# Equation of standard error of a sample mean

In the equation of the standard error of a sample mean, it's given by: standard deviation divided by root-n.

My major concern is: is it the standard deviation of population or the sample on which this is based?

In some books, it is mentioned to be standard deviation of population while some books mentioned it is standard deviation of sample. In Wikipedia, it is the standard Deviation of the population which is impossible to know, hence standard deviation of sample is used.

In the usual situation, the actual standard error of the mean is $$\sigma/\sqrt{n}$$, but you almost never know $$\sigma$$ (the population standard deviation), so you have to estimate it. Consequently, the estimated standard error of the mean is usually taken to be $$s/\sqrt{n}$$.

Some sources call that "the standard error of the mean", but strictly it's an estimate of that.

• In this what is sigma ?Is it the standard deviation of mean of many samples taken from population ? If yes than how is is approximated by standard deviation of a sample...There might be a big difference in the two standard deviations – Manoj Gupta Aug 28 '19 at 9:26
• Following the usual convention, it's the population standard deviation. Yes, an estimate of it might be some distance from the population standard deviation (particularly with small samples). You can only use the information you have. – Glen_b -Reinstate Monica Aug 28 '19 at 15:45

When you divide by n, it's the "population". But that's theoretical hogwash, you never know the SD of a population or if you did why the heck wouldn't you know the mean?

Using the sample SD is fine, but the SD/root-n estimator is biased due to using the same data to estimate two quantities. The degrees-of-freedom correction: SD / root(n-1) gives the minimum variance unbiased estimator.

• Actually, the VARIANCE estimate is unbiased. It's difficult to get an unbiased SD; it can be done, but it's usually not worth the hassle. – Sheridan Grant Aug 28 '19 at 4:24
• In this what is sigma ?Is it the standard deviation of mean of many samples taken from population ? If yes than how is is approximated by standard deviation of a sample...There might be a big difference in the two standard deviations – Manoj Gupta Aug 28 '19 at 9:32
• Re "hogwash:" Sometimes I know the precision of a physical measurement system (and thereby its SD) but do not know the value I am measuring. – whuber Aug 28 '19 at 14:07
• @ManojGupta sigma is the SD of the population distribution. The standard deviation of [the] mean[s] of many samples taken from [the] population is what we call the standard error. They are related but not the same. – AdamO Aug 28 '19 at 14:36
• @SheridanGrant you'll have to explain that one. SD is the root of variance. Both variance and SD estimates are biased when of the form SSE/n or root-SSE/n respectively. If you meant an unbiased estimate of the standard error, you're right: the formula is correct in either the limit when CLT applies or when the population (distribution) is normal. – AdamO Aug 28 '19 at 14:38

It was already answered in two previous answers, but as a comment, notice that what Wikipedia says is:

## Estimate

Since the population standard deviation is seldom known, the standard error of the mean is usually estimated as the sample standard deviation divided by the square root of the sample size (assuming statistical independence of the values in the sample). $${\sigma}_\bar{x}\ \approx \frac{s}{\sqrt{n}}$$

So the definition is given in terms of the unknown population parameter, but the estimator (function to estimate it from the data you have) is given in terms of sample estimate.

• In this what is sigma ?Is it the standard deviation of mean of many samples taken from population ? If yes than how is is approximated by standard deviation of a sample...There might be a big difference in the two standard deviations – Manoj Gupta Aug 28 '19 at 9:32
• @ManojGupta what sigma? $s$ is sample standard deviation, as explained in Wikipedia. The quote is rather self-explanatory... – Tim Aug 28 '19 at 9:54
• What is σ ? I mean what is actual standard deviation to be used in standard error equation? – Manoj Gupta Aug 28 '19 at 10:08
• @ManojGupta all the three answers answered this question, so I don't really understand what is unclear for you. – Tim Aug 28 '19 at 10:11