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I have a question that is possibly more about language than math, but still it concerns me a lot. I understand that this question may irritate many (because it's stupid, and apparently because I am stupid too), but still I ask not to hate me too much.

We all remember the definition of sample mean:

$ {\displaystyle A={\frac {1}{n}}\sum _{i=1}^{n}a_{i}.} $

It is sum divided by size. Okay. Note that we're not talking about population mean, we're talking about sample mean.

So, today a colleague of mine uttered the following sentence:

Sample mean does not depend on sample size.

Now I'll try to explain my hesitation. As I see that there's some $ n $ that is not constant in this formula, I want to say that sample mean depends on $ n $. But as this $ n $ is fully defined by the sample itself, there is, seemingly, a sense in which sample mean does not depend on $ n $.

So the question is: is my colleague's statement true? And what's that important thing that I don't get about sample mean that makes me so confused about such a basic thing?

oopcode
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    Strictly speaking the sample mean depends on the sample size. "Morally" it doesn't, in the sense that when $n$ is large enough the sample mean of a collection of independent samples will be very close to the population mean with large probability. For instance, if you are trying to estimate the proportion of the population with some property that they either have or don't (for example, whether they are going to vote or not), and at least 10% of the population has the property and at least 10% don't have the property, then the mean of 3000 samples and the mean of 30000 samples will be (Cont.) – Ian Aug 04 '16 at 17:15
  • (Cont.) essentially the same (even if you measure relative error). The difficulty is mainly in justifying the assumption that your samples are independent or sufficiently so. For instance, you may have found yourself only sampling from a group where the property is unusually common or rare, or you may have entirely missed a group which is small (but large enough to be significant in the population as a whole) where the property is unusually common or rare. – Ian Aug 04 '16 at 17:17

2 Answers2

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I think that, without a more specific qualification of that statement, it is difficult to say what was really meant.

For example, one interpretation could be that, for a simple random sample, the sample mean is an unbiased estimator of the population mean, and this property is independent of the sample size. But this interpretation goes well beyond what was literally said.

heropup
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A clear answer would be "NO". It depends on sample size and as the number of samples goes to infinity, your sample mean will approach to the population mean.

Ian
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Seyhmus Güngören
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