Empirical Rule Calculator

68 – 95 – 99.7 Rule

Enter a mean (μ) and standard deviation (σ) to compute the exact interval boundaries for the 68-95-99.7 empirical rule and visualise the three shaded sigma regions on a bell curve.

μ ± 1σ → 68.27% μ ± 2σ → 95.45% μ ± 3σ → 99.73% Bell Curve Diagram Normal Distribution

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Enter a mean and standard deviation above to generate the bell curve and 68-95-99.7 intervals.

What Is the Empirical Rule?

The empirical rule — also called the 68-95-99.7 rule or the three-sigma rule — is a statistical principle that describes how data is distributed around the mean in a normal distribution (bell curve).

The rule states that for a normally distributed dataset with mean μ and standard deviation σ:

1 Approximately 68.27% of data falls within 1 standard deviation of the mean (between μ−σ and μ+σ)
2 Approximately 95.45% of data falls within 2 standard deviations (between μ−2σ and μ+2σ)
3 Approximately 99.73% of data falls within 3 standard deviations (between μ−3σ and μ+3σ)

The empirical rule only applies to normally distributed data. A normal distribution is symmetric and bell-shaped — data clusters near the mean and becomes increasingly rare further away. Many natural phenomena follow this pattern: human heights, IQ scores, measurement errors, and process outputs in manufacturing.

Empirical Rule Formula — Step by Step

Applying the empirical rule requires only addition and subtraction. Given a dataset with mean μ and standard deviation σ:

1σ Rule (68%)

μ − σ to μ + σ

68.27% of data lives here

2σ Rule (95%)

μ − 2σ to μ + 2σ

95.45% of data lives here

3σ Rule (99.7%)

μ − 3σ to μ + 3σ

99.73% of data lives here

Worked example — IQ Scores (μ = 100, σ = 15)

Band	Calculation	Range	% Within
±1σ	100±15	85 – 115	68.27%
±2σ	100±30	70 – 130	95.45%
±3σ	100±45	55 – 145	99.73%

Interpretation: About 68% of people have an IQ between 85 and 115. Only 0.27% of the population falls outside the 55–145 range (roughly 1 in 370 people).

Visual Diagram — 68-95-99.7 Bell Curve

The three coloured regions show what proportion of normally distributed data falls within each sigma band. The innermost region (dark teal) is 1σ; amber wings are the 3σ outer region.

±1σ — 68.27%

±2σ — 95.45%

±3σ — 99.73%

What about data beyond 3σ?

Only 0.27% of normally distributed data falls beyond ±3σ — roughly 1 in 370 observations. This is why "3-sigma events" are considered extremely rare in quality control. Each outer tail (beyond +3σ or below −3σ) contains just 0.135% of data.

Empirical Rule in Real Life

The empirical rule applies to any dataset that is approximately normally distributed. Here are common real-world examples where the 68-95-99.7 rule is routinely used:

Dataset	μ	σ	68% range (±1σ)	99.7% range (±3σ)
IQ Scores	100	15	85 – 115	55 – 145
Adult male height (cm)	178	7	171 – 185	157 – 199
Exam scores	72	8	64 – 80	48 – 96
Battery life (hrs)	12	1.5	10.5 – 13.5	7.5 – 16.5
Manufacturing part (mm)	50	0.2	49.8 – 50.2	49.4 – 50.6

Quality Control (Six Sigma)

Manufacturing uses the empirical rule to set tolerances. A "Six Sigma" process aims for defect rates below 3.4 per million by pushing the acceptable range well beyond ±3σ. Control charts use σ boundaries to flag unusual process variation.

Outlier Detection

Data points beyond ±2σ occur only ~4.55% of the time in a normal distribution. Points beyond ±3σ are extremely rare (0.27%). These thresholds are used in anomaly detection to flag potential errors, fraud, or genuinely exceptional values.

Empirical Rule vs. Chebyshev's Theorem

The empirical rule only works for normally distributed data. Chebyshev's theorem is the more general alternative — it applies to any dataset regardless of shape, but gives weaker (less precise) bounds.

Band	Empirical Rule (normal only)	Chebyshev's Theorem (any distribution)
±1σ	68.27%	No guarantee (formula gives 0%)
±2σ	95.45%	At least 75%
±3σ	99.73%	At least 88.9%

Chebyshev's formula: at least (1 − 1/k²) × 100% of data falls within k standard deviations of the mean, for any k > 1. Because it makes no assumption about distribution shape, its bounds are conservative. The empirical rule gives much tighter bounds, but requires a normally distributed dataset for those bounds to be accurate.

When NOT to use the empirical rule

The empirical rule should not be applied to skewed distributions (income data, waiting times, house prices), bimodal distributions (test scores with two distinct groups), or datasets with heavy tails (financial returns, earthquake magnitudes). Always verify approximate normality using a histogram or Q-Q plot before applying the rule.

How to Apply the Empirical Rule: 4 Steps

1
Verify the data is approximately normal
Plot a histogram or check the skewness statistic. A symmetric, bell-shaped distribution is required. Many natural measurements qualify; count data, financial returns, and heavily skewed data usually do not.
2
Find the mean (μ) and standard deviation (σ)
Calculate or obtain these summary statistics. Use sample standard deviation (Bessel's correction, n−1) if working with a sample rather than the full population.
3
Compute the interval boundaries
Apply the three formulas:
• 1σ interval: [μ − σ, μ + σ]
• 2σ interval: [μ − 2σ, μ + 2σ]
• 3σ interval: [μ − 3σ, μ + 3σ]
4
Interpret using the 68-95-99.7 percentages
State what percentage of your population falls within each range, and what percentage falls outside. Use this for quality control limits, probability estimates, and identifying outliers beyond ±2σ or ±3σ.

Empirical Rule FAQ

What is the empirical rule in statistics?

The empirical rule (68-95-99.7 rule) states that for a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. It is derived from the properties of the standard normal distribution and provides quick probability estimates without requiring calculus or normal distribution tables.

What does 68-95-99.7 mean?

The three numbers refer to the percentage of data contained within each sigma band: 68% within 1σ, 95% within 2σ, and 99.7% within 3σ. The exact values are 68.2689%, 95.4500%, and 99.7300%. The rule is called "68-95-99.7" as a memorable shorthand — "68 percent, 95 percent, 99.7 percent."

What percentage of data falls within 2 standard deviations?

In a normal distribution, 95.45% of data falls within 2 standard deviations of the mean (between μ−2σ and μ+2σ). This means 4.55% falls outside this range — 2.275% in each tail. In practice, "within 2 standard deviations" is commonly rounded to "95%" and used as a threshold for identifying outliers: any data point beyond ±2σ is considered unusual (occurring only about 1 in 22 observations).

Does the empirical rule work for skewed data?

No. The empirical rule only applies to normally distributed data. For skewed distributions (such as income, house prices, or waiting times), the actual percentages within 1σ, 2σ, and 3σ will differ — sometimes dramatically — from 68%, 95%, and 99.7%. For any distribution (normal or not), use Chebyshev's theorem instead, which guarantees at least 75% within ±2σ and at least 88.9% within ±3σ regardless of shape.

What falls outside 3 standard deviations?

Only 0.27% of normally distributed data falls beyond ±3σ — about 1 in 370 data points. Each individual tail (above +3σ or below −3σ) contains just 0.135%. Values beyond 3σ are called outliers or extreme values. In manufacturing and finance, a "three-sigma event" (a 3σ deviation) is considered highly unusual. However, in heavy-tailed distributions (e.g., financial markets), extreme events occur far more frequently than 0.27%.

What is the difference between the empirical rule and Chebyshev's theorem?

The empirical rule gives exact percentages for normal distributions (68%, 95%, 99.7%). Chebyshev's theorem gives minimum guaranteed percentages for any distribution, using the formula: at least (1 − 1/k²) × 100%. At k=2, Chebyshev guarantees at least 75% (vs the empirical rule's 95.45%). At k=3, Chebyshev guarantees at least 88.9% (vs 99.73%). Empirical rule is more informative when normality holds; Chebyshev is safer when the distribution is unknown.

How is the empirical rule used in quality control?

Statistical process control (SPC) uses control charts with ±3σ control limits. Any measurement falling outside these limits triggers an investigation since only 0.27% of normal process variation naturally falls there. The "Six Sigma" quality methodology aims for process variation so small that specification limits are at ±6σ from the mean, allowing for only 3.4 defects per million opportunities even accounting for process drift.

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External Resources

Wikipedia — 68–95–99.7 rule: mathematical derivation and applications NIST/SEMATECH e-Handbook — Normal Distribution and the empirical rule in data analysis