Sampling distribution of a sample mean

• Shape of sampling distribution of depends on the population distribution.

– If population distribution is normal, then so is the distribution of sample mean.

• If a population has the N(,) distribution, then the sample mean x of n independent observations has the N(, ) distribution.

n

x

Central Limit Theorem

• What happens when the population distribution is not normal?

• As we increase the number of observations that we use to draw our sample, the sampling distribution of x changes its shape

N = 1 N = 2

N = 10 N = 25

Central Limit Theorem

• Draw a simple random sample of size n from any population with mean and standard deviation .

• When n is large, the sampling distribution of the sample mean x is approximately normal N(, )

• Applies to sum or averages

n