Statistics and Distributions

What percentage falls within 1, 2, and 3 standard deviations of the mean?

Quick answer

In any normal distribution, approximately 68% of data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations. This principle is known as the Empirical Rule, or the 68-95-99.7 rule.

The topic explained

The 68-95-99.7 rule explained

A normal distribution forms a symmetric, bell-shaped curve centered directly over its mean. When asking what percentage falls within 1, 2, and 3 standard deviations, statisticians use the Empirical Rule. This rule states that roughly 68% of all values lie within one standard deviation above or below the mean, about 95% lie within two, and about 99.7% lie within three.

Because the curve is perfectly symmetrical, half of each percentage sits on the left side of the mean and half sits on the right. This allows you to split the distribution into distinct vertical slices.

How to break down the slices step by step

To find the percentage between any whole standard deviations, divide each section into equal halves. Between the mean and one standard deviation above it sits 34% (half of 68%). Between one and two standard deviations above the mean sits 13.5% (half of 95% minus 68%).

Between two and three standard deviations above the mean lies 2.35% (half of 99.7% minus 95%). Beyond three standard deviations lies the remaining 0.15% in each tail. Add or subtract these slice percentages to calculate values above, below, or between any whole standard deviations.

A concrete example with test scores

Suppose an exam has a mean score of 70 points and a standard deviation of 10 points. One standard deviation spans from 60 to 80 points, covering about 68% of test takers. Two standard deviations cover scores from 50 to 90 points, containing roughly 95% of students.

If 1,000 students took the exam, you can find how many scored above 80. Above 80 is more than one standard deviation above the mean. That region holds 13.5% + 2.35% + 0.15% = 16% of the curve. Multiply 1,000 by 0.16 to get 160 students.

Free printable worksheet

Applying the Empirical Rule in a Normal Distribution

Designed for 11th-grade statistics students. Use the 68-95-99.7 rule to determine the percentage or count of individuals for each scenario.

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Applying the Empirical Rule in a Normal DistributionName:

What percentage of data in a normal distribution lies between the mean and 1 standard deviation above the mean?
What percentage of data falls outside of 2 standard deviations from the mean (combined across both tails)?
A sample has a mean of 100 and a standard deviation of 15. Between which two values does approximately 95% of the data fall?
Using the same distribution (mean = 100, standard deviation = 15), what percentage of values is greater than 115?
What percentage of values falls between 1 standard deviation below the mean and 2 standard deviations above the mean?
In a group of 500 adult heights with a mean of 66 inches and standard deviation of 3 inches, approximately how many people are taller than 72 inches?

Common questions

Can I use the 68-95-99.7 rule for skewed distributions?

No. The Empirical Rule applies only to bell-shaped, symmetric normal distributions. For skewed or irregular data, use Chebyshev's inequality instead.

What happens if a value is not a whole number of standard deviations from the mean?

When a value is not an exact integer number of standard deviations, convert it into a z-score and look up the exact percentage using a standard normal distribution table or a statistical calculator.

Why do the percentages add up to 99.7% instead of 100%?

The normal curve extends infinitely in both directions without ever touching the horizontal axis. The remaining 0.3% is split evenly between the extreme outer tails, leaving roughly 0.15% beyond three standard deviations on each side.

What is the difference between the Empirical Rule and z-scores?

The Empirical Rule is a mental-math shortcut for values that lie exactly 1, 2, or 3 standard deviations from the mean. Z-scores provide a precise way to calculate percentages for any decimal distance from the mean.

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