Biased estimators
In statistics, we mostly use estimators that are unbiased — their error distributions are centred on zero. However you may sometimes meet estimators that are biased — the mean of their error distribution (the bias) is not zero.
Although bias is undesirable, a little bias may be acceptable in an estimator if its standard error is also small.
Estimating the mean of a skew distribution
The diagram below compares the error distributions of the sample mean and median when estimating the population mean of a very skew distribution. Such skew distributions are often appropriate for lifetimes of biological organisms and manufactured products.
For all distributions, whether or not they are skew, the sample mean is an unbiased estimator of the population mean. However the sample mean is further into the tail of a skew distribution than the median, so the sample median tends to under-estimate the population mean — it has negative bias.
The diagram above shows the error distributions for the sample mean and the median. Observe that:
Observe also that when the sample size is large, the bias of the median results in errors that are considerably further from zero.
(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.)