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In statistics we can calculate a mean of a sample and the standard error of the mean. Let's say the mean of a sample is 2 and the standard error 1.5. If we repeatedly sampled from this population, it would be expected that the average distance of the sample mean from the population mean is 1.5. Up until now I think I've subconsciously thought that 2 is assumed to be the centre of the distribution, or in other words the population mean. But am I right in saying that 2 is just as likely to be in the tails of the distribution as it is the centre of the distribution?

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I understand your example such that The 2 is the population ("true") mean. An unbiased estimate of the mean would result in a sampling distribution which is centered about that population value.

However, in practice we don't know the population value. What we see is the mean in the sample, and that mean could have any possible value. It is more likely to be near the population mean, but there is no guarantee that that is the case in your particular sample. If you perform many statistical tests (as most of us will do in their lifetime) you will most of the time be near the population mean, but some of the times you will be way off.

Taken in isolation, you have no way of knowing when you are reasonably near and when not. However, usually we participate in a debate where multiple people/teams try different ways of answering the same or "sameish" question. In time it will become clear which study is the outlier and which study is not.


As an aside: You are right that a standard error is the standard deviation of the sampling distribution. However, a standard deviation is not the average deviation from the mean but the square root of the average squared deviations. Since squaring is a non-linear transformation: $\sqrt{\mathrm{E}(dev^2)} \neq \mathrm{E}(dev)$. In fact, in case of the mean we know that $\mathrm{E}(dev)= 0$, and the estimate of central tendency that minimizes $\mathrm{E}(|dev|)$ is the median.

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