Frequency table. Lecture 12: Relationships Between Categorical Variables. Contingency table. Bar plots

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1 Frequency table Contingency tables Lecture 12: Relationships Between Categorical Variables Statistics 10 Colin Rundel February 27, 2012 A common way to summarize categorical variables is with a frequency table. Major Frequency Arts and Humanities 35 Natural Sciences 60 Social Sciences 118 Total 213 If we replaced the counts with percentages or proportions, the table would be called a relative frequency table. Major Proportion Arts and Humanities 35/213 = Natural Sciences 60/213 = Social Sciences 118/213 = Total 213/213 = 1 Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Contingency tables Contingency table Bar plots A table that summarizes data for two categorical variables is called a contingency table. A bar plot is a common way to display a single categorical variable. A bar plot where proportions instead of frequencies are shown is called a relative frequency bar plot. Arts and Natural Social Humanities Sciences Sciences Total International Midwest Northeast South West Total column totals row totals international midwest northeast south west international midwest northeast south west Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Would you prefer to be reported counts or proportions if you re looking for

2 Proportions: relationship vs. major Probability Relationship Single Other Arts and Humanities Natural Sciences Social Sciences What proportion of social sciences majors are single? What proportion of those in a relationship are social sciences majors? The proportions that we have been calculating also serve as estimates of the probability that a member of the population belongs to a particular category. Marginal Probability - proportion of population with trait A P(trait A) = # with trait A total Joint Probability - proportion of population with trait A and B P(trait A and B) = # with trait A and B total Conditional Probability - proportion of individuals with trait B who also have trait A P(trait A trait B) = # with trait A and B # with trait B = P(A and B) P(B) Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Florida voters and illegal immigration Marginal probability 1,000 randomly sampled Florida voters were asked if they thought workers who have illegally entered the US should be allowed to keep their jobs and apply for US citizenship, allowed to keep their jobs as temporary guest workers but not allowed to apply for US citizenship, or lose their jobs and have to leave the country. The results of the survey by political ideology is shown below. Note: 910 respondents answered both questions. SurveyUSA poll from Jan 27-29, 2012, client/ PollReport.aspx?g=60d6fa c51-a5f8-714f40976df2. Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 What is the probability that a Florida voter is conservative? is in favor of illegal immigrants working in the US staying and applying for citizenship? P(conservative) = P(apply for citizenship) = marginal probabilities Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18

3 Joint probability Conditional probability What is the probability that a Florida voter is conservative and is in favor of illegal immigrants working in the US staying and applying for citizenship? P(conservative and apply for citizenship) = joint probability If we know that a randomly selected Florida voter is conservative, what is the probability that s/he is in favor of illegal immigrants working in the US staying and applying for citizenship? P(apply for citizenship conservative) = conditional probability Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Conditional probability - another example Same survey asked if whether voters who are familiar with the DREAM act support or oppose it. 32% of the respondents are Democrats, 51% of the respondents support the DREAM act, and 21% of the respondents are Democrats and support the DREAM act. If we randomly select a respondent who supports the DREAM act, what is the probability that s/he is a Democrat? What is the probability that a Democrat supports the DREAM act? A contingency table of proportions support oppose total Democrat non-democrat total P(Democrat) = 0.32 P(Support) = 0.51 P(Democrat and Support) = 0.21 P(Democrat Support) = P(Support Democrat) = Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18

4 Independence Proportions: relationship vs. major We often want to know if two variables are related to each other, this is assessed based on the property of independence. Question to ask yourself is: Does knowing someone has trait A tell you anything about whether they also have trait B? If traits A and B are independent then all of the following will be true: P(A and B) = P(A)P(B) P(A B) = P(A) P(B A) = P(B) Are being in a relationship and major associated? Relationship Single Other Total Arts and Humanities Natural Sciences Social Sciences Total Note - if one of the above is true the other two are also true Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Segmented bar plots in a relationshipit's complicated / other single What is the difference between the two segmented bar plots below? Which plot is preferable for analyzing if being in a relationship and major are associated? arts and humanities in a relationship it's complicated / other single natural sciences social sciences arts and humanities natural sciences social sciences arts and humanities natural sciences social sciences Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18

5 Simpson s Paradox - Berkeley Gender Bias Case Simpson s Paradox - Berkeley Gender Bias Case, cont. Berkeley was sued in the 1970s for having a bias against admitting women based on the following: Admit Deny Total Male Female Total P(Admit Female) = 1494/4321 = P(Admit Male) = 3738/8842 = Dept. A Dept. B Dept. C Dept. D Total Total Admit Deny Admit Deny Admit Deny Admit Deny Admit Deny Total Male Female Total P(Admit Dept. A Male) = 512/825 = 0.62 P(Admit Dept. A Female) = 89/108 = 0.82* P(Admit Dept. B Male) = 353/560 = 0.63 P(Admit Dept. B Female) = 17/25 = 0.68* P(Admit Male) = 1123/2127 = 0.53* P(Admit Female) = 439/1101 = 0.40 P(Admit Dept. C Male) = 120/325 = 0.37* P(Admit Dept. C Female) = 202/593 = 0.34 P(Admit Dept. D Male) = 138/825 = 0.33 P(Admit Dept. D Female) = 131/108 = 0.35* Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18 Statistics 10 (Colin Rundel) Lecture 12 February 27, / 18