Sanderson M. Smith

SIMPSON'S PARADOX: A SIMPLE EXAMPLE

One should be cautious with statistical data when two or more groups are combined to form a single group. This is especially true when percentages are involved.

Consider two individuals, A and B. The following table displays their respective success ratios for both halves of a given year.

 SUCCESSES ATTEMPTS % SUCCESSES SUCCESSES ATTEMPTS % SUCCESSES A: First Half 80 100 80% B: First Half 78 100 78% A: Second Half 20 40 50% B: Second Half 2 5 40%

As the table clearly indicates, A's percentage of successes was greater than B's for both the first half and the second half of the year.

Does it follow that A's percentage of successes was greater than B's for the whole year?

Here are the combined totals....

 SUCCESSES ATTEMPTS % SUCCESSES A: Total for year 100 140 71.4% B: Total for year 80 105 76.2%

Here are some related algebraic thoughts:

If a/b > c/d and e/f > g/h, then it is true that a/b + e/f > c/d + g/h.

a/b + c/d is not equal to (a+c)/(b+d).

a/b + c/d is equal to (ad + bc)/(bd).