What is the estimation error?
When a sample statistic (such as or p ) is used
to estimate a population parameter, (µ or π)
there is an error,
error = (estimate − parameter )
Since this definition involves the unknown population parameter,
We cannot find the numerical value of the error in practical problems.
The estimation error is a random quantity — it would be different if we took a different random sample. The error therefore has a distribution.
Error distribution
If we knew the underlying population distribution (including the value of the parameter of interest, µ or π), we could take several random samples from it and evaluate the error that would occur from using each sample. This would give information about the distribution of errors.
Of course, in practice we cannot conduct this type of simulation since we do not know the population distribution. However there are alternative ways to find the error distribution that will be described in the following sections.
For many types of estimate, we can find the error distribution or an approximation to it.
The error distribution describes the accuracy of estimates.
Simulation: Errors when estimating a mean
The diagram below selects samples of 20 values from a normal population. We have called the values 'weights' (e.g. from a sample of reptiles) but they could equally be rainfalls, yields of rice from fields, ...
A single sample is shown and the resulting sample mean is an estimate of the mean population weight.
In a practical application, we would not be able to find the error in this estimate — there is no information even about whether the error is likely to be positive or negative. However since this is a simulation, we do know the underlying population mean. Click Peek at population to see the population distribution (and population mean) and evaluate the estimation error.
Click Another sample several times to take further random samples from the population and build up the distribution of the estimation error.
The error distribution describes how far the estimate (sample mean) is likely to be from the parameter being estimated (population mean).
In the next section, we will show that the error distribution can be approximated, even when the population distribution is unknown.