Average enough of anything, and a bell appears.
Take samples from any shape, flat, lopsided or two-humped, and plot each sample’s average. The averages pile into a normal curve, narrower as the samples grow. That’s the central limit theorem, and it’s why confidence intervals and A/B tests work.
Guess first
A die is flat: each face, 1 to 6, comes up as often as the others. Roll it 30 times and take the average; do that 1,000 times. What shape do the 1,000 averages make?
The 1,000 averages pile into a bell around 3.5: the middle 95% fall between 2.87 and 4.10. Their spread, 0.30, is the die’s own spread, 1.71, divided by √30, as the theorem says. Averaging cancels the highs against the lows, whatever the shape.
Try other shapesThe population
Values in each sample
Draw
The population
This sample Draw a sample to see it here.
0 averages of 30 valuesAverages The normal curve
Draw samples, and watch where their averages land.
Values in each sample
Confidence
Samples
100 samples, each with its interval for the meanCatches it Misses
How it’s worked out: every sample drawn here, in your browser, from the shapes shown (a fair die; waiting times as an exponential with mean 1; two humps as two normal curves, 60% and 40%; U-shaped as an arcsine). The normal curve: the population’s mean, and its spread divided by the square root of the sample size. The intervals: the sample’s mean, plus and minus Student’s t times its standard error.
Why averages behave.
Any shape, the same bell
Averages of independent values from almost any population with a finite spread tend to a normal curve as the samples grow. Lopsided shapes need bigger samples; for a die, 30 is plenty.
Narrower by √n
The averages spread by the population’s spread divided by the square root of the sample size: four times the data halves the uncertainty. That’s the standard error.
What 95% confidence means
It’s a promise about the method, not one interval: of many samples, about 95 in 100 of their intervals catch the true mean. Here, 958 of 1,000 intervals from 30 rolls did. Any one interval either caught it or didn’t.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, “Confidence Limits for the Mean”: the interval with Student’s t.
- Diez, Çetinkaya-Rundel and Barr, “OpenIntro Statistics”, 4th edition, 2019 (free, CC BY-SA): the central limit theorem and confidence intervals.