Industrial & Systems
Central Limit Theorem Lab
The Central Limit Theorem is the reason the normal distribution is everywhere, and this lab makes it happen in front of you. Pick a population — flat uniform, or a deliberately right-skewed one — set the sample size n, and the lab keeps drawing n values, averaging them, and dropping that average into a histogram. However un-normal the population is, the pile of sample means climbs toward a bell curve centred on the population mean with standard deviation σ/√n (the standard error). The predicted normal curve is drawn on top so you can watch the agreement build, and the readouts compare the observed spread with the σ/√n prediction. Crank up n and the curve — and the real histogram — get narrower in exact step. It's the foundation under confidence intervals, control charts and Six Sigma.
0
Sample means
0.000
Observed mean
0.000
Observed SD
0.129
Predicted σ/√n
The method, with your numbers
Averages converge to normal
- 1
Know the population you're drawing from
μ = E[X], σ = √Var(X)
X ~ Uniform(0,1): μ = 1/2 = 0.5, σ = √(1/12) = 0.289
A flat population: every value in [0, 1] equally likely. No bell in sight yet.
- 2
Centre of the sample-mean distribution
E[X̄] = μ
E[X̄] = 0.5
Averaging never moves the centre — the dashed line on the canvas stays at μ no matter what n is.
- 3
Standard error — averaging shrinks the spread
SE = σ / √n
SE = 0.289 / √5 = 0.289 / 2.236 = 0.129
The √n is the catch: to halve the spread you need 4× the sample size.
- 4
The CLT's prediction (the curve on the canvas)
X̄ ≈ N(μ, SE²) for large n
X̄ ≈ N(μ = 0.5, SE = 0.129)
This is why so many real-world measurements look normal: each one is an average of many small independent effects.
Drag any control above — every number here recalculates. Want this method for any problem? Step Sheets →
Each bar is one average of n draws. However un-normal the population — even the skewed one — the histogram of sample means piles up into the bell curve the CLT predicts: centred on the population mean with standard deviation σ/√n. Raise n and watch the predicted curve (and the spread) get narrower.
How to use this simulation
The Central Limit Theorem is the reason the normal distribution is everywhere, and this lab makes it happen in front of you. Pick a population — flat uniform, or a deliberately right-skewed one — set the sample size n, and the lab keeps drawing n values, averaging them, and dropping that average into a histogram. However un-normal the population is, the pile of sample means climbs toward a bell curve centred on the population mean with standard deviation σ/√n (the standard error). The predicted normal curve is drawn on top so you can watch the agreement build, and the readouts compare the observed spread with the σ/√n prediction. Crank up n and the curve — and the real histogram — get narrower in exact step. It's the foundation under confidence intervals, control charts and Six Sigma.
Everything runs in your browser — no sign-up, no download. Change a value and the result updates instantly, so you can build a feel for how each input shapes the outcome. It pairs with Crameleon's practice exams and step sheets when you want to go from intuition to working the problems.