In research situations, it is often useful to be able to compare "apples and oranges."
Suppose, for example, that we measured a Bantu woman to be six feet (84 inches) in height and an American woman to be 5'11" (82 inches) in height. The Bantu woman is clearly taller than the American woman. But relative to Bantu women in general, is our Bantu woman "taller" or "shorter" than the American woman? Perhaps surprisingly, a standard statistical transformation will allow us to compare the two women.
In order to carry out the comparison, we must know the mean (average) and standard deviation for the heights of both American women and Bantu women. Suppose that the average height for American women was 79 inches with a standard deviation of 3 inches. Similarly, suppose that the average height for Bantu women was 83 inches with a standard deviation of 2 inches.
We can calculate "standard scores" (or Z-scores)
for both heights as follows:
The Z-score equals the mean minus the actual measure divided
by the standard deviation.
For the American woman, her standardized height is:
The result indicates that, when compared with "normal" American women, our specific American woman is taller than the Bantu woman (compared with the normal height of Bantu women).
By calculating Z-scores, there are many comparisons that are possible to make.