What is the 68-95-99.7 rule?
The rule turns two numbers, a mean and a standard deviation, into a quick map of where data sits. If you know the average and the spread of a bell-shaped dataset, you can predict the percentage of values inside any one, two, or three standard deviation band without doing integration or reading a table.
The word "approximately" matters. The figures 68, 95, and 99.7 are rounded. The exact values are 68.27%, 95.45%, and 99.73%, which come from the area under the normal distribution curve. For homework, exams, and quick estimates the rounded numbers are standard.
- 1 standard deviation: μ ± 1σ holds about 68% of the data
- 2 standard deviations: μ ± 2σ holds about 95% of the data
- 3 standard deviations: μ ± 3σ holds about 99.7% of the data
- Exact values: 68.27%, 95.45%, 99.73%
- Beyond 3σ: only about 0.3% of values, often treated as outliers
- Applies to: data that is approximately normal (symmetric and bell-shaped)
The Empirical Rule formula
The rule is built from the mean and the standard deviation. The mean sets the centre of the bell curve, and the standard deviation sets the step size you move out in each direction.
μ = mean (centre of the curve)
σ = standard deviation (spread)
μ±kσ = interval k steps from the mean
Each band is two-sided. The 95% interval runs from two standard deviations below the mean to two standard deviations above it, not just on one side. This is a common point of confusion, so it is worth saying plainly: μ ± 2σ means the stretch from μ − 2σ up to μ + 2σ.
The 34, 13.5, 2.35 slice breakdown
Because the curve is symmetric, you can split each percentage in half and look at one side. Reading outward from the centre on either side, the slices are about 34.1%, then 13.6%, then 2.1%, then 0.15% in the far tail. These smaller pieces let you answer questions the headline numbers cannot, such as the percentage between one and two standard deviations.
The bell curve, shaded by region
The diagram below shows a normal curve with the mean at the centre and the standard deviation marks spaced evenly on each side. The colour of each band matches the percentage of data it contains.
68-95-99.7 on the normal curve
Adding each pair of outer bands raises the coverage from 68% to 95% to 99.7%. You can build the same diagram for your own numbers with the bell curve generator.
Why the percentages stay the same
The three numbers do not change from one dataset to another. A normal curve of heights, test scores, or measurement errors gives the same 68-95-99.7 split. The reason is that every normal distribution has the same shape once you measure distance in standard deviations rather than raw units.
When you rescale any normal variable into standard deviation units, you get the standard normal curve, the one with mean 0 and standard deviation 1. The percentage of area between −1 and +1 on that curve is 68.27%, between −2 and +2 is 95.45%, and between −3 and +3 is 99.73%. These areas come from the normal probability density function, and they are fixed. That is why the rule transfers across topics: a value one standard deviation above the mean sits at the same point on the curve whether the data is SAT scores or rainfall.
A normal curve is a probability distribution, so the total area beneath it equals 1 (or 100%). The Empirical Rule simply names the areas at the three round-number cut points. The normal distribution calculator returns the area for any cut point, not just 1, 2, and 3.
How to use the Empirical Rule, step by step
Most Empirical Rule questions follow the same four steps. The worked example below applies them to a set of exam scores.
A class sits an exam with a mean of 78 and a standard deviation of 6. The scores are roughly normal. Between which scores do about 95% of students fall?
Confirm the data is normal. The problem states the scores are roughly normal and bell-shaped, so the rule applies.
Identify the mean and standard deviation. Here μ = 78 and σ = 6.
Build the interval. The 95% band is μ ± 2σ = 78 ± (2 × 6) = 78 ± 12.
Read the range. 78 − 12 = 66 and 78 + 12 = 90.
About 95% of students score between 66 and 90. A score above 90 is in the top 2.5%, since 5% sits outside the band and the curve is symmetric.
The 3-sigma ladder: a way to remember it
If the numbers slip away from you, picture three rungs of a ladder climbing out from the centre of the curve. Each rung you climb widens the interval by one standard deviation and raises the coverage.
One standard deviation
The middle band. Most of the data sits here. Think of it as "the typical range."
Two standard deviations
Almost all of the data. This is the band most researchers report, and it lines up with a 95% interval.
Three standard deviations
Nearly everything. Only about 3 values in 1,000 fall outside this band, so points beyond it are flagged as outliers.
Climb 1, 2, 3 rungs and read off 68, 95, 99.7. For the half-side slices, remember "34, 13.5, 2.35" from the centre outward.
Empirical Rule calculator
Enter a mean and a standard deviation to get the three ranges. The other tabs convert a single value to a z-score and estimate the probability between two values using the normal distribution.
Empirical Rule Calculator
Enter the mean and standard deviation of a normal distribution to get the 68%, 95%, and 99.7% ranges.
Enter a value, the mean, and the standard deviation to find how many standard deviations the value sits from the mean.
Enter a mean, a standard deviation, and two bounds to estimate the probability that a value falls between them.
For more detailed work, the site also has a dedicated z-score calculator, a standard deviation calculator, and a full normal distribution calculator.
Worked datasets: SAT, IQ, and height
The table below applies the rule to three datasets that are close to normal. Each row gives the mean, the standard deviation, and the three ranges, so you can see how the same arithmetic plays out on different scales.
| Dataset | Mean (μ) | SD (σ) | 68% range | 95% range | 99.7% range |
|---|---|---|---|---|---|
| SAT total score | 1050 | 200 | 850 – 1250 | 650 – 1450 | 450 – 1650 |
| IQ score | 100 | 15 | 85 – 115 | 70 – 130 | 55 – 145 |
| Adult male height (in) | 70 | 3 | 67 – 73 | 64 – 76 | 61 – 79 |
| Resting heart rate (bpm) | 72 | 8 | 64 – 80 | 56 – 88 | 48 – 96 |
When real data deviates from the rule
The rule holds when the data is close to normal. Plenty of real data is not, and then the 68-95-99.7 split breaks down in predictable ways:
- Skew. Income, house prices, and reaction times have a long right tail. The mean sits to the right of the peak, so the symmetric μ ± kσ intervals no longer hold the stated percentages.
- Heavy tails. Daily stock returns put more than 0.3% of values beyond three standard deviations. Using the rule here understates how often extreme moves happen.
- Small samples. A sample of 15 values can look lumpy or skewed by chance even when the underlying population is normal. Check a histogram before trusting the rule.
Before you apply the rule, look at the distribution. A quick histogram or a check of skew tells you whether the data is close enough to normal. If it is clearly skewed, switch to Chebyshev's theorem, covered below.
Empirical Rule vs Chebyshev's theorem
Both rules tell you how much data sits near the mean, but they apply in different situations. The Empirical Rule needs a normal distribution and gives specific percentages. Chebyshev's theorem works for any distribution but gives only a minimum guarantee.
| Feature | Empirical Rule | Chebyshev's theorem |
|---|---|---|
| Applies to | Normal (bell-shaped) data only | Any distribution, any shape |
| Type of result | Specific percentages | Minimum bound (at least) |
| Within 2σ | About 95% | At least 75% |
| Within 3σ | About 99.7% | At least 88.9% |
| Within 1σ | About 68% | No useful bound |
| Best for | Test scores, heights, measurement error | Skewed or unknown distributions |
Chebyshev's theorem says that for any k greater than 1, at least 1 − 1/k² of the data falls within k standard deviations of the mean. At k = 2 that gives 1 − 1/4 = 75%. The Empirical Rule's 95% is higher because it uses the extra information that the data is normal. When you do not know the shape, the weaker Chebyshev bound is the safe choice.
How the rule connects to z-scores
A z-score measures how many standard deviations a value sits from the mean: z = (x − μ) / σ. The Empirical Rule is just the z-score scale read at its round-number marks.
A value with z = 1 sits one standard deviation above the mean, at the right edge of the 68% band. A value with z = 2 sits at the right edge of the 95% band, which puts it in the top 2.5%. A value with z = 3 is at the edge of the 99.7% band. So if you can compute a z-score, you can place any value on the bell curve and read its percentile against the z-table. The rule is the quick version; the z-table is the full version for cut points that are not whole numbers.
Real-world case studies
Case Study 1 — Education
Standardised test scores
An exam has μ = 500 and σ = 100. A student scores 700. That is z = (700 − 500) / 100 = 2, the edge of the 95% band, so the student is in roughly the top 2.5%. Admissions offices use this kind of placement to compare scores across different tests.
Case Study 2 — Human biology
Adult height
Adult male height has μ = 70 inches and σ = 3 inches. The rule says 95% of men fall between 64 and 76 inches. A height of 79 inches (6 feet 7 inches) sits at three standard deviations, in the far 0.15% tail, which matches how rarely you see it.
Case Study 3 — Manufacturing
Quality control and Six Sigma
Factories treat parts more than three standard deviations from the target size as defects, because only about 0.3% of a normal process should land there. The Six Sigma method pushes the tolerance band out to six standard deviations, which is why it targets a defect rate of about 3.4 parts per million.
Case Study 4 — Finance
Investment returns
Suppose a fund has a mean annual return of 8% and a standard deviation of 15%. The rule suggests 68% of years land between −7% and +23%. The caution: market returns have heavier tails than a normal curve, so large losses happen more often than the rule predicts. This is a case where the model and the data part ways.
Case Study 5 — Medicine
Reference ranges
Labs often define a "normal" range for a blood measurement as the central 95% of healthy people, which is μ ± 2σ. A result outside that band is flagged for review, not because it is necessarily a problem, but because it falls in the 5% that warrants a second look.
Practice problems with answers
Work through these before checking the answers. They move from direct application to reverse problems.
Problem 1
A dataset is normal with μ = 50 and σ = 5. What range holds about 68% of the values?
Answer: μ ± 1σ = 50 ± 5, so 45 to 55.
Problem 2
Scores are normal with μ = 500 and σ = 100. What percentage of scores are above 700?
Answer: 700 is z = 2. The 95% band leaves 5% outside, split into two tails, so about 2.5% score above 700.
Problem 3
You are told only that, for a normal dataset, 95% of values lie between 60 and 90. Find the mean and the standard deviation.
Answer: the mean is the midpoint, (60 + 90) / 2 = 75. The 95% band spans 4 standard deviations (2 on each side), so 90 − 60 = 30 = 4σ, giving σ = 7.5.
Free downloads
These reference sheets cover the normal distribution and the values you need alongside the Empirical Rule.
The bell curve, the rule, and key formulas on one page.
Cumulative areas for any z-score, printable for exams.
A longer reference covering the curve, z-scores, and the rule.
Formula and entity glossary
| Term | Formula / Notation | Definition |
|---|---|---|
| Empirical Rule | μ±1σ→68% | The percentage of normal data within 1, 2, and 3 standard deviations of the mean |
| Normal Distribution | N(μ, σ²) | A symmetric, bell-shaped distribution defined by its mean and standard deviation |
| Mean | μ = Σx / N | The arithmetic average; the centre of the curve |
| Standard Deviation | σ = √(Σ(x−μ)²/N) | The average distance of values from the mean; the spread |
| Variance | σ² | The square of the standard deviation |
| Z-Score | z = (x−μ)/σ | How many standard deviations a value sits from the mean |
| Probability Distribution | ∫ f(x) dx = 1 | A function whose total area equals 1, giving probabilities over ranges |
| Chebyshev's Theorem | ≥ 1 − 1/k² | The minimum fraction of any dataset within k standard deviations of the mean |
| Central Limit Theorem | x̄ ~ N(μ, σ²/n) | Sample means approach a normal distribution as the sample size grows |
Common mistakes and how to avoid them
| Mistake | ❌ Incorrect | ✓ Correct |
|---|---|---|
| Using the rule on skewed data | Applying 68-95-99.7 to income data | Income is right-skewed; use Chebyshev's theorem instead |
| Treating the band as one-sided | 95% lies below μ + 2σ | 95% lies between μ − 2σ and μ + 2σ |
| Reading 99.7% as "all" | Every value is within 3 standard deviations | About 0.3% of values fall beyond 3σ |
| Forgetting to halve the tail | 5% of scores are above μ + 2σ | 2.5% are above and 2.5% below; the 5% is split |
| Confusing the exact and rounded values | Exactly 95% within 2σ | 95.45% exact, rounded to 95% for everyday use |
Where the rule leads next
The Empirical Rule is one piece of the normal distribution toolkit. Here is how it connects to the next topics on Statistics Fundamentals:
Z-Scores
The z-score generalises the rule to any cut point, not just 1, 2, and 3 standard deviations.
Standard Deviation
The rule is built on the standard deviation. This page shows how to compute it from data.
Confidence Intervals
The 95% band is the idea behind a 95% confidence interval for a mean.
Sampling Distributions
The Central Limit Theorem explains why so many sample-based quantities turn out normal.
You can also work through the full normal distribution topic, review variance and outliers, or browse all calculators.
Academic sources and further reading
NIST/SEMATECH e-Handbook of Statistical Methods
The National Institute of Standards and Technology hosts a free statistics reference. Section 1.3.6 covers the normal distribution and the proportion of data within one, two, and three standard deviations: itl.nist.gov/div898/handbook.
Khan Academy — Empirical Rule
Khan Academy's statistics unit walks through the rule with video lessons and practice questions on the normal distribution: khanacademy.org/math/statistics-probability.
OpenStax — Introductory Statistics
OpenStax publishes a peer-reviewed, openly licensed statistics textbook. The chapter on the normal distribution covers the Empirical Rule with examples: openstax.org/details/books/introductory-statistics-2e.