Empirical Rule Calculator (68–95–99.7) and Chebyshev's Inequality
The range within k standard deviations of the mean, the share of data inside it if the data are normal, and the weaker share guaranteed for any shape of data.
Results
The dashed line marks the value you entered.
What this tool does
For normally distributed data, about 68.3 % lie within one standard deviation of the mean, 95.4 % within two and 99.7 % within three — the 68–95–99.7 rule. Chebyshev's inequality needs no assumption about the shape: at least 1 − 1/k² of any data with a finite variance lie within k standard deviations, so at least 75 % within two and 89 % within three. It is much looser, and the chart shows the gap; to guarantee 95 % without assuming normality you need k = 4.47 standard deviations instead of 1.96. The normal figure uses an approximation accurate to about 0.00001 of a percentage point.
Formula
normal data: within k s of the mean = 1 − 2(1 − Φ(k)) (68.27 %, 95.45 %, 99.73 % for k = 1, 2, 3) · any data: at least 1 − 1 ÷ k² (k > 1)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
m | Mean of the data | — |
s | Standard deviation of the data | — |
k | Number of standard deviations (k) | — |
cv | Coverage you want guaranteed | % |
NR | Share inside if the data are normal | % |
CH | Share inside guaranteed for any data (Chebyshev) | % |
LO | Lower limit (mean − k s) | — |
HI | Upper limit (mean + k s) | — |
KC | k that Chebyshev needs to guarantee that coverage | — |
Worked example
- Mean of the data100
- Standard deviation of the data15
- Number of standard deviations (k)2
- Coverage you want guaranteed95 %
- Share inside if the data are normal95.45 %
- Share inside guaranteed for any data (Chebyshev)75.00 %
- Lower limit (mean − k s)70.000
- Upper limit (mean + k s)130.000
- k that Chebyshev needs to guarantee that coverage4.47
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.