Categorical Data!• Frequency Table

– Records the totals (counts or percentage of observations) for each category. If percentages are shown, it is a relative frequency table.

• Distribution– Shows the possible values of the variable and

the relative frequency of each value

Types of Graphs

• Picture – Remember…

– Ex: Stem & Leaf

• Bar Chart – counts for comparison

• Pie Chart – pieces for proportion

Segmented Bar Chart

Contingency Table• Displays counts and sometimes percentages of

individuals falling into named categories on two or more variables. Shows how individuals are distributed along each variable contingent on the value of another variable.– Marginal distribution: distribution of either variable by

itself (last row or column)– Conditional Distribution: distribution of 1 variable for

those individuals satisfying a condition on another variable. Considers a smaller group isolated from the whole.

Independence???• Independence…think impact!

– Variables are independent (one variable does not influence the value of the other variable) if the conditional distribution of 1 variable is the same for each category of the other.

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Contingency Tables

• A contingency table allows us to look at two categorical variables together.

• It shows how individuals are distributed along each variable, contingent on the value of the other variable.– Example: we can examine the class of ticket and whether

a person survived the Titanic:

Slide 3 - 9

Contingency Tables (cont.)

• The margins of the table, both on the right and on the bottom, give totals and the frequency distributions for each of the variables.

• Each frequency distribution is called a marginal distribution of its respective variable.– The marginal distribution of Survival is:

Slide 3 - 10

Contingency Tables (cont.)

• Each cell of the table gives the count for a combination of values of the two values.– For example, the second cell in the crew column tells us

that 673 crew members died when the Titanic sunk.

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Conditional Distributions

• A conditional distribution shows the distribution of one variable for just the individuals who satisfy some condition on another variable.– The following is the conditional distribution of

ticket Class, conditional on having survived:

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Conditional Distributions (cont.)

– The following is the conditional distribution of ticket Class, conditional on having perished:

Python Eggs

Cold Neutral Hot

Number of Eggs

27 56 104

Number hatched

16 38 75

1) What percent of cold eggs hatched?

2) Neutral eggs hatched represent what percent of the total?

3) Are temperature and eggs hatched independent?

Smoking by Students & Parents

Neither Parent Smokes

One Parent Smokes

Both Parents Smoke

Student Does Not Smoke

1168 1823 1380

Student Smokes

188 416 400

Neither Parent Smokes

One Parent Smokes

Both Parents Smoke

Student Does Not Smoke

1168 1823 1380

Student Smokes

188 416 400

4371

1004

1356 2239 1780 5375

The Marginal Distribution

• What percent of children have no parent that smokes?

• Students and both parents that smoke compose what percent of the total?

• Of 1 parent smokers, what is the percentage of children who smoke?

5375

1356

5375

400

2239

416

Is student smoking preference independent of parent choice?

Neither Parent Smokes

One Parent Smokes

Both Parents Smoke

Student Does Not Smoke

1168

86.1%

1823

81.4%

1380 77.5%

Student Smokes

188

13.9%

416 18.6%

400 22.5%

4371

1004

1356 2239 1780 5375

YearResponse of families 1991 2001

Both work full time 142 131

1 full time, 1 part time 274 244

1 full time, 1 work from home 152 173

1 full time, 1 homemaker 396 416

No Opinion 51 51

Is there evidence of a change in working patterns in families???

Conditional distribution by hospital size

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How I get to school

Slide 2 - 20

Simpson’s Paradox

• When averages “pooled” together AND taken across different groups appear to be contradictory. Example: Baseball

Player Overall vs LHP vs RHP

A 33/103 (.320) 28/81 (.346) 5/22 (.227)

B 45/151 (.298) 12/32 (.375) 33/119 (.277)