Sampling Distribution Explorer

For now, let's assume we know σ: the population standard deviation

That is, we'll assume our sample SD is a perfect estimator of the population SD, so the only source of variability is the sample mean itself. This is never actually the case, but let's pretend 😉.

Watch what happens to the distribution of sample means as you draw more samples. Does the shape of the resulting sampling distribution depend on the population shape? Peruse the Explorer to find out!

Notation reminder: Greek letters and uppercase N describe the population: μ (mu) = population mean, σ (sigma) = population standard deviation, N = population size. Roman letters and lowercase n describe the sample : x̅ ("x-bar") = sample mean, s = sample SD, n = sample size.

Controls

Population shape

μ = 0, σ = 1

Sample size n

Standard error (SD of the sampling distribution of means) = σ/√n = 0.316

Draw samples

Population distribution

Sampling distribution of x̅

Here's how this Explorer works.

BUT. We never actually know σ.

The CLT above assumed you know the true population SD. In real life, you estimate it from your sample. That extra source of uncertainty is exactly what makes the t-distribution fatter in the tails.

z-statistic

(x̄ − μ) / (σ / √n)

The gold line (average of all drawn z-statistics) converges on 0 as draws increase.

Sample SD (s)

distribution of sample SDs

gold line → σ = 1.00 as draws increase

t-statistic

(x̄ − μ) / (s / √n)

The gold line (average of all drawn t-statistics) converges on 0 as draws increase.

Sample size n

df (degrees of freedom) = n − 1 = 9