Standard Error of Median The following question is from Modern Mathematical Statistics (Devore).
Consider the following observations:
0.83, 0.88, 0.88, 1.04, 1.09, 1.12, 1.29, 1.31, 1.48, 1.49, 1.59, 1.62, 1.65, 1.71, 1.76, 1.83
Assume that the distribution of the observations is normal.
b. Calculate a point estimate of the median of the observations and state which estimator you used.
e. What is the estimated standard error of the estimator that you used in part (b).
Here are my responses:
$$\\$$b. Given the assumption of normality, it is reasonable to use $\bar{x}$ as an estimate of the population median.In this case, $\bar{x}=1.3481$
$$\\$$e. Here I am unsure how to proceed. Is there a general formula for the standard error of a point estimate?
 A: At this point your estimate of the standard error is $\frac{s}{\sqrt{n}}$, with $s$ being the sample standard deviation and $n$ the sample size.
The term "standard error" refers to the standard deviation of a statistic used to estimate the value of a parameter. If a random variable has a standard deviation, statistics such as the sample mean may also have a standard deviation, which we call the standard error.
Since you are using the sample mean to estimate the median of a Normal distribution (which is the same as the mean of a Normal distribution), the population standard error would be $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation; we use $s$ instead of $\sigma$ (usually presumed to be unknown) to estimate the standard error.
If you were using the median instead of the mean to estimate the population median (which would not be wise for Normally distributed data as the mean is a better estimator for what is ultimately the same quantity; the mean and the median are equal), you would have a different standard error, a larger one. I don't know if there is an exact formula for the standard error but there is an asymptotic (in other words, approximate) one that is more complicated and more difficult to get and compute.
