## Estimating Population Mean and Variance From a Sample

In practice most of statisitics involves taking and analysing samples. The most important statistical measures for any sample are the meanand variance(or the standard deviation). We usually do not know the mean and standard deviation of the population, and use the sample to obtain estimators for these.

To estimate these values we take a sampleof size

An estimate for the population mean can be quickly found from the mean ofd the sample.

The sample mean is a good estimator for the population mean. It is unbiased so that E( bar x ) = %mu , and of all estimators for the mean using the sample, the sample mean has the smallest variance.

Using the sample varianceas an estimator for the population variance is not such a good idea. The sample variance is biased, so

Instead we use the estimatorwhich is not biased.

For the sample

an estimator for the mean is

and an estimator for the variance is