty about the exact date is reflected in the confidence interval provided by our doctor.
Of course, we cannot assume that these patterns didn’t simply occur by chance. How confident can we be that the findings presented in the table did not occur by chance? This is where tests of statistical significance come in handy. Statistical significance tells us the likelihood that the relationships we observe could be caused by something other than chance. While your statistics class will give you more specific details on tests of statistical significance and reading quantitative tables, the important thing to be aware of as a non-expert reader of tables is that some of the relationships presented will be statistically significant and others may not be. Tables should provide information about the statistical significance of the relationships presented. When reading a researcher’s conclusions, pay attention to which relationships are statistically significant and which are not.
In Table 3.2, you may have noticed that a p value is noted in the very last column of the table. A p value is a statistical measure of the probability that there is no relationship between the variables under study. Another way of putting this is that the p value provides guidance on whether or not we should reject the null hypothesis. The null hypothesis is simply the assumption that no relationship exists between the variables in question. In Table 3.2, we see that for the first behavior listed, the p value is 0.623. This means that there is a 62.3% chance that the null hypothesis is correct in this case. In other words, it seems likely that any relationship between observed gender and experiencing threats to safety at work in this sample is simply due to chance.
In the final row of the table, however, we see that the p value is 0.039. In other words, there is a 3.9% chance that the null hypothesis is correct. Thus, we can be somewhat more confident than in the preceding example that there may be some relationship between a person’s gender and their experiencing the behavior noted in this row. Statistical significance is reported in reference to a value, usually 0.05 in the social science. This means that the probability that the relationship between gender and experiencing staring or invasion of personal space at work is due to random chance is less than 5 in 100. Social science often uses 0.05, but other values are used. Studies using 0.1 are using a more forgiving standard of significance, and therefore, have a higher likelihood of error (10%). Studies using 0.01 are using a more stringent standard of significance, and therefore, have a lower likelihood of error (1%).
Notice that I’m hedging my bets here by using words like somewhat and may be. When testing hypotheses, social scientists generally phrase their findings in terms of rejecting the null hypothesis rather than making bold statements about the relationships observed in their tables. You can learn more about creating tables, reading tables, and tests of statistical significance in a class focused exclusively on statistical analysis. For now, I hope this brief introduction to reading tables will improve your confidence in reading and understanding the quantitative tables you encounter while reading reports of social science research.
A final caveat is worth noting here. The previous discussion of tables and reading the results section is applicable to quantitative articles. Quantitative articles will contain a lot of numbers and the results of statistical tests demonstrating association between those numbers. Qualitative articles, on the other hand, will consist mostly of quotations from participants. For most qualitative articles, the authors want to put their results in the words of their participants, as they are the experts. The results section may be organized by theme, with each paragraph or subsection illustrating through quotes how the authors interpret what people in their study said.
Key Takeaways
- Reading a research article requires reading beyond the abstract.
- In tables presenting causal relationships, the independent variable is typically presented in the table’s columns while the dependent variables are presented in the table’s rows.
- When reading a research report, there are several key questions you should ask yourself for each section of the report.
Glossary
Abstract- the short paragraph at the beginning of an article that summarizes its main point
Confidence interval- a range of values in which the true value is likely to be
Null hypothesis- the assumption that no relationship exists between the variables in question
p-value- a statistical measure of the probability that there is no relationship between the variables under study
Statistical significance- the likelihood that the relationships that are observed could be caused by something other than chance
Table- a quick, condensed summary of the report’s key findings
Image Attributions
CSAF releases 2009 reading list by Master Sgt. Steven Goetsch public domain
