Post History
#3: Post edited
by
r~~
·
2023-09-08T19:55:43Z (over 1 year ago)
Embarassing; it's the chi-square distribution for stddev, not t
- 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 are probably thinking about the sample standard deviation as an estimator for the population standard deviation. As such, the \(t\)-value is relevant in computing a confidence interval around the sample standard deviation for where the population standard deviation might lie.A 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](https://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution) 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 $t$-value.
- 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 \(t\)-value is multiplied by the sample standard deviation as part of finding the confidence interval for the population mean, as you've probably seen in your research.
- 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](https://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution) 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 $t$-value.
#2: Post edited
by
r~~
·
2023-09-08T19:41:33Z (over 1 year ago)
- 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 are probably thinking about the sample standard deviation as an estimator for the population standard deviation. As such, the \(t\)-value is relevant in computing a confidence interval around the sample standard deviation for where the population standard deviation might lie.
A 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 multiplied by a correction factor. Wikipedia has [a table of coefficients](https://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution) for a normal distribution. But this factor isn't the $t$-value.
- 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 are probably thinking about the sample standard deviation as an estimator for the population standard deviation. As such, the \(t\)-value is relevant in computing a confidence interval around the sample standard deviation for where the population standard deviation might lie.
- A 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](https://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution) 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 $t$-value.
#1: Initial revision
by
r~~
·
2023-09-08T19:37:19Z (over 1 year ago)
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 are probably thinking about the sample standard deviation as an estimator for the population standard deviation. As such, the \(t\)-value is relevant in computing a confidence interval around the sample standard deviation for where the population standard deviation might lie. A 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 multiplied by a correction factor. Wikipedia has [a table of coefficients](https://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution) for a normal distribution. But this factor isn't the $t$-value.