Empirical Rule and Chebyshev's Theorem

68–95–99.7 intervals for bell-shaped data, and Chebyshev's minimum percentages for any shape.

Inputs

About 81.5%

For bell-shaped data, about 81.5% of values are between 9.2 and 11.6.

Empirical Rule (bell-shaped data)
k	Interval mean ± k·SD	Percent of data
1	9.2 to 10.8	about 68%
2	8.4 to 11.6	about 95%
3	7.6 to 12.4	about 99.7%

Show steps

Rule. For bell-shaped data, about 68% / 95% / 99.7% lie within 1 / 2 / 3 standard deviations of the mean. Each half of the curve splits into 34%, 13.5%, 2.35%, and 0.15%.
Locate the endpoints. 9.2 is 1 SD below the mean; 11.6 is 2 SD above the mean.
Add the pieces of the curve between those points: about 81.5%.

Which rule should I use?

Bell-shaped data (or the problem says "approximately normal"): use the Empirical Rule. About 68%, 95%, and 99.7% of the data fall within 1, 2, and 3 standard deviations of the mean.
Skewed or unknown shape: use Chebyshev's Theorem. At least 1 − 1/k² of the data fall within k standard deviations of the mean. It works for any shape, but it only gives a minimum.

Why Chebyshev's numbers are smaller

Chebyshev has to work for every possible shape, including very lopsided data, so it can only promise "at least 75%" within 2 standard deviations. Knowing the data are bell-shaped lets us say more: about 95%.