Statistical Methods for Psychology

deviations of each score from the mean. Granted, these deviations have been squared,

summed, and so on, but at heart they are still deviations. And even though we have di-

vided by (N 2 1) instead of N, we still have obtained something very much like a mean

or an “average” of these deviations. Thus, we can say without too much distortion that

attractiveness ratings for Set 4 deviated, on the average, 0.66 unit from the mean,

whereas attractiveness ratings for Set 32 deviated, on the average, only 0.07 unit from

the mean. This way of thinking about the standard deviation as a sort of average devia-

tion goes a long way toward giving it meaning without doing serious injustice to the

concept.

These results tell us two interesting things about attractiveness. If you were a subject in

this experiment, the fact that computer averaging of many faces produces similar compos-

ites would be reflected in the fact that your ratings of Set 32 would not show much

variability—all those images are judged to be pretty much alike. Second, the fact that those

ratings have a higher mean than the ratings of faces in Set 4 reveals that averaging over

many faces produces composites that seem more attractive. Does this conform to your

everyday experience? I, for one, would have expected that faces judged attractive would be

those with distinctive features, but I would have been wrong. Go back and think again

about those faces you class as attractive. Are they really distinctive? If so, do you have an

additional hypothesis to explain the findings?

We can also look at the standard deviation in terms of how many scores fall no more

than a standard deviation above or below the mean. For a wide variety of reasonably

symmetric and mound-shaped distributions, we can say that approximately two-thirds

of the observations lie within one standard deviation of the mean (for a normal distribu-

tion, which will be discussed in Chapter 3, it is almost exactly two-thirds). Although

there certainly are exceptions, especially for badly skewed distributions, this rule is still

useful. If I told you that for elementary school teachers the average starting salary is

expected to be $39.259 with a standard deviation of $4,000, you probably would not be

far off to conclude that about two-thirds of graduates who take these jobs will earn

between $25,000 and $43,000. In addition, most (e.g., 95%) fall within 2 standard

deviations of the mean.

Computational Formulae for the Variance

and the Standard Deviation

The previous expressions for the variance and the standard deviation, although perfectly

correct, are incredibly unwieldy for any reasonable amount of data. They are also prone to

rounding errors, because they usually involve squaring fractional deviations. They are ex-

cellent definitional formulae, but we will now consider a more practical set of calculational

formulae. These formulae are algebraically equivalent to the ones we have seen, so they

will give the same answers but with much less effort.

The definitional formula for the sample variance was given as

A more practical computational formula is

s^2 X=

aX

22 Aa

XB^2

N

N 21

s^2 X= a

(X 2 X)^2

N 21

42 Chapter 2 Describing and Exploring Data