week5 - Lecture 11 Nancy Pfenning Stats 1000 Chapter 6:...

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Lecture 11 Nancy Pfenning Stats 1000 Chapter 6: Relationships Between Categorical Variables Some variables are categorical by nature (eg. sex, race, major, political party); others are created by grouping quantitative variables into classes (eg. age as child, adolescent, young adult, etc.). We analyze categorical data by recording counts or percents of cases occurring in each category. In this course, we mainly consider just two categorical variables at a time. Example We can construct a two-way table (also called a contingency table) showing the relationship between gender (row variable) and lenswear (column variable) of statistics class members during a previous semester. None Glasses Contacts Total Male 65 36 37 138 Female 110 32 91 233 Total 175 68 128 371 One natural question to ask is, which group tends to wear glasses more, males or females? The counts are quite close for males and females (36 vs. 32) but since there are relatively few males in the class, the percentage is actually much higher for the males. Thus, to compare lenswear of males and females, since there are way more females in the classes than males, it is better to report percentages instead of counts, concentrating on one gender category at a time. This tells the conditional distribution of lenswear, given gender. It suggests we are thinking of gender as the explanatory variable. NONE GLASSES CONTACTS TOTAL Cond.dist.of lens, given male 65 138 = 47% 36 138 = 26% 37 138 = 27% 100% Cond.dist.of lens, given female 110 233 = 47% 32 233 = 14% 91 233 = 39% 100% To use a bar graph to display a conditional distribution, label the horizontal axis with the explanatory variable—in this case, males and females. Over the male label would be bars of height 47%, 26%, and 27%, for percentages wearing none, contacts, or glasses. Over the female label would be bars of height 49%, 14%, and 39%, for percentages wearing none, contacts, or glasses. This impresses on us visually the di±erence between males and females: males have a tendency to wear glasses, females to wear contacts. Alternatively, (but perhaps less intuitively) we may choose to think of lenswear as the explanatory variable and consider the distribution of gender separately for each lens category: Cond.Dist.of Gender, Cond.Dist.of Gender, Cond.Dist.of Gender, Lenswear given none given glasses given contacts Male 65 175 = 37% 36 68 = 53% 37 128 = 29% Female 110 175 = 63% 32 68 = 47% 91 128 = 71% TOTAL 100% 100% 100% We could display this distribution by labeling three lenswear categories on the horizontal axis, and drawing bars with heights to represent the percentages of males and females in each. 47
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Example Of the male students in the classes, what percentage wear glasses? The condition of being male is given; we look at the conditional distribution of lenswear, given that a person is male, and Fnd the percentage wearing glasses to be 26%. Example
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week5 - Lecture 11 Nancy Pfenning Stats 1000 Chapter 6:...

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