4-1 How do we describe data using three measures of central tendency, and what is the relative usefulness of the two measures of variation?

Once researchers have gathered their data, they may use descriptive statistics to organize that data meaningfully. One way to do this is to convert the data into a simple bar graph, as in FIGURE 4.1, which displays a distribution of different brands of trucks still on the road after a decade. When reading statistical graphs such as this, take care. It’s easy to design a graph to make a difference look big (Figure 4.1a) or small (Figure 4.1b). The secret lies in how you label the vertical scale (the y-axis).

The point to remember: Think smart. When viewing graphs, read the scale labels and note their range.

The next step is to summarize the data using some measure of central tendency, a single score that represents a whole set of scores. The simplest measure is the mode, the most frequently occurring score or scores. The most familiar is the mean, or arithmetic average—the total sum of all the scores divided by the number of scores. The midpoint—the 50th percentile—is the median. On a divided highway, the median is the middle. So, too, with data: If you arrange all the scores in order from the highest to the lowest, half will be above the median and half will be below it.

Measures of central tendency neatly summarize data. But consider what happens to the mean when a distribution is lopsided, when it’s skewed by a few way-out scores. With income data, for example, the mode, median, and mean often tell very different stories (FIGURE 4.2). This happens because the mean is biased by a few extreme scores. When Microsoft co-founder Bill Gates sits down in an intimate café, its average (mean) customer instantly becomes a billionaire. But the customers’ median wealth remains unchanged. Understanding this, you can see how a British newspaper could accurately run the headline “Income for 62% Is Below Average” (Waterhouse, 1993). Because the bottom half of British income earners receive only a quarter of the national income cake, most British people, like most people everywhere, make less than the mean. Mean and median tell different true stories.

The point to remember: Always note which measure of central tendency is reported. If it is a mean, consider whether a few atypical scores could be distorting it.

Knowing the value of an appropriate measure of central tendency can tell us a great deal. But the single number omits other information. It helps to know something about the amount of variation in the data—how similar or diverse the scores are. Averages derived from scores with low variability are more reliable than averages based on scores with high variability. Consider a basketball player who scored between 13 and 17 points in each of the season’s first 10 games. Knowing this, we would be more confident that she would score near 15 points in her next game than if her scores had varied from 5 to 25 points.

The range of scores—the gap between the lowest and highest—provides only a crude estimate of variation. A couple of extreme scores in an otherwise uniform group, such as the $950,000 and $1,420,000 incomes in Figure 4.2, will create a deceptively large range.

The more useful standard for measuring how much scores deviate from one another is the standard deviation. It better gauges whether scores are packed together or dispersed, because it uses information from each score. The computation (see TABLE 4.1 for an example) assembles information about how much individual scores differ from the mean. If your college or university attracts students of a certain ability level, their intelligence scores will have a relatively small standard deviation compared with the more diverse community population outside your school.

You can grasp the meaning of the standard deviation if you consider how scores tend to be distributed in nature. Large numbers of data—heights, weights, intelligence scores, grades (though not incomes)—often form a symmetrical, bell-shaped distribution. Most cases fall near the mean, and fewer cases fall near either extreme. This bell-shaped distribution is so typical that we call the curve it forms the normal curve.

As FIGURE 4.3 shows, a useful property of the normal curve is that roughly 68 percent of the cases fall within one standard deviation on either side of the mean. About 95 percent of cases fall within two standard deviations. Thus, about 68 percent of any group of people taking an intelligence test will score within ±15 points of 100. About 95 percent will score within ±30 points.