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 




B: First Half 



A: Second Half 




B: Second Half 



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 



B: Total for year 



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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