Unbiased and Biased Estimators

population size

4

sample size

2

population values

value 1

1

value 2

2

value 3

3

value 4

4

value 5

5

value 6

6

statistic

mean

population:

{1,2,3,4}

value of parameter:

μ = 2.5

mean of the sampling

distribution:

μ

x

= 2.5

sample	x
{1,2}	1.5
{1,3}	2.
{1,4}	2.5
{2,3}	2.5
{2,4}	3.
{3,4}	3.5

A statistic is called an unbiased estimator of a population parameter if the mean of the sampling distribution of the statistic is equal to the value of the parameter. For example, the sample mean,

x

, is an unbiased estimator of the population mean,

μ

. In symbols,

μ

x

=μ

. On the other hand, since

μ

s

≠σ

, the sample standard deviation,

s

, gives a biased estimate of

σ

.

For a small population of positive integers, this Demonstration illustrates unbiased versus biased estimators by displaying all possible samples of a given size, the corresponding sample statistics, the mean of the sampling distribution, and the value of the parameter. Note: for the sample proportion, it is the proportion of the population that is even that is considered.