What is the formula for sample standard deviation of a small sample size?
The formula for sample standard deviation is given by:
Am I right that when the sample size is small (
Here,
The reason I am asking is the following. I had no doubts whatsoever that I must multiply standard deviation by the Student's
3 answers
The following users marked this post as Works for me:
User | Comment | Date |
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Ivan Nepomnyashchikh |
Thread: Works for me Thank you for the clarification! |
Sep 8, 2023 at 20:03 |
The sample standard deviation (with Bessel's correction) is defined to be the first formula in your post. It doesn't ‘become’ anything else.
You were possibly remembering using the sample standard deviation in an estimator for the population mean. The
A somewhat related notion is correcting for the fact that the sample standard deviation is consistently an underestimate of the population standard deviation for small population sizes, even after Bessel's correction. There's no one formula for an unbiased estimate of the population standard deviation for an arbitrary distribution, but for particular distributions, the sample standard deviation can be made unbiased by multiplying by a correction factor. Wikipedia has a table of coefficients for a normal distribution (if you use these values, note that they are meant to be divisors, not multiplicands—they are all < 1). But this factor isn't the
This popular meme of
But none of that has anything at all to do with any definition of the "sample standard deviation."
Conventionally the "sample variance" is defined in a way in which one divides by the sample size minus 1. That makes the "sample variance" an unbiased estimator of the population variance. Unbiasedness is overrated, and at any rate the so-called "sample standard deviation" is not an unbiased estimator of the population standard deviation.
The formula in which one multiplies by
In many contexts, one uses a capital
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From the little that I know ...
-
If the sample = population (census)
where is the size of the population and is the population mean. The variance is and the standard deviation then is -
If the sample is smaller than the population (any study except a census)
where is the size of the sample and is the sample mean. This is called the unbiased variance and the standard deviation is .
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