Sanderson M. Smith

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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).

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