An estimatorfora statistical parameterissaid to be unbiased if
The mean of a sampleisan unbiased estimator of the population meanThiscan be used in a less obvious way in the following example.
A bag has 500 beans of twocolours, black and white. A sample of twenty beads is randomlyselected, with each bead put back in the bag immedicately afterselection, and the number of white beads is counted. Suppose we countn white beads is this sample. The proportion of white beads in thesample is then
To estimate the actualnumber of white beads, w, in the bag, calculate
The probability of pickinga white bead from the bag is sopicking a sample of twenty beads, replacing each one in the bag afterselection, gives a binomial probability distribution for the numberof white beads of
Thus the average number ofwhite beads is a sample is
The estimated number ofwhite beads in the bag using our single sample isso
soisan unbiased estimater for the mean.