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# Standard Error of the Median

Providing certain assumptions are made, the standard error of the median can be estimated by multiplying the standard error of the mean by a constant:

#### Algebraically speaking -

SE (median) = 1.2533 × SE( ) where:

• SE (median) is the standard error of the median,
• SE ( ) is the standard error of the mean.

The assumptions are:

1. the sample size is large
2. the sample is drawn from a normally distributed population

Since the median is usually only used when the data are not drawn from a normally distributed population, this rather limits the usefulness of this formula, and it is rarely used. A better approach is to use simulation.

For example, using R, it is simple enough to calculate the mean and median of 1000 observations selected at random from a normal population (μx=0.1 & σx=10). Repeating this calculation 5000 times, we found the standard deviation of their 5000 medians (0.40645) was 1.25404 times the standard deviation of their means. - In good agreement with both the (approximate) formula above - and with the estimated standard error for such a mean (using σx/√n).

Following an identical procedure, sampling a slightly skewed population, the standard deviation of their medians was only 1.19698 times the standard deviation - and when we sampled a highly skewed population, the standard deviation of their medians dropped to just 1/1018 of the standard deviation of their means.

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