# Unbiased Estimators

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An estimator for a statistical parameteris said to be unbiased if

The mean of a sampleis an unbiased estimator of the population meanThis can be used in a less obvious way in the following example.

A bag has 500 beans of two colours, black and white. A sample of twenty beads is randomly selected, with each bead put back in the bag immedicately after selection, and the number of white beads is counted. Suppose we count n white beads is this sample. The proportion of white beads in the sample is then

To estimate the actual number of white beads, w, in the bag, calculate

The probability of picking a white bead from the bag is so picking a sample of twenty beads, replacing each one in the bag after selection, gives a binomial probability distribution for the number of white beads of

Thus the average number of white beads is a sample is

The estimated number of white beads in the bag using our single sample isso

sois an unbiased estimater for the mean.