The Empirical Rule

A factory fills bottles that are supposed to hold 500\,\text{ml}, but no machine is perfect — some come out at 498, some at 503. If you knew only two numbers — the average fill and how much it typically wobbles — could you say what fraction of bottles will be badly under-filled, without measuring every single one? For anything with a bell-shaped spread, the answer is yes, and it takes just three percentages you can carry in your head.

For any (approximately) normal distribution, the data clusters around the mean in a strikingly regular way. The empirical rule — also called the 68–95–99.7 rule — says how much of the data falls within one, two, and three standard deviations of the mean:

Almost everything — 99.7\% — sits within three standard deviations of the centre. Values further out than that are genuinely rare. That is the whole rule: three numbers, 68, 95, 99.7, and a picture that makes them obvious.

Three nested bands

The picture below is a standard normal curve with three shaded bands. The innermost band spans \mu \pm 1\sigma and holds 68\% of the area; widening to \mu \pm 2\sigma captures 95\%; and \mu \pm 3\sigma captures 99.7\%.

Because the curve is symmetric, each percentage splits evenly across the two tails. For example, 95\% inside 2\sigma leaves about 5\% outside — roughly 2.5\% in each tail. It is worth memorising the slices between the bands, because they let you answer far more than the three headline questions:

Worked example 1 — read a range straight off

Suppose a class takes a test whose scores are normal with mean \mu = 70 and standard deviation \sigma = 10. About what fraction of students score between 50 and 90?

First, measure the range in standard deviations. The mean is 70; 50 is 20 below it and 90 is 20 above — that is exactly 2\sigma on each side. So we want the fraction within 2\sigma of the mean, which the rule gives immediately as about 95\%. No integrals, no tables — just the distance in \sigma and one memorised number.

Worked example 2 — work backward to a range

Now go the other way. For the same test (\mu = 70, \sigma = 10), what range of scores captures the middle 68% of the class?

The rule says 68\% sits within 1\sigma of the mean. One standard deviation is 10 points, so the band runs from 70 - 10 = 60 up to 70 + 10 = 80. The middle 68\% of students score between 60 and 80. Widening to the middle 95\% would stretch the band to 50–90 instead.

Worked example 3 — use symmetry for a one-sided tail

Same test again. What fraction of students score above 80 — that is, above \mu + 1\sigma?

The rule gives the two-sided fact: 68\% lies within 1\sigma, so 100\% - 68\% = 32\% lies outside — split evenly into the two tails. By symmetry, half of that 32\% is in the upper tail:

\frac{100\% - 68\%}{2} = \frac{32\%}{2} = 16\%.

So about 16\% of the class scores above 80. The same trick handles "below 50" (below \mu - 2\sigma): (100\% - 95\%)/2 = 2.5\%. Every one-sided question is just a two-sided rule cut in half.

The empirical rule is not a law of all data — it is a fact about the normal (bell-shaped) distribution specifically. Reach for "95% within 2\sigma" on a skewed or heavy-tailed dataset and you will get the wrong answer. Household incomes, for instance, have a long right tail: the mean is dragged up by a few very high earners, and far fewer than 95\% of households actually fall within 2\sigma of that inflated mean.

There is a rule that holds for any distribution — Chebyshev's inequality — which guarantees that at least 1 - 1/k^2 of the data lies within k standard deviations. For k = 2 that is only 1 - 1/4 = 75\%, and for k = 3 just 1 - 1/9 \approx 88.9\%. Those bounds are true no matter the shape, but they are much looser than 95\% and 99.7\%. So: bell shape → use the tight empirical numbers; unknown shape → fall back to Chebyshev's weaker guarantee.

Those tail percentages turned into industrial jargon. In manufacturing, a "3-sigma" process keeps its output within 3\sigma of the target — sounds great, until you realise 0.3\% of parts still fail. On a line making a million parts, that is 3{,}000 defects. So companies chase "Six Sigma": squeezing the process until the specification limits sit a full 6\sigma from the mean, where defects become astronomically rare (allowing for real-world drift, the famous target works out to about 3.4 defects per million opportunities).

Particle physicists take it even further in the other direction. Before announcing a discovery, they demand "5-sigma" evidence — a result so extreme that, if it were just a random fluke, the odds of seeing it are about 1 in 3.5 million. That is the bar the Higgs boson had to clear in 2012 before anyone was allowed to say the word "discovery". Same bell curve as your test scores, just read out at its far, far tail.

See it explained