Chapter 3: Displaying and Describing Categorical Data Sarah Lovelace and Alison Vicary Period 2

● Frequency table: table with category names and counts of totals for each category o gives frequencies that each category occurs ● Relative frequency table: frequency table with percentage each category occurs out of the total number of events/items o If all possible categories are given, relative frequencies (percents) should add up to 100%. o Makes it easier to compare two tables with the same categories but different overall totals ● Distribution: Distribution of a variable describes the variable’s possible values and the relative frequency of each value. o can be seen in a relative frequency table Vocabulary

● Area principle: The area occupied by a part of the graph should correspond to the size of the value it represents. ● Bar chart: displays the distribution of a categorical variable using bars o Allows visual comparison of frequencies o Has small spaces between the bars o Could be a relative frequency bar chart ● Pie chart: displays the distribution of variable with slices of a circle proportional to the categories’ relative frequencies o must include piece of circle for all categories so relative frequencies add up to 100% Vocabulary

● Contingency table: table with counts or percentages of individuals falling into categories of multiple variables o Shows number of individuals in a category, contingent on being in another category o Percents can be included in each cell, giving percent of row, column, or entire table o Marginal distribution: frequency distribution of a single variable from a contingency table  may be written in the margin of the contingency table with totals o Conditional distribution: distribution of one variable just for individuals that satisfy a condition of another variable  Example: distribution of grade level, conditional on being in Statistics  Segmented bar chart: bars for each category are divided into segments with sizes proportional to the relative frequencies of another variable conditional on being in that bar’s category Vocabulary

● Independence: Variables are considered independent if the distribution of one variable is the same for all categories of the other o See if variables are independent by comparing conditional distributions of one variable for each value of the the other ● Simpson’s paradox: happens when averages taken across different groups seem to contradict the overall averages o Better to compare percentages or averages of one variable within each category of the other instead of finding the overall average Vocabulary

In July 1991 and again in April 2001 the Gallup Poll asked random samples of 1015 adults about their opinions on working parents. The table summarizes responses to this question: “Considering the needs of both parents and children, which of the following do you see as the ideal family in today’s society?” Based upon these results, do you think there was a change in people’s attitudes during the 10 years between these polls? Explain. Problem 29 (page 41)

Problem 29, continued Both work full time One works full time, other part time One works, other works at home One works, other stays home for kids No opinion51

Relative frequency table: percent of total responses Problem 29, continued Both work full time14.0%12.9% One works full time, other part time27.0%24.0% One works, other works at home15.0%17.0% One works, other stays home for kids39.0%41.0% No opinion5.0%

Relative frequency bar chart Problem 29, continued

Answer: People’s opinions about working parents and the ideal family did not change over the 10 years from 1991 to Slight differences between relative frequencies of the two years are minimal and may be attributed to sampling variability. Problem 29, continued

A company held a blood pressure screening clinic for its employees. The results are summarized in the table below by age group and blood pressure level. a.Find the marginal distribution of blood pressure level. b.Find the conditional distribution of blood pressure level within each group. c.Compare these distributions with a segmented bar graph. d.Write a brief description of the association between age and blood pressure among these employees. e.Does this prove that people’s blood pressure increases as they age? Explain. Problem 31 (page 41) Under Over 50 Low Normal High235173

Problem 31, continued Under Over 50Total LowCount % of Row % of Column % of Table % 27.6% 5.7% % 20.7% 7.8% % 15.7% 6.5% % 20.0% NormalCount % of Row % of Column % of Table % 49.0% 10.1% % 50.8% 19.2% % 47.2% 19.6% % 48.9% HighCount % of Row % of Column % of Table % 23.5% 4.9% % 28.5% 10.8% % 37.1% 15.4% % 31.0% TotalCount % of Row % of Column % of Table % 100% 20.7% % 100% 37.8% % 100% 41.6% %

Problem 31, continued Under Over 50Total Low % % % % Under Over 50Total Normal % % % % Under Over 50Total High % % % %

Problem 31, continued

d.with an increase in age, there is also an increase in blood pressure. The numbers for high and low blood pressure are fairly equal with lower age, but as age increases, the percentage of adults with low blood pressure falls and the percentage of adults with high blood pressure. e.No, these data do not necessarily prove that blood pressure increases with age. They indicate that there is a possibility that they are related, but there may be a lurking variable leading to higher blood pressure with increased age. Problem 31, continued