A New View of Statistics

© 1997 Will G Hopkins

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Summarizing Data:
Simple Statistics: THE SPREAD continued
 Standard Deviation
The standard deviation is usually the best measure of spread. It has a complicated definition: take the distance of each number from the mean, square it, average the result, then take the square root. In short, it's the root mean square of the distances (or differences) from mean. It's usually abbreviated as SD in scientific journals and as s in stats books and stats journals.

Actually, when you take the mean or average of the squares, you have to divide by n - 1 (one less than the sample size). Dividing by n gives you a biased estimate. Obviously for large n it doesn't matter whether you use n or n-1, but for n<20 it starts to make a difference. When using a calculator to work out a standard deviation, press the sn-1 button, not sn.

The figure shows how big one SD looks on a frequency distribution for a normally distributed variable like height. I've shown the frequencies (number of subjects) as a continuous curve rather than as discrete points for each value of height. The best way to think about the SD is that about two-thirds of the values of a variable are found within one SD each side of the mean.

The standard deviation is sometimes expressed as a percent of the mean, in which case it's known as a coefficient of variation. When the SD and mean come from repeated measurements of a single subject, the resulting coefficient of variation is an important measure of reliability. This form of within-subject variation is particularly valuable for sport scientists interested in the variability an individual athlete's performance from competition to competition or from field test to field test. The coefficient of variation of an individual athlete's performance is typically a few percent.

A measure of spread closely related to the SD is the variance, which is simply the square of the SD. I can't show you variance on a diagram. Statisticians prefer it to the SD, but it's not much use for researchers.

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