1. Section 4.3 2-Way Tables and Marginal Distributions
AP Statistics
October 20, 2010

2. What about Categorical Data?
Whenever we have looked at data so far, it has quantitative vs. quantitative
What tools do we use if we are faced with categorical data?
AP Statistics, Section 4.3, Part 1

3. AP Statistics, Section 4.3, Part 1
2-Way Tables
First, think of data where each member of the distribution belongs to only one category.
Education Level:
Did not complete high school
Completed high school
1 to 3 years of college
4 or more years of college
AP Statistics, Section 4.3, Part 1

4. AP Statistics, Section 4.3, Part 1
2-Way Tables
First, think of data where the member of the distribution belongs to only category.
Age:
25 to 34 years
35 to 54 years
55+ years
AP Statistics, Section 4.3, Part 1

5. Sample 2-Way Table Education is considered Age is considered
Each number represents a “count”, or the number of data points that meet the criteria
Sample 2-Way Table
Education is considered
“row variables”
Age is considered
“column variables”
25 to 34
35 to 54
55+
Total
Did not complete HS
4,474
9,155
14,224
27,853
Completed HS
11,546
26,481
20,060
58,087
1 to 3 years of college
10,700
22,618
11,127
44,445
4+ years of college
11,066
23,183
10,596
44,845
37,786
81,435
56,008
175,230
AP Statistics, Section 4.3, Part 1

6. Calculating Marginal Distributions
When we do a marginal distribution, we only look at totals (the values found on the right margin or bottom margin)
In marginal distributions, we divide by “Grand Total”
25 to 34
35 to 54
55+
Total
Did not complete HS
4,474
9,155
14,224
27,853
Completed HS
11,546
26,481
20,060
58,087
1 to 3 years of college
10,700
22,618
11,127
44,445
4+ years of college
11,066
23,183
10,596
44,845
37,786
81,435
56,008
175,230
AP Statistics, Section 4.3, Part 1

7. Calculating Marginal Distributions
Years of school completed, by age, (thousand of persons)
25 to 34
35 to 54
55+
Total
Did not complete HS
4,474
9,155
14,224
27,853
Completed HS
11,546
26,481
20,060
58,087
1 to 3 years of college
10,700
22,618
11,127
44,445
4+ years of college
11,066
23,183
10,596
44,845
37,786
81,435
56,008
175,230
AP Statistics, Section 4.3, Part 1

8. Calculating Marginal Distributions
Years of school completed, by age, (thousand of persons)
25 to 34
35 to 54
55+
Total
Did not complete HS
4,474
9,155
14,224
27,853
Completed HS
11,546
26,481
20,060
58,087
1 to 3 years of college
10,700
22,618
11,127
44,445
4+ years of college
11,066
23,183
10,596
44,845
37,786
81,435
56,008
175,230
AP Statistics, Section 4.3, Part 1

9. Graph of Marginal Distribution
AP Statistics, Section 4.3, Part 1

10. Calculating Conditional Distributions
25 to 34
35 to 54
55+
Total
Did not complete HS
4,474
9,155
14,224
27,853
Completed HS
11,546
26,481
20,060
58,087
1 to 3 years of college
10,700
22,618
11,127
44,445
4+ years of college
11,066
23,183
10,596
44,845
37,786
81,435
56,008
175,230
The “conditional” part sounds like
“on the condition the respondents are 35 to 54”
“among those who have completed high school but did not go to college”
“for those respondents over 55 years of age”
AP Statistics, Section 4.3, Part 1

11. Calculating Conditional Distributions
25 to 34
35 to 54
55+
Total
Did not complete HS
4,474
9,155
14,224
27,853
Completed HS
11,546
26,481
20,060
58,087
1 to 3 years of college
10,700
22,618
11,127
44,445
4+ years of college
11,066
23,183
10,596
44,845
37,786
81,435
56,008
175,230
When we look at conditional distributions, we are restricted to a particular column or row (but not “margins”)
In conditional distributions, we divide by “Total” of the column or row.
AP Statistics, Section 4.3, Part 1

12. Calculating Conditional Distributions
Years of school completed, by age, (thousand of persons)
25 to 34
35 to 54
55+
Total
Did not complete HS
4,474
9,155
14,224
27,853
Completed HS
11,546
26,481
20,060
58,087
1 to 3 years of college
10,700
22,618
11,127
44,445
4+ years of college
11,066
23,183
10,596
44,845
37,786
81,435
56,008
175,230
AP Statistics, Section 4.3, Part 1

13. Calculating Conditional Distributions
Years of school completed, by age, (thousand of persons)
25 to 34
35 to 54
55+
Total
Did not complete HS
4,474
9,155
14,224
27,853
Completed HS
11,546
26,481
20,060
58,087
1 to 3 years of college
10,700
22,618
11,127
44,445
4+ years of college
11,066
23,183
10,596
44,845
37,786
81,435
56,008
175,230
AP Statistics, Section 4.3, Part 1

14. AP Statistics, Section 4.3, Part 1
Round-off Error
Occurs when tables round numbers in each entry and the totals don’t actually match up to what they really are.
AP Statistics, Section 4.3, Part 1

15. Graph of Conditional Distribution
AP Statistics, Section 4.3, Part 1

16. AP Statistics, Section 4.3, Part 1
Simpson’s Paradox
A paradox is “a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true”
Simpson’s Paradox involves percentages…
AP Statistics, Section 4.3, Part 1

17. Which Hospital Will You Choose?
Among the patients of Hospital B, what percentage survived?
On the condition you were admitted to Hospital A, what is your chance of surviving?
Hospital A
Hospital B
Total
Died 63 16 79
Survived
2037
784
2821
2100
800
2900
SR=97%
SR=98%
AP Statistics, Section 4.3, Part 1

18. What if you are in “good” condition?
Amongst Hospital B patients in good condition, what percentage lived?
If you were admitted to Hospital A in good condition, what is your chance of surviving?
Hospital A
Hospital B
Total
Died 6 8 14
Survived
594
592
1186
600
1200
SR=99.0%
SR=98.6%
AP Statistics, Section 4.3, Part 1

19. What if you are in “poor” condition?
Among Hospital B patients in poor condition, what percentage lived?
If you were admitted to Hospital A in poor condition, what is your chance of living?
Hospital A
Hospital B
Total
Died 57 8 65
Survived
1443
192
1635
1500
200
1700
SR=96.2%
SR=96.0%
AP Statistics, Section 4.3, Part 1

20. AP Statistics, Section 4.3, Part 1
Simpson’s Paradox
Refers to the reversal of the direction of a comparison or an association when data from several groups are combined to form a single group.
The lurking variables in Simpson’s Paradox are categorical.
This happens very rarely!
AP Statistics, Section 4.3, Part 1

21. AP Statistics, Section 4.3, Part 1
Assignment
Complete Worksheets
4.2A and 4.2B
AP Statistics, Section 4.3, Part 1