A bell curve is not only a shape — it is a promise about where the values will be. If a set of measurements is roughly bell-shaped, then about 68% of them sit within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Two numbers, a mean and an SD, and you can map an entire dataset. Drag the boundary below and the area under the curve is integrated live; then draw a real sample and count how many values actually landed inside the shading.
The model, and the slice you have shaded
The data, against the shading
landed inside the shadinglanded outsidethe normal model
How the coverage grows with k
bell curve (exact)this sampleguaranteed for any shape
The empirical ruledrag the boundary outward
±1.00 SD
300
Send the boundary to
Shape the sample is drawn from
Boundary k1.00
Lower edge—
Upper edge—
Area shaded—
Within
Rule of thumb
Exact bell
This sample
1 SD
68%
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2 SD
95%
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3 SD
99.7%
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The small figure under each sample percentage is its gap from the exact bell value. A gap turns red only when it is too large to pass as ordinary sampling noise at this n — so shrink the sample far enough and even a badly non-bell shape stops looking guilty, which is itself worth noticing.
The rule is worth learning in both directions. Forwards, it turns two numbers into a map: give me a mean and a standard deviation for bell-shaped data and I can tell you roughly where any fraction of the values lie, without ever seeing them. Backwards — and this is the direction that gets used far more in practice — it turns a single measurement into a judgement about how ordinary it is. A value one SD from the mean is thoroughly unremarkable; roughly a third of all values are further out than that. A value two SDs out is uncommon, about one in twenty-two. A value three SDs out is genuinely striking, about one in 370, which is precisely why manufacturing lines and laboratory quality controls draw their alarm limits at three. Notice what makes this possible: the standard deviation has become a ruler. Once you count distance in SDs rather than centimetres or grams, height and birth weight and bottle volume all collapse onto the same curve, and one table of percentages serves every one of them.
Five things that are easy to get wrong. First, this is a rule about bell-shaped data, not about all data. Switch the shape below to Right-skewed or Two clumps and watch the sample column of the table walk away from 68–95–99.7 while the mean and the SD stay perfectly valid. What holds for any shape whatsoever is much weaker — at least 75% within 2 SDs and at least 88.9% within 3, and nothing at all guaranteed within 1. Second, 68–95–99.7 are rounded. The exact areas are 68.27%, 95.45% and 99.73%, computed in the table beside you, and it is 1.96 SDs rather than 2 that cuts off exactly 95%. Third, the percentages are two-tailed. “95% within 2 SD” leaves 5% outside, but that 5% is split between the two ends: only about 2.3% lie above 2 SDs, not 5%. Fourth, unusual is not the same as wrong. A value beyond 3 SDs is rare under the model, not impossible and not evidence of a mistake — in a bell-shaped sample of 1,000 you should expect about three of them, and finding none would itself be a little odd. Fifth, the rule describes a distribution, not a person. That a measurement sits inside the middle 95% says it is common; it does not say it is healthy, correct, or acceptable, and those are separate questions the arithmetic cannot answer.