How empirical rule calculator works
The empirical rule (also called the 68-95-99.7 rule or three-sigma rule) is a property of bell-shaped data: about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Give the tool a mean and a standard deviation and it draws all three bands as concrete ranges — for heights averaging 170 cm with SD 7 cm, that is 163–177 cm, 156–184 cm, and 149–191 cm.
The word 'about' hides real precision, so each band also shows the exact normal-distribution coverage: 68.27%, 95.45%, and 99.73%, computed from the error function rather than the rounded mnemonic. The complements are shown too — 0.27% of data lies outside three standard deviations, which is why 3σ control limits dominate manufacturing: a process drifting beyond them is usually broken, not unlucky.
A custom multiple field lets you test any k: μ ± 1.96σ covers exactly 95.0004%, the source of the famous 1.96 in confidence intervals, and μ ± 2.5σ covers 98.76%. The rule belongs to roughly bell-shaped data — heights, measurement errors, exam averages of large classes — and misleads on skewed data like incomes, where the long tail pushes more than 0.3% beyond three standard deviations.