The following are desirable properties for statistics that estimate population parameters:

**Unbiased**: on average the estimate should be equal to the population parameter, i.e.*t*is an unbiased estimator of the population parameter*τ*provided*E*[*t*] =*τ*.**Consistent**: the accuracy of the estimate should increase as the sample size increases**Efficient**: all things being equal we prefer an estimator with a smaller variance

**Property 1**: The sample mean is an unbiased estimator of the population mean

Proof: If we repeatedly take a sample {*x _{1}, x_{2}, …, x_{n}*} of size

*n*from a population with mean

*µ*, then the sample mean can be considered to be a random variable defined by

Since each of the *x _{i}* are random variables from the same population,

*E*[

*x*] =

_{i}*μ*for each

*i*, from which it follows by Property 1 of Expectation that:

**Property 2**: If we repeatedly take a sample {*x _{1}, x_{2}, …, x_{n}*} of size

*n*from a population with variance

*σ*

^{2}, then

Proof: Since the *x _{i}* are independent with variance

*σ*

^{2}, by Property 6.3b and 6.4b of Expectation

**Observation**: The proof of Property 2 depends on the sample members being distributed independently of each other. This may not be the case where the sample is not a small fraction of the population. In this case it can be shown that

where *N* = the size of the population.

**Observation**: If we repeatedly take a sample {*x _{1}, x_{2}, …, x_{n}*} of size

*n*from a population, then the variance

*s*

^{2}of the sample is a random variable defined by

**Property 3**: The sample variance is an unbiased estimator of the population variance.

Click here for a proof of Property 3.