Heights of adults. Errors in a measurement. The IQ scale. Exam scores. Stock returns over short windows. They're all the same shape — a bell, peaked in the middle, thinning at the edges. The universe loves this curve.
The normal (or Gaussian) distribution is the bell-shaped curve that shows up everywhere — heights, exam scores, measurement errors. It's symmetric, centered on the mean, and described entirely by mean and standard deviation.
Standardised test scaling, quality control, statistical inference, financial risk, A/B testing — modern statistics is built on this curve and the central limit theorem behind it.
0 balls dropped through 10 rows of pegs. The dashed line is the theoretical normal curve — the histogram converges to it as you drop more.
The 68-95-99.7 rule
- About 68% of values fall within ±1 standard deviation of the mean
- About 95% within ±2 standard deviations
- About 99.7% within ±3 standard deviations
Central Limit Theorem — average enough independent things and the distribution of those averages goes normal, no matter the original shape. That's why the bell appears everywhere.
If exam scores are normal with mean 70 and SD 10, what fraction scored between 60 and 80?
Within ±1 SD → about 68%.
On the same exam, what fraction scored above 90?
Not everything is normal. Income, city sizes, file lengths, network sizes — these are right-skewed with long tails. Pretending they're normal makes you wildly underestimate extreme events.
- Normal = symmetric bell, defined by mean (centre) and SD (spread).
- 68-95-99.7 rule gives instant tail estimates without tables.
- Many things look normal after averaging (CLT) — but raw data often isn't.