A five-number summary for the heights in inches of the women who participated in the survey in Case Study 1.1 is as shown.

a. What is the median height for these women?

b. What is the range of heights—that is, the difference in heights between the shortest woman and the tallest woman?

c. What is the interval of heights containing the shortest one-fourth of the women?

d. What is the interval of heights containing the middle one-half of the women?

Case Study 1.1

Who Are Those Speedy Drivers?

A survey taken in a large statistics class at Penn State University contained the question “What’s the fastest you have ever driven a car? mph.” The data provided by the 87 males and 102 females who responded are listed here.

From these numbers, can you tell which sex tends to have driven faster and by how much? Notice how difficult it is to make sense of the data when you are simply presented with a list. Even if the numbers had been presented in numerical order, it would be difficult to compare the two groups.

Your first lesson in statistics is how to formulate a simple summary of a long list of numbers. The dotplot shown in Figure 1.1 helps us see the pattern in the data. In the plot, each dot represents the response of an individual student. We can see that the men tend to claim a higher “fastest ever driven” speed than do the women.

The graph shows us a lot, and calculating some statistics that summarize the data will provide additional insight. There are a variety of ways to do so, but for this example, we examine a five-number summary of the data for males and females. The five numbers are the lowest value; the cut-off points for one-fourth, one half, and three-fourths of the data; and the highest value. The three middle values of the summary (the cutoff points for one-fourth, one-half, and three fourths of the data) are called the lower quartile, me

dian, and upper quartile, respectively. Five-number summaries can be represented like this:

Some interesting facts become immediately obvious from these summaries. By looking at the medians, you see that half of the men have driven 110 miles per hour or more, whereas the halfway point for the women is only 89 miles per hour. In fact, three-fourths of the men have driven 95 miles per hour or more, but only one fourth of the women have done so. These facts were not at all obvious from the original lists of numbers.

Moral of the Story: Simple summaries of data can tell an interesting story and are easier to digest than long lists.

Definitions: Data is a plural word referring to numbers or nonnumerical labels (such as male/female) collected from a set of entities (people, cities, and so on). The median of a numerical list of data is the value in the middle when the numbers are put in order. For an even number of entities, the median is the average of the middle two values. The lower quartile and upper quartile are (roughly) the medians of the lower and upper halves of the data.

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