Conducting an experiment is a major accomplishment, but it has little impact if researchers cannot devise an effective way to present their data. If they just display raw data in a table, others will find it difficult to draw any useful conclusions. Imagine you have collected the data presented in Table A.1, which represents the number of minutes of REM sleep (the dependent variable) each of your 44 participants (n = 44; n is the symbol for sample size) had during one night spent in your sleep lab. Looking at this table, you can barely tell what variable is being studied.

A common and simple way to display data is to use a frequency distribution, which shows how often the various values in a data set are present. In Table A.2, we have displayed the data in seven classes, or groups, of equal width. The frequency for each class is tallied up and appears in the middle column. The first class goes from 4 to 24 minutes, and in our sample of 44 participants, only 3 had a total amount of REM in this class (18, 20, 22 minutes). The greatest number of participants experienced between 67 and 87 minutes of REM sleep. By looking at the frequency for each class, you begin to see patterns. In this case, the greatest number of participants had REM sleep within the middle of the distribution, and fewer appear on the ends. We will come back to this pattern shortly.

A-5

Frequency distributions can also be presented with a histogram, which displays the classes of a variable on the x-axis and the frequency of the data on the y-axis (portrayed by the height of the vertical bars). The values on the y-axis can be either the raw frequency (actual number) or the relative frequency (proportion of the whole set; see Table A.2, right column). The example portrayed in Figure A.1 is a histogram of the minutes of REM sleep, with the classes representing the number of minutes in REM on the x-axis and the raw frequency on the y-axis. Looking at a histogram makes it easier to see how the data are distributed across classes. In this case you can see that the most frequent duration of REM is in the middle of the distribution (the 67- to 87-minute class), and that the frequency tapers off toward both ends. Histograms are often used to display quantitative variables that have a wide range of numbers that would be difficult to interpret if they weren’t grouped in classes.

Similar to a histogram is a frequency polygon, which uses lines instead of bars to represent the frequency of the data values. The same data displayed in the histogram (Figure A.1) appear in the frequency polygon in Figure A.2. We see the same general shape in the frequency polygon, but instead of raw frequency we have used the relative frequency to represent the proportion of participants in each of the classes (see Table A.2, right column). Thus, rather than saying 12 participants had 67 to 87 minutes of REM sleep, we can state that the proportion of participants in this class was approximately .27, or 27%. Relative frequencies are especially useful when comparing data sets with different sample sizes. Imagine we wanted to compare two different studies examining REM sleep: one with a sample size of 500, and the other with a sample size of 44. The larger sample might have a greater number of participants in the 67- to 87-minute group (let’s say 50 participants out of 500 [.10] versus the 12 out of the 44 participants [.27] in the smaller sample), making the raw frequency of this group (50) in the larger sample greater than the raw frequency of this group (12) in the smaller sample. But the proportion for the smaller sample would still be greater (smaller sample = .27 versus larger sample = .10). The relative frequency makes it easier to detect these differences in proportion.

Another common way to display quantitative data is through a stem-and-leaf plot, which uses the actual data values in its display. The stem is made up of the first digits in a number, and the leaf is made up of the last digit in each number. This allows us to group numbers by 10s, 20s, 30s, and so on. In Figure A.3, we display the REM sleep data in a stem-and-leaf plot using the first part of the number (either the 10s and/or the 100s) as the stem, and the ones column as the leaf. In the top row, for example, 8 is from the ones column of the smallest number in the data set, 18; 0 and 2 in the second row represent the ones column from the numbers 20 and 22; the 0 in the bottom row comes from 150.

Once the data have been displayed on a graph, researchers look very closely at the -distribution shape, which is just what it sounds like—how the data are spread along the x-axis (that is, the shape is based on the variable represented along the x-axis and the frequency of its values portrayed by the y-axis). A symmetric shape is apparent in the histogram in Figure A.4a, which shows a distribution for a sample that is high in the middle, but tapers off at the same rate on each end. The bell-shaped or normal curve on the right (b) also has a symmetric shape, and this type of curve is fairly typical in psychology (curves generally represent the distribution of the entire population). Many human characteristics have this type of distribution, including cognitive abilities, personality characteristics, and a variety of physical characteristics such as height and weight. Through many years of study, we have found that measurements for the great majority of people fall in the middle of the distribution, and a smaller proportion have characteristics represented on the ends (or in the tails) of the distribution. For example, if you look at the IQ scores displayed in Figure A.4b, you can see that 68% of people have scores between 85 and 115, about 95% have scores between 70 and 130, around 99.7% fall between 55 and 145, and only a tiny percentage (0.3%) are below 55 or above 145. These percentages are true for many other characteristics.

A-6

Some data will have a skewed distribution, which is not symmetrical. As you can see in Figure A.5a, a negatively skewed or left-skewed distribution has a longer tail to the left side of the distribution. A positively skewed or right-skewed distribution (Figure A.5b) has a longer tail to the right side of the distribution. Determining whether a distribution is skewed is particularly important because it informs our decision about what type of statistical analysis to conduct. Later, we will see how certain types of data values can play a role in skewing the distribution.

A-7

Thus far, we have discussed several ways to represent quantitative data. With qualitative data, a frequency distribution lists the various categories and the number of members in each. For example, if we wanted to display the college major data on 44 students interviewed at the library, we could use a frequency distribution (Table A.3).

Another common way to display qualitative data is through a bar graph, which displays the categories of interest on the x-axis and their frequencies on the y-axis. Figure A.6 shows how the data collected in the library can be presented in a bar graph. Bar graphs are useful for comparing several different populations on the same variable (for example, perhaps comparing college majors by gender or ethnicity).

Pie charts can also be used to display qualitative data, with pie slices representing the proportion of the data set belonging to each category (Figure A.7). As you can see, the biggest percentage is nursing (27%), followed by culinary arts (18%), and undecided (18%). The smallest percentage is shared by biology and psychology (both 7%). Often researchers use pie charts to easily display data for which it is important to know the relative proportion of each category (for example, a psychology department trying to gain support for funding its courses might want to be able to display the relative number of psychologists in particular subfields; see Figure 1.1, page 4).

With any type of data display, one must be on the lookout for misleading portrayals. In Figure A.6a, we display data for the 44 students interviewed in the library. Notice that, while Figures A.6b and A.6c look different, they display the same data for the same campus of 4,400 students. Quickly look at (b) and (c) of the figure and decide, if you were head of the psychology department, which bar chart you would use to demonstrate the popularity of the psychology major. In (b), the size of the department (as measured by number of students) looks fairly small compared to that of other departments, particularly the nursing program. But notice that the scale on the y-axis starts at 250 in (b), whereas it begins at 0 in (c). In this third bar chart, it appears that the student count for the psychology program is not far behind that for other programs like chemistry and English, for example. An important aspect of critical thinking is being able to evaluate the source of evidence, something one must consider when reading graphs and charts. (For example, does the author of the bar chart in Figure A.6b have a particular agenda to reduce funding for the psychology and biology departments?) It is important to recognize that manipulating the presentation of data can lead to faulty interpretations (the data on the 4,400 students are valid, but the way they are presented is not).