About the Empirical Rule Calculator
This empirical rule calculator applies the 68-95-99.7 rule (also called the three-sigma rule) to a normal distribution. Enter the mean and standard deviation and it gives the ranges that contain about 68% of values (within one standard deviation), 95% (within two) and 99.7% (within three), with a bell-curve chart marking each band.
Students use it for statistics homework, teachers to interpret test scores, and analysts for quick sanity checks — for example, to judge whether a measurement is unusual, or to set rough quality-control limits. Optionally enter a specific value to see its z-score and the exact percentage of a normal distribution that falls below it.
The rule is an approximation that holds for data that are roughly bell-shaped and symmetric. For skewed data, Chebyshev’s inequality gives a guaranteed (but looser) bound: at least 75% of any distribution lies within two standard deviations, and at least 88.9% within three.
How to use the empirical rule calculator
- 1Enter the mean of your data.
- 2Enter the standard deviation.
- 3Read the ranges holding 68%, 95% and 99.7% of values.
- 4Optionally enter a value to see its z-score and percentile.
- 5Use the chart to visualise where each band sits on the bell curve.
Formula and method
For a normal distribution with mean μ and standard deviation σ, the proportions inside each band depend only on how many standard deviations wide the band is. The exact figures are 68.27%, 95.45% and 99.73%, which the empirical rule rounds to 68, 95 and 99.7. Each band is found by simply adding and subtracting k × σ from the mean.
The z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean, and the percentage below it is the standard normal cumulative probability Φ(z). The rule applies to data that are approximately normal; heavily skewed data or data with outliers can differ substantially.
- μ
- Mean of the distribution
- σ
- Standard deviation
- z
- z-score: (x − μ) ÷ σ
- Φ(z)
- Standard normal cumulative probability
Worked examples
IQ scores (mean 100, SD 15)
About 68% of people score between 85 and 115, 95% between 70 and 130 and 99.7% between 55 and 145. A score of 130 is 2 standard deviations above the mean, higher than about 97.7% of people.
Adult height (illustrative: mean 69 in, SD 3 in)
One SD either side gives 66–72 inches (68%), two SDs 63–75 inches (95%) and three SDs 60–78 inches (99.7%). A height of 74 inches is about 1.67 SDs above the mean, taller than roughly 95.2% of the group if heights are normally distributed.
Machine fill weights (mean 500 g, SD 4 g)
Nearly all fills (99.7%) should weigh between 488 g and 512 g. A 488 g pack is exactly 3 standard deviations low — only about 0.135% of packs should be that light or lighter, so it may signal a problem.
Frequently asked questions
What is the empirical rule?+
The empirical rule (68-95-99.7 rule) states that for a normal distribution about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. It is a quick way to judge how typical or unusual a value is.
What are the exact percentages?+
The exact normal-distribution values are 68.27% within 1σ, 95.45% within 2σ and 99.73% within 3σ. The empirical rule rounds these to 68%, 95% and 99.7% for easy mental math.
When can I use the empirical rule?+
Use it when your data are approximately normal — unimodal, symmetric and bell-shaped, like heights, test scores or measurement errors. For skewed data such as incomes, use Chebyshev’s theorem or the actual percentiles instead.
What percentage of data is above one standard deviation?+
About 16%. Since 68% lies within ±1σ, the remaining 32% is split equally between the two tails, so roughly 16% is above μ + σ and 16% is below μ − σ. Above μ + 2σ is about 2.5%.
How is the empirical rule different from Chebyshev’s theorem?+
The empirical rule applies only to normal (bell-shaped) data and gives approximate percentages. Chebyshev’s theorem applies to any distribution and gives minimums: at least 1 − 1/k² of values lie within k standard deviations — 75% within 2σ and 88.9% within 3σ.