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Sample and population Variance and SD
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I am reposting this part of an answer I gave in order to get an explanation from a stats expert (hopefully).
My confusion is with this wikipedia entry where variance is said to be unbiased whereas SD is biased. The entry says
"...the sample variance is an unbiased estimator for the population variance, but its square root, the sample standard deviation, is a biased estimator for the population standard deviation.
My understanding is that sample bias comes about due to the increased probability that a sample is near the mean in a sample of a distribution than is the case with a full population. The n-1 compensates for that.
In the wikipedia entry I can't see how the variance can be unbiased whilst its root is then biased. Is this correct? In stats it is so often a case of what words mean rather than what numbers mean.
My confusion is with this wikipedia entry where variance is said to be unbiased whereas SD is biased. The entry says
"...the sample variance is an unbiased estimator for the population variance, but its square root, the sample standard deviation, is a biased estimator for the population standard deviation.
My understanding is that sample bias comes about due to the increased probability that a sample is near the mean in a sample of a distribution than is the case with a full population. The n-1 compensates for that.
In the wikipedia entry I can't see how the variance can be unbiased whilst its root is then biased. Is this correct? In stats it is so often a case of what words mean rather than what numbers mean.
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(Original post by nerak99)
I am reposting this part of an answer I gave in order to get an explanation from a stats expert (hopefully).
My confusion is with this wikipedia entry where variance is said to be unbiased whereas SD is biased. The entry says
"...the sample variance is an unbiased estimator for the population variance, but its square root, the sample standard deviation, is a biased estimator for the population standard deviation.
I am reposting this part of an answer I gave in order to get an explanation from a stats expert (hopefully).
My confusion is with this wikipedia entry where variance is said to be unbiased whereas SD is biased. The entry says
"...the sample variance is an unbiased estimator for the population variance, but its square root, the sample standard deviation, is a biased estimator for the population standard deviation.
It's a nice problem in mathematical statistics (which is sometimes used to torture students) to show what the expected value for the sample standard deviation is when dealing with Normal random variables.
My understanding is that sample bias comes about due to the increased probability that a sample is near the mean in a sample of a distribution than is the case with a full population. The n-1 compensates for that.
In the wikipedia entry I can't see how the variance can be unbiased whilst its root is then biased. Is this correct? In stats it is so often a case of what words mean rather than what numbers mean.
![\mathbb{E}[f(X)]](https://www.thestudentroom.co.uk/latexrender/pictures/e6/e6faf43f972347a0891b10c4a3e98383.png "\mathbb{E}[f(X)]")
need not be equal to ![f(\mathbb{E}[X])](https://www.thestudentroom.co.uk/latexrender/pictures/5d/5d7d1e255e9179ccf352563fea462ae3.png "f(\mathbb{E}[X])")
for a general function f. Just write the equations down and you'll see that there's no reason to expect them to be equal.If f is linear, all is sweetness and light, and if X is specified, it's sometimes possible to calculate how these two will differ.
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