Distribution of the sample mean Mean

the arithmetic mean of population, or population mean, denoted μ. sample mean (the arithmetic mean of sample of values drawn population) makes estimator of population mean, expected value equal population mean (that is, unbiased estimator). sample mean random variable, not constant, since calculated value randomly differ depending on members of population sampled, , consequently have own distribution. random sample of n observations distributed population, sample mean distribution distributed mean , variance follows:

x
¯



∼
n

{
μ
,



σ

2


n


}

.


{\displaystyle {\bar {x}}\thicksim n\left\{\mu ,{\frac {\sigma ^{2}}{n}}\right\}.}

often, since population variance unknown parameter, estimated mean sum of squares; when estimated value used, distribution of sample mean no longer normal distribution rather student s t distribution n − 1 degrees of freedom.