What is a normal distribution? The bell curve, explained

A normal distribution is a pattern of values that forms a symmetric bell curve: most values cluster around the middle, and fewer and fewer appear the farther you go out on either side. It is fixed by just two numbers, the mean, which sets where the peak sits, and the standard deviation, which sets how wide the bell is. It is also called the Gaussian distribution, after the mathematician Carl Friedrich Gauss.

In short

• Two numbers describe the whole curve: the mean (its center) and the standard deviation (its width).
• About 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.
• It appears whenever a quantity is the sum of many small, independent effects, which is what the central limit theorem says, and it fails badly for things that aren't.

What does the bell curve show?

A distribution describes how often each value turns up. Picture measuring something many times, such as the weight of loaves from one bakery, and stacking up the results in a bar chart. If the result is a normal distribution, the bars make a hump with its peak at the mean, the two sides are mirror images, and the tails thin out quickly but never quite reach zero.

Because the curve is symmetric, its mean, median and mode are the same value. Because it is set by two numbers, two normal distributions with the same mean and the same standard deviation are identical, and any normal distribution can be turned into any other by shifting and stretching it. That is why one table, or one calculator function, works for all of them.

The curve has an exact formula, which the NIST/SEMATECH e-Handbook gives for the standard case. In general, the height of the curve at a value x is f(x) = (1 ÷ (σ√(2π))) × e^(−(x − μ)² ÷ (2σ²)). Here μ (mu) is the mean, σ (sigma) is the standard deviation, π is 3.14159…, and e is about 2.718. The key part is the exponent: the height falls off with the square of the distance from the mean, which is why the tails thin out so fast.

How do you read a normal distribution? A worked example with IQ

The working tool is the z-score, the number of standard deviations a value sits from the mean: z = (x − μ) ÷ σ. Every normal distribution, turned into z-scores, becomes the standard normal distribution, with mean 0 and standard deviation 1, and its areas are known.

IQ tests are scaled so that scores in the population have a mean of 100 and a standard deviation of 15, as the note on what IQ measures explains. Question: what share of people score above 130?

1. Find the z-score. (130 − 100) ÷ 15 = 2. A score of 130 is 2 standard deviations above the mean.
2. Use the 68-95-99.7 rule. About 95% of values lie within 2 standard deviations, between 70 and 130. That leaves about 5% outside, split equally between the two tails because the curve is symmetric.
3. Take one tail. About 2.5% score above 130. The exact figure for the standard normal curve is 2.28%, about 23 people in every 1,000.

The same steps give the other landmarks:

IQ score	z-score	Share of people above it, if scores are exactly normal
100	0	50%
115	1	about 15.9%
130	2	about 2.3%
145	3	about 0.13%, roughly 1 in 740

Real tests are normed on samples of a few thousand people, so the far tail is estimated rather than measured, and the last row is the model's answer, not a count.

Why does the bell curve show up so often?

Because of the central limit theorem, one of the deepest results in probability. Roughly stated: add up many independent random effects, none of which dominates, and the total is close to normally distributed, whatever the shape of the individual effects. The theorem needs the effects to have a finite spread; its early form, that the normal curve approximates coin-toss counts, is the de Moivre–Laplace theorem (Wikipedia).

Dice show it happening. Exact counts for fair dice:

Number of dice	Shape of the totals	Detail
1	Flat	Each total from 1 to 6 has a 1 in 6 chance
2	A triangle	7 is likeliest (6 ways in 36); 2 and 12 have 1 way each
3	A rounded hump	10 and 11 are likeliest (27 ways in 216 each); 3 and 18 have 1 way each
10	Very close to a bell	Mean 35, standard deviation about 5.4; 68.7% of totals fall within 1 standard deviation and 95.0% within 2, against 68.3% and 95.4% for an exact normal curve

A single die is as un-bell-like as possible, every face equally likely, yet ten of them added together come within a fraction of a percentage point of the normal curve's 68% and 95%. Many real measurements work like the dice. A measuring error is the sum of many small disturbances; a characteristic shaped by many genes plus environment adds up many small pushes. That is why errors in repeated measurements, and many biological measurements within one group of people, come out roughly normal.

The history follows the same thread. Abraham de Moivre found the curve in 1733 as an approximation for coin-toss counts. In 1809 Gauss used it to justify the method of least squares for fitting data to astronomical observations, and Francis Galton later built his "bean machine", a board of pins that drops balls into a bell-shaped pile (Wikipedia).

When is data not normally distributed?

Whenever the conditions behind the central limit theorem break: when effects multiply instead of adding, when one effect can dominate, or when values are bounded on one side. Three common cases:

• Incomes and wealth. These can't go below zero but can reach enormous sizes, so the distribution has a long right tail. Mean income sits well above the median, which is why incomes are reported as medians.
• Waiting times and lifetimes. How long until the next bus, or how long a light bulb lasts, are skewed: many short values and a few long ones.
• Financial returns. Day-to-day changes in stock prices look roughly bell-shaped in the middle but have far fatter tails: extreme days happen much more often than a normal curve allows. On 19 October 1987 the Dow Jones Industrial Average fell 22.6% in one day (Wikipedia). If daily moves were normal with a standard deviation of around 1%, a fall like that would be a 20-standard-deviation event, with a chance so small it would not be expected once in the age of the universe. The mathematician Benoit Mandelbrot argued from the 1960s that heavy-tailed models fit markets better.

The practical lesson: the 68-95-99.7 rule is a statement about the normal curve, not about data in general. Treating fat-tailed data as normal makes rare disasters look impossible.

What does the normal distribution lead to?

Much of standard statistics is built on it. Confidence intervals and many significance tests, including the ones behind most p-values, assume that averages of samples are approximately normal, an assumption the central limit theorem often justifies even when the raw data aren't. It also explains a pattern people often misread: when a measurement mixes skill with luck, an extreme result is usually followed by one closer to the mean, the pattern called regression to the mean.

Check yourself

• Without looking back, explain why a score of 130 on an IQ test puts someone in roughly the top 2.5%.
• Why does the total of ten dice look like a bell curve when a single die doesn't?
• Name one kind of data that isn't normally distributed, and say which condition of the central limit theorem it breaks.

Sources

• Normal Distribution, in the e-Handbook of Statistical Methods — NIST/SEMATECH
• The Standard Normal Distribution, in Introductory Statistics 2e — OpenStax, Rice University
• Central limit theorem — Wikipedia
• Normal distribution — Wikipedia
• Black Monday (1987) — Wikipedia