Unbiased Estimators

a. What is an unbiased estimator?

b. For the following statistics, identify those that are unbiased estimators: mean, median, range, variance, proportion.

c. Determine whether the following statement is true or false: “The sample standard deviation is a biased estimator, but the bias is relatively small in large samples, so \(s\) is often used to estimate \(\sigma \)”

The definition of an unbiased estimator is considered. A set of estimators is provided.

a.

A sample statistic is an unbiased estimator of the corresponding population parameter if the sample statistic's mean value (or expected value) over all possible samples of size \(n\) is equal to the corresponding population parameter.

b.

The following statistics are considered to be unbiased estimators of their respective population parameters:

• Proportion
• Mean
• Variance
• Thus, out of the given list of estimators, mean, variance, and proportion are unbiased estimators. On the other hand, median and range are biased estimators.

c.

The sample standard deviation \(s\) is often used to estimate the population standard deviation \(\sigma \)because the bias is small in large samples. Thus, this statement is true.