The average / centre of your distribution
Enter a valid mean value
Must be greater than zero
Enter a positive standard deviation
Appended to results for clarity
68โ95โ99.7 Rule Results
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What Is the Empirical Rule?
The empirical rule (also called the 68โ95โ99.7 rule) describes how data is distributed in a normal (bell-shaped) distribution. It states that:
- 68% of data falls within 1 standard deviation of the mean (ฮผ ยฑ 1ฯ)
- 95% of data falls within 2 standard deviations (ฮผ ยฑ 2ฯ)
- 99.7% of data falls within 3 standard deviations (ฮผ ยฑ 3ฯ)
68% range: ฮผ โ ฯ to ฮผ + ฯ
95% range: ฮผ โ 2ฯ to ฮผ + 2ฯ | 99.7% range: ฮผ โ 3ฯ to ฮผ + 3ฯ
💡 When to use it: The empirical rule only applies to normal distributions. Before applying it, verify your data is approximately bell-shaped using a histogram or normality test.
Real-World Applications
| Field | Example Use |
|---|---|
| Education | IQ scores (ฮผ=100, ฯ=15) โ 68% score between 85โ115 |
| Manufacturing | Quality control โ products outside 3ฯ are defects |
| Finance | Stock returns โ 95% of annual returns fall within 2ฯ of mean |
| Healthcare | Blood pressure ranges, birth weights, height distributions |
| Six Sigma | Process quality โ 6ฯ means only 3.4 defects per million |
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Frequently Asked Questions
The empirical rule states that for a normal distribution, 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
No โ it only applies to data that follows a normal (bell-curve) distribution. Skewed distributions, bimodal data, or data with heavy tails require different statistical methods.
A value more than 3 standard deviations from the mean is an outlier โ it occurs less than 0.3% of the time in a normal distribution. In manufacturing, these are defects. In finance, these are extreme market events.
Standard deviation (ฯ) is the square root of variance (ฯยฒ). Standard deviation is expressed in the same units as the data, making it more interpretable. The empirical rule uses standard deviation.
Six Sigma quality means a process produces fewer than 3.4 defects per million opportunities โ equivalent to 6 standard deviations from the mean. The empirical rule's 3ฯ (99.7%) becomes the baseline; Six Sigma extends this to 6ฯ (99.99966%).
The empirical rule only applies to normally distributed data. It does not apply to: skewed distributions (income, home prices); bimodal distributions; uniform distributions; or any dataset with extreme outliers distorting the mean. Before applying, verify normality with a histogram, Q-Q plot, or normality test (Shapiro-Wilk). For non-normal data, use Chebyshev's theorem instead.
Chebyshev's theorem works for any distribution: at least 1โ1/kยฒ of values fall within k standard deviations of the mean. At k=2: at least 75% (vs 95.4% for normal). At k=3: at least 88.9% (vs 99.7%). Chebyshev gives weaker but universal bounds; the empirical rule gives tighter bounds for confirmed normal distributions. Use Chebyshev when you can't verify normality.