1. 4.4 Statistical Paradoxes

2. Total SAT Subpopulation Scores by Ethnic Group
SAT Scores
Below is a table of SAT scores broken down by ethnic group for 1976 and All of the non-white groups saw their scores increase or hold the same while overall scores declined for the population as a whole. Why?
Total SAT Subpopulation Scores by Ethnic Group
Year
White
Black
Asian
American
Indian
Mexican
Puerto
Rican
1976
944
686
932
808
781
765
1990
933
737
938
825
809
764
Berliner, D. (1993) Educational Reform in an Era of Disinformation. Educational Policy Analysis Archives

3. Better in Each Case, But Worse Overall
It is possible for something to appear better in each of two or more group comparisons but actually be worse overall. This occurs because of the way in which the overall results are divided into unequally sized groups.

4. Simpson’s Paradox
The general case in which a set of data gives different results for each of several group comparisons than it does when the groups are taken together.
Named for Edward Simpson who described it in 1951.
The same idea also described by Scottish statistician George Yule around 1900.

5. Who has the better overall shooting percentage?
2-pt. FG
3-pt. FG
DeMarcus Cousins
Kentucky Wildcats
0.558
0.167
Andy Rautins
Syracuse
0.582
0.403

6. Same Decision? 2-pt. FG 3-pt. FG Total FG 192/344 0.558 1/6 0.167 ?
DeMarcus Cousins
Kentucky Wildcats
192/344
0.558
1/6
0.167 ?
Andy Rautins
Syracuse
32/55
0.582
94/233
0.403

7. Which treatment for kidney stones is better?
Treatment A
Treatment B
Small Stones
81/87
234/270
Large Stones
192/263
55/80
Total
Which treatment would you recommend? Why?
C. R. Charig, D. R. Webb, S. R. Payne, O. E. Wickham (29 March 1986). "Comparison of treatment of renal calculi by operative surgery, percutaneous nephrolithotomy, and extracorporeal shock wave lithotripsy". Br Med J (Clin Res Ed) 292 (6524): 879–882. doi: /bmj PMID  PMC 

8. Drug Tests
The following chart represents drug tests of applicants for a job.
Find the percent of false positive tests.
Find the percent of false negative tests.
Find the percent of positive tests.
Find the percent of drug users.
Should the test be set-up this way?
Drug Users
Nonusers
Positive test result
297 34
Negative test result 3
1666