Distribution of the sample mean

A population mean, µ, is usually estimated by the mean from a random sample.

If the population standard deviation is σ, has a distribution with

 =  μ
 = 

and is approximately normal if the population is normal or n is reasonably large.

Distribution of the estimation error

Since the error in the estimate is simply the sample mean minus a constant (µ), its distribution has the same shape but is centred on zero,

error   =    − µ   ~   normal (0,   )