# Citation: Sample mean as consistent and unbiased estimator of the expected value

A reviewer asked for a citation that the sample mean is a consistent and unbiased estimator of the expected value and therefore converges towards the expected value. I know I can easily do the calculations myself based on the variance of the sample mean etc. However, since the reviewer specifically asked for a citation, I am not sure a proof instead would be acceptable. Also, I don't like adding proofs that have almost nothing to do with the actual subject of the article.

In most books I looked at, this fact is either only provided for the normal distribution (i.e. sample mean is an estimator of the parameter $$\mu$$), or included as an exercise.

Is anybody aware of a citation that I could use to support the claim that the sample mean is an unbiased estimator for any distribution (assuming of course expected value exists and variance is finite).

Unbiasedness: Casella and Berger, Statistical Inference, 2nd Ed., Theorem 5.2.6 p213-214

Consistency: Casella and Berger, Statistical Inference, 2nd Ed., Example 10.1.2 et seq., p468-469

• Casella was my first thought, too. I wonder if it would be necessary, however, to go to peer-reviewed, primary literature.
– Dave
Jul 31, 2020 at 10:36
• @Dave I don't really know why the reviewer wants a citation for this, as I considered it general knowledge. So I just hope they are satisfied with this citation. The paper isn't about statistics and it is just a small part of the error estimation of the analysis, so I really believe this is sufficient. Jul 31, 2020 at 10:58
• @Dave Good luck with that. Jul 31, 2020 at 17:12
• @Glen_b that’s what I was thinking, that it would be quite difficult to find it in the primary literature.
– Dave
Jul 31, 2020 at 17:24
• I think a textbook will be adequate for what's essentially undergraduate work; it's not like physicists expect you to cite Philosophiæ Naturalis Principia Mathematica and/or give the quote "Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur." when you just want to write $F=ma$. Jul 31, 2020 at 17:34