The use of the term " n " " 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation ( the square root of variance ).
22.
With the correction, the unbiased sample variance is unbiased, while the corrected sample standard deviation is still biased, but less so, and both are still consistent : the correction factor converges to 1 as sample size grows.
23.
The square root is a concave function and thus introduces negative bias ( by Jensen's inequality ), which depends on the distribution, and thus the corrected sample standard deviation ( using Bessel's correction ) is biased.
24.
However this quantity is again uncertain due to the limited number of samples, and I would like to have an idea of the confidence I can put into this sample standard deviation found by this more robust fractile difference based approach.
25.
However, if we have only a sample from the population, we know only the sample mean \ hat { \ mu } and sample standard deviation \ hat { \ sigma }, which are only estimates of the true parameters.
26.
One way is by dividing by a measure of scale ( statistical dispersion ), most often either the population standard deviation, in standardizing, or the sample standard deviation, in studentizing ( e . g ., Studentized residual ).
27.
The "'standard error of the mean "'( SEM ) is the standard deviation of the sample standard deviation ) divided by the square root of the sample size ( assuming statistical independence of the values in the sample ):
28.
For example, the square root of the unbiased estimator of the population variance is " not " a mean-unbiased estimator of the population standard deviation : the square root of the unbiased sample variance, the corrected sample standard deviation, is biased.
29.
For example, for data drawn from the normal distribution, the MAD is 37 % as efficient as the sample standard deviation, while the Rousseeuw Croux estimator " Q " " n " is 88 % as efficient as the sample standard deviation.
30.
For example, for data drawn from the normal distribution, the MAD is 37 % as efficient as the sample standard deviation, while the Rousseeuw Croux estimator " Q " " n " is 88 % as efficient as the sample standard deviation.
