Post History
#2: Post edited
by
Peter Taylor
·
2024-07-17T07:12:08Z (3 months ago)
Fix typos
This popular meme of $\text{“}N<30\text{”}$ is only a popular meme (except that it's mentioned in statistics courses for the mathematically unschooled), and is about the central limit theorem, which says (loosely speaking) that the sample mean or the sample sum is approximately normally distributed if the sample size is big. However, first not that "big" can be reasonably construed as perhaps $N\ge 12$ if the population distribution is not particularly skewed, whereas with a very skewed distribution $N=100$ may not be enough.- 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 $\text{“}t_{N-1,\text{confidence}}\text{”}$ has nothing to do with how the sample standard deviation is defined, but is used in finding confidence intervals.
In many contexts, one uses a capital $N$ to denot the size of the population and a lower-case $n$ to denote the size of the sample, so by that standard, you should use the lower-case $n$ here.
- This popular meme of $\text{“}N<30\text{”}$ is only a popular meme (except that it's mentioned in statistics courses for the mathematically unschooled), and is about the central limit theorem, which says (loosely speaking) that the sample mean or the sample sum is approximately normally distributed if the sample size is big. However, first note that "big" can be reasonably construed as perhaps $N\ge 12$ if the population distribution is not particularly skewed, whereas with a very skewed distribution $N=100$ may not be enough.
- 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 $\text{“}t_{N-1,\text{confidence}}\text{”}$ has nothing to do with how the sample standard deviation is defined, but is used in finding confidence intervals.
- In many contexts, one uses a capital $N$ to denote the size of the population and a lower-case $n$ to denote the size of the sample, so by that standard, you should use the lower-case $n$ here.
#1: Initial revision
by
Michael Hardy
·
2024-07-17T05:07:15Z (3 months ago)
This popular meme of $\text{“}N<30\text{”}$ is only a popular meme (except that it's mentioned in statistics courses for the mathematically unschooled), and is about the central limit theorem, which says (loosely speaking) that the sample mean or the sample sum is approximately normally distributed if the sample size is big. However, first not that "big" can be reasonably construed as perhaps $N\ge 12$ if the population distribution is not particularly skewed, whereas with a very skewed distribution $N=100$ may not be enough.
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 $\text{“}t_{N-1,\text{confidence}}\text{”}$ has nothing to do with how the sample standard deviation is defined, but is used in finding confidence intervals.
In many contexts, one uses a capital $N$ to denot the size of the population and a lower-case $n$ to denote the size of the sample, so by that standard, you should use the lower-case $n$ here.