This note is just a quick follow-up to our last note on correcting the bias in estimated standard deviations for binomial experiments.
For normal deviates there is, of course, a well know scaling correction that returns an unbiased estimate for observed standard deviations.
It (from the same source):
… provides an example where imposing the requirement for unbiased estimation might be seen as just adding inconvenience, with no real benefit.
Let’s make a quick plot comparing the naive estimate of standard deviation (“forgetting to use
n-1 in the denominator”) and the Bessel corrected estimate (the square-root of the Bessel corrected variance). It is well known that the naive estimate is biased-down and under-estimates both the variance and standard deviation. The Bessel correction deliberately inflates the variance estimate to get the expected value right (i.e., to remove the bias). However, as we can see in the following graph: for the standard deviation the correction is too much. The square-root of the Bessel corrected variance is systematically an over-estimate of the standard deviation.
We can show this graphically as follows.
The above graph is portraying, for different sample sizes (
n), the ratio of the expected values of the various estimates to the true value of the standard deviation (for observations from an i.i.d. normal random source). So: an unbiased estimate would lie on the line
Notice the Bessel corrected is further away from the true value of the standard deviation than the naive estimate was (just biased in the opposite direction). So from the standard-deviation point of view the Bessel correction isn’t really better than the naive estimate.
All work is shared here.