Pl. I
Explorable
Pl. I · Explorable · Statistics, seen 1/3
Statistics, seen · 1 of 3
Sampling, seen
Take a sample, work out its average, and do it again a few hundred times. The averages pile up into a bell curve, even when the numbers they come from look nothing like one.
§1 · Try it
Try it
The top chart is a population: every value you could possibly draw, and how common it is. Each sample takes n values from it at random and averages them. The bottom chart collects those averages. Start with the skewed population, press Run, and watch the pile of averages grow.
- Mean of the means
- 2.02 μ = 2.00
- Spread of the means
- 0.61 σ/√n = 2.00/√10 = 0.63
§2 · Notice
Four things to notice
- The averages centre on the population mean μ, whatever the population's shape. A sample mean is an unbiased estimate.
- They are far less spread out than single values. Their spread, the standard error, is σ/√n. Quadruple the sample size and the spread halves; the readout under the chart shows the predicted and observed values side by side.
- Their shape turns into a bell, even for the skewed and the two-humped populations. That is the central limit theorem. How fast it happens depends on the population: compare n = 2 with n = 30 for the skewed one.
- At n = 1 nothing happens. Each "average" is a single draw, so the bottom chart simply copies the population. The bell only appears once you average several values.
§3 · Why
Why this matters
Most of statistics rests on this pattern. A poll of 1000 people, a clinical trial or an A/B test all report an average from one sample, and the central limit theorem is what lets them say how far that average is likely to be from the truth. It is typically within one standard error, and rarely more than two.
It also explains why bigger samples help less and less. The error shrinks with the square root of n, so each halving of the error costs four times as much data.