Define sampling distribution of mean

Sampling distribution of the mean: the probability distribution of the sample mean x̄ obtained from all possible random samples of the same size n drawn from a population. A sample statistic varies from one sample to another, and this distribution describes how.

Properties (population mean μ, standard deviation σ):

1. Mean of x̄: μ(x̄) = μ (x̄ is an unbiased estimator of μ).
2. Standard error (SD of x̄): σ(x̄) = σ/√n for infinite population or sampling with replacement; σ(x̄) = (σ/√n)√((N − n)/(N − 1)) for sampling without replacement from a finite population of size N.
3. Shape: if the population is normal, x̄ is normal for any n; for other populations, x̄ is approximately normal when n is large (central limit theorem).
4. As n increases, the spread σ/√n decreases.

Example: population 2, 4, 6, with samples of size 2 taken with replacement give 9 samples and sample means 2, 3, 4, 3, 4, 5, 4, 5, 6.

Mean of x̄ = 4 = μ, and SE = √(4/3) = 1.1547 = σ/√2 with σ = 1.633.