Comparing Batches of Categorical Data

In topics D2 and D3, we looked at the country of origin of Major League baseball players who were born in the year 1975. We found that approximately two-thirds of the players were born in the United States, but a sizeable proportion of players came from Latin America.

Looking at these data more carefully, it is natural to ask: do players from other countries excel in particular positions in baseball? Specifically, are there differences in the country of birth distribution between pitchers and nonpitchers?

To answer the question, we revisit our data and divide the 205 players into two groups - the 108 players who are pitchers and the 97 players who are nonpitchers. For each group, we categorize the players by the country of origin (USA, Latin American, or Other). We present these two frequency tables below. In addition, for each group, we find the proportion and percentage for each category.

Pitchers			Country	Nonpitchers
Frequency	Proportion	Percentage		Frequency	Proportion	Percentage
78	0.722	72	USA	57	0.588	59
24	0.222	22	Latin America	38	0.392	39
6	0.056	6	Other	2	0.021	2
108	1	100	TOTAL	97	1	100

To compare the country of origins of the pitchers and nonpitchers, we'd like first to construct a graph. In topic D2, we used a bar chart and a segmented bar chart to display a single frequency table of categorical data. These same graphs can be used to compare several frequency tables.

We first show side-by-side bar charts of the pitcher and nonpitcher tables. Since the numbers of pitchers and nonpitchers are not equal, it is best to plot the proportions (or percentages) of the two tables - in that way, the total proportion is equal to one for both groups and it is easier to make comparisons. We see differences - there are over twice as many USA pitchers than Latin American pitchers, while the proportions of USA and Latin American nonpitchers are more similar in size. It appears that pitchers are more likely to be American than nonpitchers.

A similar graphical comparison can be made by use of segmented bar charts. In the figure below, we show side-by-side segmented bar charts of the category proportions. As in the earlier display, it is best to graph the proportions (instead of the frequencies) since the sum of proportions is equal to one for both tables. From this display, we clearly see the differences in the country of birth between the pitchers and nonpitchers.

Now that we have noticed that there is a difference in the country of birth in the two groups of players, how can we summarize this difference? If we focus on the percentages in the table, we read that 22% of the pitchers are Latin American in origin, and 39% of the nonpitchers are Latin American. If we look at the difference, we can say that the proportion of Latin Americans among the nonpitchers exceeds the proportion of Latin Americans among the pitchers by 39 - 22 = 17%.

Practice: Comparing Batches of Categorical Data

In the book Cities Rated & Ranked, 331 metropolitan areas are ranked from best to worst on the basis of quality of life. The top 30 and the bottom 30 metropolitan areas are listed in the table below. In addition, the table gives the population density (LOW or HIGH compared to the U.S. average) and the population growth in the 1990-2002 (LOW or HIGH compared to the U.S. average).This data is stored in the StatCrunch data file quallity.of.life.txt.

1. For each group of metropolitan areas, find the number of areas that have a low and high population density and place your frequencies in the table below.

	TOP AREAS		BOTTOM AREAS
Population density	Frequency	Proportion	Frequency	Proportion
LOW
HIGH

2. Construct side-by-side bar charts of the proportions of low and high density areas in the "top" and "bottom" groups.

3. By computing proportions, investigate if the population density differs between the "top" areas and the "bottom" areas.

4. For each group of areas, find the proportion that have a low and high population growth rate. Construct two segmented bar charts to compare the proportions of low and high growth in the two groups.

5. Are the population growth rates different for the "top" and "bottom" metropolitan areas? (Compare proportions to answer the question.)

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