Statistics that are used to organize and summarize the information so that the researcher can see what happened during the research study and can also communicate the results to others are called descriptive statistics.Let us assume that the data are quantitative and consist of scores on one or more variables for each of several study participants. Although in most cases the primary research question will be about one or more statistical relationships between variables, it is also important to describe each variable individually. We will look at some of the most common techniques for describing single variables including:

• Frequency distributions
• Measures of Central Tendency
• Measures of Dispersion

The first step in understanding data is using tables, charts, graphs, plots, and other visual tools to see what our data look like.  This is known as data visualization.

We will begin with frequency distributions which are visual representations and include tables and graphs. We will conclude with some tips for making graphs some principles for good data visualization!

Data Visualization

On January 28, 1986, the Space Shuttle Challenger exploded 73 seconds after takeoff, killing all 7 of the astronauts on board. As when any such disaster occurs, there was an official investigation into the cause of the accident, which found that an O-ring connecting two sections of the solid rocket booster leaked, resulting in failure of the joint and explosion of the large liquid fuel tank (see figure 1).^[1]

The investigation found that many aspects of the NASA decision-making process were flawed, and focused in particular on a meeting between NASA staff and engineers from Morton Thiokol, a contractor who built the solid rocket boosters. These engineers were particularly concerned because the temperatures were forecast to be very cold on the morning of the launch, and they had data from previous launches showing that performance of the O-rings was compromised at lower temperatures. In a meeting on the evening before the launch, the engineers presented their data to the NASA managers, but were unable to convince them to postpone the launch. Their evidence was a set of hand-written slides showing numbers from various past launches.

The visualization expert Edward Tufte has argued that with a proper presentation of all of the data, the engineers could have been much more persuasive. In particular, they could have shown a figure like the one in Figure 2, which highlights two important facts. First, it shows that the amount of O-ring damage (defined by the amount of erosion and soot found outside the rings after the solid rocket boosters were retrieved from the ocean in previous flights) was closely related to the temperature at takeoff. Second, it shows that the range of forecasted temperatures for the morning of January 28 (shown in the shaded area) was well outside of the range of all previous launches. While we can’t know for sure, it seems at least plausible that this could have been more persuasive.

Graphing Qualitative & Quantitative Variables

We’ll learn some general lessons about how to graph data that fall into a small number of categories. A later section will consider how to graph numerical data in which each observation is represented by a number in some range. Qualitative variables can be summarized by frequency (how often) and researchers can then use frequency tables and bar charts to show frequencies for categorized responses, but we are limited in graphing them due to the data not be numerically based. The key point about the qualitative data is they do not come with a pre-established ordering (the way numbers are ordered). 

We are focused on quantitative variables.  Quantitative data, such as a person’s weight, are naturally ordered with respect to people of different weights. Often we wish to know if there are any scores that might look a bit out of place. A frequency distribution is a way to take a disorganized set of scores and places them in order from highest to lowest and at the same time grouping everyone with the same score. Frequency distributions can help researchers identify outliers. An outlier is an observation of data that does not fit the rest of the data. An outlier is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes (for example, writing down 50 instead of 500) while others may indicate that something unusual is happening.

Frequency Tables

All of the graphical methods shown in this section are derived from frequency tables. Table 1 shows a frequency table for the results of the iMac study; it shows the frequencies of the various response categories. It also shows the relative frequencies, which are the proportion of responses in each category. For example, the relative frequency for “none” of 0.17 = 85/500.

Previous Ownership	Frequency	Relative Frequency
None	85	0.17
Windows	60	0.12
Macintosh	355	0.71
Total	500	1

Table 1. Frequency Table for the iMac Data.

Graphs

A statistical graph is a tool that helps you learn about the shape or distribution of a sample or a population. A graph can be a more effective way of presenting data than a mass of numbers because we can see where data clusters and where there are only a few data values. Statisticians often graph data first to get a picture of the data; then, more formal tools may be applied.

Some of the types of graphs that are used to summarize and organize quantitative data are the dot plot, the bar graph, the histogram, the stem-and-leaf plot, the frequency polygon (a type of broken line graph), the pie chart, and the box plot. In this lesson, we will briefly look at bar graphs, histograms, and frequency polygons.

Figure 2. Bar chart of iMac purchases as a function of previous computer ownership.

Often we need to compare the results of different surveys, or of different conditions within the same overall survey. In this case, we are comparing the “distributions” of responses between the surveys or conditions. Bar charts are often excellent for illustrating differences between two distributions. Figure 3 shows the number of people playing card games at the Yahoo website on a Sunday and on a Wednesday in the spring of 2001. We see that there were more players overall on Wednesday compared to Sunday. The number of people playing Pinochle was nonetheless the same on these two days. In contrast, there were about twice as many people playing hearts on Wednesday as on Sunday. Facts like these emerge clearly from a well-designed bar chart.

Comparing Distributions

Figure 3. A bar chart of the number of people playing different card games on Sunday and Wednesday.

The bars in Figure 3 are oriented horizontally rather than vertically. The horizontal format is useful when you have many categories because there is more room for the category labels. We’ll have more to say about bar charts when we consider numerical quantities later in this chapter.

Some graphical mistakes to avoid with bar charts

Don’t get fancy! People sometimes add features to graphs that don’t help to convey their information. See the examples below as things not to do! Three-dimensional figures are less clear than 2-d.  Further, don’t get creative as show below! Use plain bars, as tempting as it is to substitute meaningful images. The MacIntosh is out of proportion to the None and Windows categories. Edward Tufte coined the term “lie factor” to refer to the ratio of the size of the effect shown in a graph to the size of the effect shown in the data. If a graphic has a lie factor near 1, then it is appropriately representing the data, whereas lie factors far from one reflect a distortion of the underlying data. The computer monitor bar figure has a lie factor of about 8! He suggests that lie factors greater than 1.05 or less than 0.95 produce unacceptable distortion-so just keep it simple with plain bars!

Figures 4 & 5. A three-dimensional version of Figure 2 and a redrawing of Figure 2 with disproportionate bars.

Another distortion in bar charts results from setting the baseline to a value other than zero. The baseline is the bottom of the Y-axis, representing the least number of cases that could have occurred in a category. Normally, but not always, this number should be zero. Figure 7 shows the iMac data with a baseline of 50. Once again, the differences in areas suggests a different story than the true differences in percentages. The number of Windows-switchers seems minuscule compared to its true value of 12%.

Figure 7. A redrawing of Figure 2 with a baseline of 50.

Finally, we note that it is a serious mistake to use a line graph when the X-axis contains merely qualitative (or categorical) variables. A line graph is essentially a bar graph with the tops of the bars represented by points joined by lines (the rest of the bar is suppressed). Figure 8 inappropriately shows a line graph of the card game data from Yahoo. The drawback to Figure 8 is that it gives the false impression that the games are naturally ordered in a numerical way when, in fact, they are ordered alphabetically.

Figure 8. A line graph used inappropriately to depict the number of people playing different card games on Sunday and Wednesday.

Recap

Bar charts can be effective methods of portraying qualitative data. Bar charts are better when there are more than just a few categories and for comparing two or more distributions. Be careful to avoid creating misleading graphs.

Graphing Quantitative Variables

As discussed in the section on variables in Chapter 1, quantitative variables are variables measured on a numeric scale. Height, weight, response time, subjective rating of pain, temperature, and score on an exam are all examples of quantitative variables. Quantitative variables are distinguished from categorical (sometimes called qualitative) variables such as favorite color, religion, city of birth, favorite sport in which there is no ordering or measuring involved.

There are many types of graphs that can be used to portray distributions of quantitative variables. We already reviewed bar charts. The upcoming sections cover the following types of graphs: (1) histograms, (2) frequency polygons, (3) stem and leaf displays, (4) box plots, (5) more bar charts, (6) line graphs, and (7) scatter plots (discussed in a different chapter). Some graph types such as stem and leaf displays are best suited for small to moderate amounts of data, whereas others such as histograms are best- suited for large amounts of data. Graph types such as box plots are good at depicting differences between distributions. Scatter plots are used to show the relationship between two variables.

Histograms

A histogram is a graphic version of a frequency distribution. It helps to display the shape of a distribution. The graph consists of bars of equal width drawn adjacent to each other and has both a horizontal axis and a vertical axis. The horizontal axis (x-axis) is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The histogram shows the distribution of the values including the highest, middle, and lowest values.

Sometimes we need to group scores if the data has a large distribution. For example, if I wanted to create a frequency distribution of 642 students’ scores on a psychology test, that would be a big frequency table. For reference, the test consists of 197 items each graded as “correct” or “incorrect.” The students’ scores ranged from 46 to 167. A simple frequency table would be too big, containing over 100 rows. To simplify the table, we group scores together as shown in Table 4. A basic rule for grouping data is to make sure each group (or class) has the same grouping amount (in this example it is grouped in 10s), and to make sure you have the lowest category including your lowest value to make sure all scores are included.

Interval’s Lower Limit	Interval’s Upper Limit	Class Frequency
39.5	49.5	3
49.5	59.5	10
59.5	69.5	53
69.5	79.5	107
79.5	89.5	147
89.5	99.5	130
99.5	109.5	78
109.5	119.5	59
119.5	129.5	36
129.5	139.5	11
139.5	149.5	6
149.5	159.5	1
159.5	169.5	1

Table 4. Grouped Frequency Distribution of Psychology Test Scores

To create this table, the range of scores was broken into intervals, called class intervals. The first interval is from 39.5 to 49.5, the second from 49.5 to 59.5, etc. Next, the number of scores falling into each interval was counted to obtain the class frequencies. There are three scores in the first interval, 10 in the second, etc. For this data set, class intervals of width 10 provide enough detail about the distribution to be revealing without making the graph too “choppy.” More information on choosing the widths of class intervals is presented later in this section. Placing the limits of the class intervals midway between two numbers (e.g., 49.5) ensures that every score will fall in an interval rather than on the boundary between intervals. If you add up all the scores for the class frequencies you will get to the total number of scores (in this example 642).

In a histogram, the class intervals are represented by bars. The height of each bar corresponds to its class frequency. A histogram of these data is shown in Figure 9.

Figure 9. Histogram of scores on a psychology test.

The histogram makes it plain that most of the scores are in the middle of the distribution, with fewer scores in the extremes. You can also see that the distribution is not symmetric: the scores extend to the right farther than they do to the left. The distribution is therefore said to be skewed. (We’ll have more to say about shapes of distributions a little later in the chapter).

In our example, the observations are whole numbers. Histograms can also be used when the scores are measured on a more continuous scale such as the length of time (in milliseconds) required to perform a task. In this case, there is no need to worry about fence sitters since they are improbable. (It would be quite a coincidence for a task to require exactly 7 seconds, measured to the nearest thousandth of a second.) We are therefore free to choose whole numbers as boundaries for our class intervals, for example, 4000, 5000, etc. The class frequency is then the number of observations that are greater than or equal to the lower bound, and strictly less than the upper bound. For example, one interval might hold times from 4000 to 4999 milliseconds. Using whole numbers as boundaries avoids a cluttered appearance, and is the practice of many computer programs that create histograms.

Histograms can be based on relative frequencies instead of actual frequencies. Histograms based on relative frequencies show the proportion of scores in each interval rather than the number of scores. In this case, the Y-axis runs from 0 to 1 (or somewhere in between if there are no extreme proportions). You can change a histogram based on frequencies to one based on relative frequencies by (a) dividing each class frequency by the total number of observations, and then (b) plotting the quotients on the Y-axis (labeled as proportion).

There is more to be said about the widths of the class intervals, sometimes called bin widths. Your choice of bin width determines the number of class intervals. This decision, along with the choice of starting point for the first interval, affects the shape of the histogram. The best advice is to experiment with different choices of width, and to choose a histogram according to how well it communicates the shape of the distribution.

Frequency Polygons

Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful for comparing sets of data. Frequency polygons are also a good choice for displaying cumulative frequency distributions.

To create a frequency polygon, start just as for histograms, by choosing a class interval. Then draw an X-axis representing the values of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle value represented by the class. Draw the Y-axis to indicate the frequency of each class. Place a point in the middle of each class interval at the height corresponding to its frequency. Finally, connect the points. You should include one class interval below the lowest value in your data and one above the highest value. The graph will then touch the X-axis on both sides.

A frequency polygon for 642 psychology test scores shown in Figure 12 was constructed from the frequency table shown in Table 5.

Lower Limit	Upper Limit	Count	Cumulative Count
29.5	39.5	0	0
39.5	49.5	3	3
49.5	59.5	10	13
59.5	69.5	53	66
69.5	79.5	107	173
79.5	89.5	147	320
89.5	99.5	130	450
99.5	109.5	78	528
109.5	119.5	59	587
119.5	129.5	36	623
129.5	139.5	11	634
139.5	149.5	6	640
149.5	159.5	1	641
159.5	169.5	1	642
169.5	170.5	0	642

Table 5. Frequency Distribution of Psychology Test Scores

The first label on the X-axis is 35. This represents an interval extending from 29.5 to 39.5. Since the lowest test score is 46, this interval has a frequency of 0. The point labeled 45 represents the interval from 39.5 to 49.5. There are three scores in this interval. There are 147 scores in the interval that surrounds 85.

You can easily discern the shape of the distribution from Figure 10. Most of the scores are between 65 and 115. It is clear that the distribution is not symmetric inasmuch as good scores (to the right) trail off more gradually than poor scores (to the left). We call this skew and we will study shapes of distributions more systematically later in this chapter.

Figure 10. Frequency polygon for the psychology test scores.

A cumulative frequency polygon for the same test scores is shown in Figure 11. The graph is the same as before except that the Y value for each point is the number of students in the corresponding class interval plus all numbers in lower intervals. For example, there are no scores in the interval labeled “35,” three in the interval “45,” and 10 in the interval “55.” Therefore, the Y value corresponding to “55” is 13. Since 642 students took the test, the cumulative frequency for the last interval is 642.

Figure 11. Cumulative frequency polygon for the psychology test scores.

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets. Figure 12 provides an example. The data come from a task in which the goal is to move a computer cursor to a target on the screen as fast as possible. On 20 of the trials, the target was a small rectangle; on the other 20, the target was a large rectangle. Time to reach the target was recorded on each trial. The two distributions (one for each target) are plotted together in Figure 15. The figure shows that, although there is some overlap in times, it generally took longer to move the cursor to the small target than to the large one.

Figure 12. Overlaid frequency polygons.

It is also possible to plot two cumulative frequency distributions in the same graph. This is illustrated in Figure 13 using the same data from the cursor task. The difference in distributions for the two targets is again evident.

Figure 13. Overlaid cumulative frequency polygons.

Figure 14. Stem and Leaf Plot

Box Plots

We have already discussed techniques for visually representing data (see histograms and frequency polygons). In this section, we present another important graph, called a box plot. Box plots are useful for identifying outliers (extreme scores) and for comparing distributions. We will explain box plots with the help of data from an in-class experiment. Students in Introductory Statistics were presented with a page containing 30 colored rectangles. Their task was to name the colors as quickly as possible. Their times (in seconds) were recorded. We’ll compare the scores for the 16 men and 31 women who participated in the experiment by making separate box plots for each gender. Such a display is said to involve parallel box plots.

There are several steps in constructing a box plot. The first relies on the 25th, 50th, and 75th percentiles in the distribution of scores. Figure 15 shows how these three statistics are used. For each gender we draw a box extending from the 25th percentile to the 75th percentile. The 50th percentile is drawn inside the box.

Therefore, the bottom of each box is the 25th percentile, the top is the 75th percentile, and the line in the middle is the 50th percentile. The data for the women in our sample are shown in Table 6.

14, 15, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19

20, 20, 20, 20, 20, 20, 21, 21, 22, 23, 24, 24, 29

Table 6. Women’s times.

For these data, the 25th percentile is 17, the 50th percentile is 19, and the 75th percentile is 20. For the men (whose data are not shown), the 25th percentile is 19, the 50th percentile is 22.5, and the 75th percentile is 25.5.

Figure 15. The first step in creating box plots is to identify appropriate quartiles.

Before proceeding, the terminology in Table 7 is helpful.

Name	Formula	Value from example
Upper Hinge	75th Percentile	20
Lower Hinge	25th Percentile	17
H-Spread	Upper Hinge – Lower Hinge	3
Step	1.5 x H-Spread	4.5
Upper Adjacent	Largest value below Upper Hinge + 1 Step	24
Lower Adjacent	Smallest value above Lower Hinge + 1 Step	14
Outside value/Outlier	Value beyond “whiskers”	29

Table 7. Box plot terms and values for women’s times.

Continuing with the box plots, we put “whiskers” above and below each box to give additional information about the spread of data. Whiskers are vertical lines that end in a horizontal stroke. Whiskers are drawn from the upper and lower hinges to the upper and lower adjacent values (24 and 14 for the women’s data), as shown in Figure 16.

Figure 16. The box plots with the whiskers drawn.

Although whiskers may not cover all data points, we still wish to represent data outside whiskers in our box plots. This is achieved by adding additional marks beyond the whiskers. Specifically, outside values are indicated by small “o’s” and outlier values are indicated by asterisks (*). In our data, there are no far-out values and just one outside value. This outside value of 29 is for the women and is shown in Figure 17.

Figure 17. The box plots with the outside value shown.

There is one more mark to include in box plots (although sometimes it is omitted). We indicate the mean score for a group by inserting a plus sign. A mean is one type of average we will learn about calculating in the next chapter. Figure 18 shows the result of adding means to our box plots.

Figure 18. The completed box plots.

Figure 18 provides a revealing summary of the data. Since half the scores in a distribution are between the hinges (recall that the hinges are the 25th and 75th percentiles), we see that half the women’s times are between 17 and 20 seconds whereas half the men’s times are between 19 and 25.5 seconds. We also see that women generally named the colors faster than the men did, although one woman was slower than almost all of the men.

The Shape of Distribution

Finally, it is useful to present discussion on how we describe the shapes of distributions, which we will revisit in the next chapter to learn how different shapes affect our numerical descriptors of data and distributions.

The primary characteristic we are concerned about when assessing the shape of a distribution is whether the distribution is symmetrical or skewed. A symmetrical distribution, as the name suggests, can be cut down the center to form 2 mirror images. Although in practice we will never get a perfectly symmetrical distribution, we would like our data to be as close to symmetrical as possible for reasons we delve into in Chapter 3. Many types of distributions are symmetrical, but by far the most common and pertinent distribution at this point is the normal distribution, shown in Figure 19. Notice that although the symmetry is not perfect (for instance, the bar just to the right of the center is taller than the one just to the left), the two sides are roughly the same shape. The normal distribution has a single peak, known as the center, and two tails that extend out equally, forming what is known as a bell shape or bell curve.

Figure 19. A symmetrical distribution

Symmetrical distributions can also have multiple peaks. Figure 20 shows a bimodal distribution, named for the two peaks that lie roughly symmetrically on either side of the center point. As we will see in the next chapter, this is not a particularly desirable characteristic of our data, and, worse, this is a relatively difficult characteristic to detect numerically. Thus, it is important to visualize your data before moving ahead with any formal analyses.

Figure 20. A bimodal distribution

Distributions that are not symmetrical also come in many forms, more than can be described here. The most common asymmetry to be encountered is referred to as skew, in which one of the two tails of the distribution is disproportionately longer than the other. This property can affect the value of the averages we use in our analyses and make them an inaccurate representation of our data, which causes many problems.