Empirical Rule Calculator
Apply the 68-95-99.7 rule to any normal distribution — enter mean and standard deviation to get the three ±σ ranges with a shaded bell-curve diagram.
The empirical rule explained, step by step
The empirical rule — also called the 68-95-99.7 rule — is a quick approximation for how values spread in an approximately normal (bell-shaped, symmetric) distribution. Each card below covers one of the three σ-bands the calculator reports, plus the assumption that makes the rule work.
1. ~68% within ±1σ — the middle of the bell
About 68% of values in a roughly normal distribution sit within one standard deviation of the mean. Shift the mean and the band moves with it; change σ and the band widens or narrows.
2. ~95% within ±2σ — the working confidence band
Adding a second σ on each side captures about 95% of the distribution. This is why 2σ appears everywhere in applied statistics — from control-chart limits to the informal '95% confidence' shortcut.
3. ~99.7% within ±3σ — 'almost everything'
Three sigmas on each side cover roughly 99.7% of the distribution — only about 0.3% of values fall outside. This is where the 'three-sigma rule' in quality control comes from, and why 3σ is a common outlier flag for normal data.
4. When the rule breaks — skew, bimodality, heavy tails
The 68-95-99.7 percentages come from a normal curve. If the data is skewed, bimodal, heavy-tailed or discrete with few values, the σ-bands stop matching those percentages. When you can't assume normality, Chebyshev's inequality (at least 1 − 1/k² inside k σ) still holds for any distribution.
Features of this calculator
- Instant μ ± 1σ, μ ± 2σ and μ ± 3σ bounds with their approximate ~68% / ~95% / ~99.7% shares.
- Shaded bell-curve diagram with the three bands in progressively lighter shades, labeled with the actual numeric boundaries from your inputs.
- Optional X input: enter a specific value to see which band it falls into and its approximate percentile from the normal CDF.
- Show/hide step-by-step working laying out μ ± kσ arithmetically for k = 1, 2, 3.
Frequently asked questions
+Why the specific numbers 68, 95 and 99.7?
They're the actual areas under a standard normal curve between ±1, ±2 and ±3 standard deviations, rounded to convenient values. The exact percentages are 68.27%, 95.45% and 99.73%.
+How do I know if my data is 'normal enough' for the rule?
Plot a histogram and look for a roughly symmetric bell shape. For a more formal check, a Q-Q plot or a normality test (Shapiro-Wilk, Anderson-Darling) is standard. Mild deviations are usually fine for a quick empirical-rule estimate.
+What's outside ±3σ called?
Values beyond ±3σ are commonly flagged as outliers in normal populations because only about 0.3% of the distribution sits out there. In quality control this is the origin of the "three-sigma control limit".
+Can I use the rule on sample data?
Yes, using the sample mean x̄ and sample standard deviation s as estimates of μ and σ. Just remember the rule is an approximation and small samples produce noisy s values, so the bands may shift as you collect more data.
+What percentage is between 1σ and 2σ?
About (95% − 68%) / 2 ≈ 13.5% on each side of the mean. So roughly 13.5% of values lie between μ + 1σ and μ + 2σ, and another 13.5% between μ − 2σ and μ − 1σ.