Comparison of estimators

Most estimators that are used in statistics have error distributions that are centred on zero — they are unbiased. The standard errors of the estimators are therefore the most important way to compare their accuracy.

Comparison of mean and median

Consider a normal population — say weights of a particular mammal — whose distribution is centred on a parameter µ. Since the normal distribution is symmetric, µ is both the distribution's mean and median.

Is it better to use the sample mean or sample median to estimate µ?

We will use a simulation to help answer this question. The diagram below takes samples of 20 values from a normal population with mean 1000.

Click Another sample several times to see the sample means and medians. Hold the Another sample button down until about 150 samples have been taken.

Observe that:

Since the errors for the sample mean tend to be closer to zero than those for the median, we conclude that:

The sample mean is better than the sample median as an estimator of the centre of a normal population.

The diagram below shows the theoretical distributions of the sample mean and median for different sample sizes.

Use the slider to see that the standard error of the sample median is always about 25% higher than that of the sample mean.

(Note: The error distribution shown for the mean is exact, but that of the median is only approximate — the exact distribution of the sample median is difficult to obtain.)