# Does the standard error assume the mean is the centre of the distribution?

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?

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.