Statistics for People Who (Think They) Hate Statistics

by Neil J. Salkind

The answer is that s (the standard deviation) is an estimate of the population standard deviation, and it is an unbiased estimate. Unbiased means that your sample estimate of the mean is just as likely to be a little higher than the population mean as it is to be a little lower. It is unbiased, though, only when we subtract 1 from n. By subtracting 1 from the denominator, we artificially force the standard deviation to be a tiny bit larger than it would be otherwise. Why would we want to do that? Because, as good scientists, we are conservative. Being conservative means that if we have to err (and there is always a pretty good chance that we will), we will do so on the side of overestimating what the standard deviation of the population is. Dividing by a smaller denominator lets us do so. Thus, instead of dividing by 10, we divide by 9. Or instead of dividing by 100, we divide by 99. Notice that with larger sample sizes, the adjustment of dividing by n − 1 doesn’t make as much difference as it does with a smaller set of scores. This pattern, that larger sample sizes lead to more accurate estimates of population statistics (like the standard deviation and the mean), is reflected throughout all statistics and throughout this book. Biased estimates are appropriate if your intent is only to describe the characteristics of the sample. But if you intend to use the sample as an estimate of a population parameter, then it’s best to calculate the unbiased statistic.