Statistics notes from Statistical Methods on Psychology by David Howell
Central Limit Theorem: Given a population with mean mu and variance sigma squared, the sampling distribution of the mean (distribution of sample means) will have a mean of mu and a variance of sigma squared divided by n. The distribution will approach the normal distribution as n, the sample size, increases. (pg170)
Note that “standard error” is the same as the “standard deviation” of a sampling distribution.
If the standard deviation of the population is known then statistical tests can use Z scores (although larger n does help as the distribution will be closer to the normal distribution.
If the population standard deviation is unknown then the distribution of the mean has a Student’s t distribution, where
Note the degrees of freedom for the test statistic is n-1 where n is the number of observations that are made to derive the sample mean.