# Bivariate Data and Analysis Part B: Contingency Tables (20 minutes)

In Part A, you examined bivariate data — data on two variables — graphed on a scatter plot. Another useful representation of bivariate data is a contingency table, which indicates how many data points are in each quadrant.

Let’s take another look at the scatter plot from Part A, with the quadrants indicated: Recall that:

 • Quadrant I has points that correspond to people with above-average arm spans and heights. • Quadrant II has points that correspond to people with below-average arm spans and above-average heights. • Quadrant III has points that correspond to people with below-average arm spans and heights. • Quadrant IV has points that correspond to people with above-average arm spans and below-average heights.

The following diagram summarizes this information: If you count the number of points in each quadrant on the scatter plot, you get the following summary, which is called a contingency table: Problem B1

Use the counts in this contingency table to answer the following:

 a. Do most people with below-average arm spans also have below-average heights? b. Do most people with above-average arm spans also have above-average heights? c. What do these answers suggest?

The column proportions and percentages are also useful in summarizing these data:

 Column proportions: Column percentages:  Note that there are 12 people with below-average arm spans. Most of them (10/12, or 83.3%) are also below average in height. Also, there are 12 people with above-average arm spans. Most of them (11/12, or 91.7%) are also above average in height. Note that the proportions and percentages are counted for the groups of arm spans only. The proportion 2/12 in the upper left corner of the table means that two out of 12 people with below-average arm spans also have above-average heights. It is important to note that the proportions across each row may not add up to 1. When we look at column proportions, we divide the values in the contingency table by the total number of values in the column, rather than in the row. In this example, there are 13 values in the first row, but there are 12 values in the column; therefore, we’re looking at proportions of 12 rather than 13. Percentages are equivalent to proportions but can be more descriptive for interpreting some results. Since 91.7% of the people with above-average arm spans are also above average in height, and 83.3% of the people with below-average arm spans are also below average in height, this indicates a strong positive association between arm span and height. Note that in this study, we’re using the word “strong” in a subjective way; we have not defined a specific cut-off point for a “strong” versus a “not strong” association.

Problem B2

Use the counts in the contingency table to answer the following:

 a. Do most people with below-average heights also have below-average arm spans? b. Do most people with above-average heights also have above-average arm spans?

Problem B3

Perform the calculations to find the row proportions and row percentages for this data, and complete the tables below. Note that there are 13 people whose heights are above average and 11 whose heights are below average; this will have an effect on the proportions and percentages you calculate. Do you find a strong positive association between height and arm span?

Row proportions:       Row Total

Row percentages:       Row Total

Tip: The proportions in the “Above Average” row will be out of 13. Once you find the proportions, use them to find the percentages.

### Solutions

Problem B1

 a. Yes. Of the 12 people with below-average arm spans, 10 have below-average heights. b. Yes. Of the 12 people with above-average arm spans, 11 have above-average heights. c. These answers suggest a positive association between arm span and height.

Problem B2

### Solution: Problem B2

 a. Yes. Of the 11 people with below-average heights, 10 have below-average arm spans. b. Yes. Of the 13 people with above-average heights, 11 have above-average arm spans.

### Solution: Problem B3

Here are the completed tables:

Row proportions: Row percentages: Since 90.9% of the people with below-average heights also have below-average arm spans, and 84.6% of the people with above-average heights also have above-average arm spans, this again indicates a strong positive association between height and arm span.