3.2 Pie Charts and Two-Way Tables

Pie Charts The distribution of a categorical variable can be described by a pie chart, which is a disk where slices represent the categories. The proportion of the total area for one slice is equal to the relative frequency for the category represented by the slice. The relative frequencies are usually written as percentages.

Example: Construct and Interpret a Pie Chart A total of 273 children were surveyed about what job they would want to do. The jobs and the percentages of the children who voted for them are shown in the table.

Example: continued 1. Use technology to construct a pie chart of the distribution. 2. Find the proportion of the observations that fall in the spy category. 3. Find the proportion of the observations that do NOT fall in the spy category. 4. Find the proportion of the observations that fall in the athlete category OR fall in the movie-star category. 5. On the basis of the pie chart, a student concludes that 6% of all American children want to be a doctor when they grow up. What would you tell the student?

1. Use technology to construct a pie chart of the distribution. Solution 1. Use technology to construct a pie chart of the distribution.

Solution 2. Find the proportion of the observations that fall in the spy category. From the red slice in the pie chart, we see 16% of the children want to be a spy. So, the proportion is 0.16.

Solution 3. Find the proportion of the observations that do NOT fall in the spy category. Method 1 We can find the proportion of observations that do NOT fall in the spy category by adding up the relative frequencies of all the other categories: 0.13 + 0.12 + 0.10 + 0.08 + 0.06 + 0.35 = 0.84 Method 2 Because the relative frequencies of all the categories of any categorical variable always add to 1, we can find the proportion of observations that do NOT fall in the spy category by subtracting 0.16 from 1 – 0.16 = 0.84

Solution 4. Find the proportion of the observations that fall in the athlete category OR fall in the movie-star category. To find the proportion of observations that fall in the athlete category OR fall in the movie-star category, we add the relative frequencies of the categories athlete and movie star: 0.12 + 0.10 = 0.22

Solution 5. On the basis of the pie chart, a student concludes that 6% of all American children want to be a doctor when they grow up. What would you tell the student? Although it is true that 6% of the 273 children surveyed want to be a doctor, we cannot conclude that 6% of all American children want to be a doctor. If the survey was not carried out well, there would be nonsampling error. And even if the study was performed well, there would be sampling error.

Example: Using a Two-Way Table to Find Proportions The table summarizes the responses from all 42 students who participated in the survey about whether they had read a novel in the past year. 1. How many of the students read a novel in the past year? 2. What proportion of the students did not read a novel in the past year?

Example: Using a Two-Way Table to Find Proportions The table summarizes the responses from all 42 students who participated in the survey about whether they had read a novel in the past year. 3. What proportion of the women read a novel in the past year? 4. What proportion of the men read a novel in the past year?

Solution 1. At the bottom of the green column in the table, we see that 30 students read a novel in the past year. 2. At the bottom of the yellow column, we see that 12 students did not read a novel. There are a total of 42 students, so the proportion of students who did not read a novel in the past year is 12/42 or about 0.286. Therefore, approximately 28.6% of the students did not read a novel in the past year.

Solution 3. To find any proportion of women, we use data only in the row for females in the table. We see 19 of 25 women read a novel in the past year. So, the proportion is 19/25 = 0.76. Therefore, 76% of the women read a novel in the past year.

Solution 4. To find any proportion of men, we use data in the Male row of the table. We see 11 of 17 men read a novel in the past year. So, the proportion is 11/17 or approximately 0.647. Therefore, approximately 64.7% of the men read a novel in the past year.