The standard deviation remains a biased estimator, but the bias is only about 1% when the sample size is as small as 20, and the remaining bias becomes smaller yet as the sample size increases. Similarly, the degree of bias that remains in the standard deviation is very small when the degrees of freedom instead of the sample size is used in the denominator. With a sample of ten values, the difference between dividing by 10 and dividing by 9 can be meaningful.
With a sample of 100 values, the difference between dividing by 100 and dividing by 99 is quite small. The larger the sample size, of course, the smaller the correction involved in using the degrees of freedom. But a little is left, and it’s usually regarded as negligible. Much of the bias in the standard deviation is in fact removed by the use of the degrees of freedom instead of N in the denominator. The square root of an unbiased estimator is not itself necessarily unbiased. It’s easy to conclude, then, that using N − 1 in the denominator of the standard deviation also removes its bias as an estimator of the population standard deviation. So doing removes the bias from the estimator. It is then biased negatively: it’s an underestimate of the variance in the population.Īs discussed in the prior section, that’s the reason to use the degrees of freedom instead of the actual sample size when you infer the population variance from the sample variance. No special adjustment is needed for to estimate μ accurately.īut when you use N, instead of the N − 1 degrees of freedom, in the calculation of the variance, you are biasing the statistic as an estimator. The absence of bias in a statistic that’s being used as an estimator is desirable. You will sometimes see and s and other statistics referred to as estimators, particularly in the context of inferring population values.Įstimators have several desirable characteristics, and one of them is unbiasedness. The main purpose of inferential statistics, is to infer population parameters such as μ and σ from sample statistics such as and s.
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