Central Limit Theorem — in motion
The Central Limit Theorem is one of the most important results in statistics. It says that no matter what shape the population has, the sampling distribution of the mean will be approximately normal — as long as your sample is large enough.
Don't take anyone's word for it. See it happen. Pick a population shape, then build up the right-hand chart gradually — click Take 1 Sample a couple of times, then Take 5, then Take 10, then Take 100. Watch the bell curve form as you go.
🔬 Sampling Distribution Simulator
Watch how sample means distribute when you take repeated samples from a population.
🎯 Population Distribution
The shape of the population you're sampling from
📊 Sampling Distribution of Means
Distribution of sample means (builds as you take samples)
👀 What to Watch For
Choose a population shape and start taking samples. Watch how the sampling distribution builds up — notice how it becomes approximately normal regardless of the population shape, especially as sample size increases.
💡 Key Observations
- The sampling distribution is narrower than the population — sample means vary less than individual scores.
- Larger samples → narrower sampling distribution — more precise estimates.
- The shape becomes normal — even from skewed or bimodal populations (especially with n ≥ 30).
- Theoretical and observed SE converge — the formula σ/√n really works.
⚠️ Why this matters
Because the sampling distribution is predictable and approximately normal, we can use it to calculate probabilities. This is what makes hypothesis testing possible.
When you run a t-test or calculate a p-value, you're essentially asking: "where does my sample result fall in the sampling distribution?"