Empirical Rule Calculator - 68-95-99.7 Rule for Normal
Apply the empirical rule (68-95-99.7 rule) to any normal distribution: enter mean and standard deviation to get the exact ranges for 68%, 95%, and 99.7% of the data.
Enter the mean (μ) and standard deviation (σ) of a normal distribution to calculate the three empirical rule intervals.
Empirical Rule Calculator - 68-95-99.7 Rule for Normal
Apply the empirical rule (68-95-99.7 rule) to any normal distribution: enter mean and standard deviation to get the exact ranges for 68%, 95%, and 99.7% of the data.
About the Empirical Rule Calculator
The empirical rule, also known as the three-sigma rule or the 68-95-99.7 rule, is a statistical shorthand that describes how data is distributed in a normal (bell-shaped) distribution. It states that approximately 68% of observations fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations. These are some of the most important numbers in applied statistics.
More precisely, the percentages are 68.27%, 95.45%, and 99.73%, derived from the cumulative distribution function of the standard normal distribution. The complement probabilities are also important: roughly 32% of data falls outside the one-sigma interval, about 5% outside the two-sigma interval, and only 0.27% (about 1 in 370) outside the three-sigma interval. This last figure is the basis of the "three-sigma limit" widely used in quality control and Six Sigma methodology.
The empirical rule only applies when the data follows — or closely approximates — a normal distribution. Many natural phenomena are approximately normal: adult heights, IQ scores, measurement errors, blood pressure readings, and many economic and financial metrics approximate normal distributions. In those cases, the empirical rule gives rapid, practical answers without any calculation beyond basic arithmetic.
To apply the rule, you need just two parameters: the mean (μ), which locates the centre of the distribution, and the standard deviation (σ), which measures the spread. The one-sigma interval is (μ − σ, μ + σ), the two-sigma interval is (μ − 2σ, μ + 2σ), and the three-sigma interval is (μ − 3σ, μ + 3σ). This calculator computes all three intervals instantly.
Practical applications are numerous. In manufacturing and quality control, a process is considered in control if the output falls within the three-sigma limits (99.73% of the time). In IQ testing with μ = 100 and σ = 15, about 68% of people score between 85 and 115, about 95% score between 70 and 130, and about 99.7% score between 55 and 145. In finance, the empirical rule is used to estimate the likelihood of extreme returns under the normality assumption, forming the basis of Value at Risk calculations. In biology and medicine, it helps identify unusual measurements: a blood pressure reading more than two standard deviations from the mean is outside the 95% interval and worth investigating.
Empirical Rule Examples
Three real-world distributions showing how the 68-95-99.7 rule gives instant insight.
| Distribution | 1σ Range (68%) | Application |
|---|---|---|
| IQ scores: μ = 100, σ = 15 | 85 to 115 | About 68% of people score 85–115, about 95% score 70–130, and about 99.7% score 55–145. A score above 130 (2σ above mean) is in the top 2.5%. |
| Adult male height: μ = 175 cm, σ = 7 cm | 168 to 182 cm | About 68% of adult males are 168–182 cm tall. About 95% fall in 161–189 cm. Heights below 154 cm or above 196 cm are outside the 3σ range (<0.3%). |
| University exam scores: μ = 78, σ = 6 | 72 to 84 | About 68% of students score 72–84. The top 2.5% (above 2σ = 90) qualify for distinction. About 99.7% score between 60 and 96. |
| Manufacturing bolt length: μ = 50 mm, σ = 0.5 mm | 49.5 to 50.5 mm | About 99.73% of bolts are within 3σ = 48.5–51.5 mm. Any bolt outside this range is flagged as defective under Six Sigma quality standards. |
How to Use the Empirical Rule Calculator
- Enter the mean (μ) of your normally distributed data in the first field. The mean can be any real number.
- Enter the standard deviation (σ) in the second field. The standard deviation must be a positive number greater than zero.
- Click Calculate. The calculator shows three coloured panels: the 68.27%, 95.45%, and 99.73% intervals.
- Each panel shows the range (lower bound to upper bound) and the percentage of data expected to fall within it.
- Use the example buttons to load well-known distributions (IQ scores, adult height, exam scores) and see the empirical rule in action.
Empirical Rule FAQ
What is the empirical rule in statistics?
The empirical rule (also called the 68-95-99.7 rule or three-sigma rule) states that for a normal distribution, about 68% of data falls within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three. It is a quick way to describe how spread out a normal distribution is and to estimate the probability of observations falling in different ranges.
Does the empirical rule apply to all distributions?
No — the empirical rule only applies to normal (Gaussian) distributions. If your data is skewed, multimodal, or has heavy tails, the percentages will be different. For non-normal distributions, Chebyshev's inequality gives a weaker but universally valid result: at least 75% of data falls within 2σ of the mean (vs 95% for normal), and at least 88.9% within 3σ (vs 99.7% for normal).
How do I know if my data is normally distributed?
Common approaches include examining a histogram (bell-shaped and symmetric?), plotting a Q-Q (quantile-quantile) plot (points should fall near a straight line for normal data), or applying formal tests like the Shapiro-Wilk or Kolmogorov-Smirnov test. For large samples (n > 30), the Central Limit Theorem means the sampling distribution of the mean is approximately normal even if the underlying data is not.
What does it mean to be 'outside two standard deviations'?
For a normal distribution, about 95.45% of data falls within 2σ of the mean, which means about 4.55% falls outside — roughly 2.275% in each tail. Being more than 2σ above the mean is statistically unusual and falls in the top 2.27% of the distribution. This threshold (often loosely stated as 5% or 1-in-20) is the basis of the conventional p < 0.05 significance level in hypothesis testing.
How is the empirical rule used in quality control?
In manufacturing and process quality, control limits are typically set at three standard deviations from the mean (the 3σ limits). Under the normality assumption, 99.73% of a process's output falls within these limits when the process is in control. Points outside the 3σ limits are treated as signals of a special cause of variation requiring investigation. This forms the foundation of Statistical Process Control (SPC) and the Six Sigma quality management methodology.
Can I use this for one-sided probabilities?
The empirical rule gives two-sided intervals centred on the mean. For one-sided probabilities, divide the complement in half. For example, about 95.45% of data falls within 2σ on both sides, so 4.55% falls outside — 2.275% above μ+2σ and 2.275% below μ−2σ. This is why a 95% two-sided confidence interval uses z = 1.96 (approximately 2σ): 2.5% is excluded from each tail.