## Student's t - Distribution

If observationsare taken from a normal distribution with mean andvariance the sample mean followsa normal distribution In practice however thepopulation variance israrely known and must also be estimated. If the variance isestimated as ( isunbiased for sothat )from a large sample of size then,the sample mean is approximately normally distributed, theaccuracy of the approximation improving with increasing until,at thet distribution is identical to the normal distribution.

When n is small however, thenormal approximation given above is not accurate enough and we mustuse Student's t – distribution:

If a random sample isselected from a normal distribution with mean andunknown variance the hasa distribution,where In fact there is not asingle t distribution. There is a whole family, each with anassociated number of degrees of freedom, hencethe label usedabove.

The distributionis an approximation to the normal distribution for each value of  Like the normaldistribution, the t – distribution is symmetrical and unimodal (theprobability density function has one peak, as in the diagram above),and also like the normal distribution, calculations invoving the tdistribution from first principles is not elementary and must be donefrom tables.