The Empirical Rule (68-95-99.7 Rule)
For bell-shaped data, about 68%, 95% and 99.7% of values fall within 1, 2 and 3 SDs of the mean. Worked examples, z-scores and when the rule fails.
Apply The Empirical Rule In Three Steps
The empirical rule states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Use it to estimate probabilities and identify outliers without a z-table. Example: on a test with a mean of 80 and a standard deviation of 5, about 95% of scores lie between 70 and 90. The rule only works for data that is roughly bell-shaped, check normality before applying it.
The Empirical Rule And The Normal Curve
The 68-95-99.7 rule maps directly onto a normal curve. Draw a symmetric bell shape; mark the mean (μ) at the centre. Place tick marks at μ ± σ, μ ± 2σ, and μ ± 3σ. The area under the curve between μ - σ and μ + σ is about 68% of the total area. Between μ - 2σ and μ + 2σ it is about 95%. Between μ - 3σ and μ + 3σ it is about 99.7%. The remaining 0.3% falls in the two tails beyond 3σ. This visual is why the rule is sometimes called the 'three-sigma rule'. OpenStax Introductory Statistics 2e, section 6.1, uses this exact diagram to introduce the standard normal distribution.
Failure case: many students apply the rule to a distribution that is not normal and get wrong coverage estimates. If your data is skewed or multimodal, the percentages will be off by a wide margin. The rule is a property of the normal distribution, not a universal law of statistics.
Worked Examples Using The 68-95-99.7 Rule
Example 1: Heights Of Adult Women
Adult female heights in a certain population are normally distributed with a mean of 165 cm and a standard deviation of 7 cm. Use the empirical rule to find the range that contains approximately 68% of heights. Answer: 165 - 7 = 158 cm and 165 + 7 = 172 cm. For 95%, the range is between 165 - 2(7) = 151 cm and 165 + 2(7) = 179 cm. About 99.7% are between 144 cm and 186 cm. What percentage of women are taller than 179 cm? Since 95% are within two standard deviations, 5% are outside that range. Half of that 5% (2.5%) are above 179 cm, so about 2.5% of women are taller than 179 cm. These figures are not from a specific survey; they illustrate the rule.
Example 2: Standardised Test Scores
A standardised test has a mean score of 500 and a standard deviation of 100. The scores are approximately normal. What score is at the 16th percentile? The 16th percentile is one standard deviation below the mean because 68% of data lies within one standard deviation, leaving 32% in the tails, and half of that (16%) is below μ - σ. So the score is 500 - 100 = 400. What percentage of test takers score between 400 and 600? That is the range within one standard deviation: about 68%.
Example 3: Identifying Possible Outliers
A manufacturer measures the length of a part. The mean is 15.0 cm and the standard deviation is 0.5 cm. The distribution is normal. A part measures 16.6 cm. That is more than three standard deviations above the mean (15.0 + 3(0.5) = 16.5). Using the empirical rule, fewer than 0.3% of parts should be this extreme. The part is flagged for review, it is a potential outlier. This does not mean it is an error, just that it warrants separate investigation.
Z-Scores Connect The Rule To The Standard Normal
The z-score formula z = (x - μ) / σ converts any value from a normal distribution to the standard normal distribution, which has a mean of 0 and a standard deviation of 1. A z-score tells you how many standard deviations a value is above or below the mean. For the empirical rule, a z-score of 1 corresponds to 68% coverage, a z-score of 2 corresponds to 95%, and a z-score of 3 corresponds to 99.7%. Using z-scores allows you to apply the rule to any normal distribution without recalculating the curve. A z-score of 1.5, for example, falls between 1 and 2 standard deviations, so you know it is beyond the 68% range but within the 95% range. OpenStax Introductory Statistics 2e, section 6.1, defines the standard normal variable z in exactly this way.
Non-Normal Data: Use Chebyshev’s Inequality Instead
When your data is not normal, the empirical rule does not apply. Use Chebyshev's theorem (also called Chebyshev's inequality) instead. For any distribution, at least 1 - 1/k² of the data lies within k standard deviations of the mean. For k = 2, at least 1 - 1/4 = 75% of data is within 2 standard deviations. For k = 3, at least 1 - 1/9 = 88.9% (often rounded to 89%) is within 3 standard deviations. Compare this to the empirical rule: for a normal distribution, about 95% is within 2 standard deviations, but Chebyshev's inequality guarantees only 75%. Chebyshev is much weaker but always true. Use it when you cannot assume normality, for example, with income data, which is typically right-skewed, or with multimodal distributions.
Failure case: students apply Chebyshev's inequality backwards, claiming that at most 75% of data is within 2 standard deviations. The inequality gives a lower bound, at least 75%, meaning the actual percentage could be higher. For a normal distribution, it is 95%, which satisfies the inequality.
Check If Your Data Is Roughly Normal Before Using The Rule
Diagnose Shape With Visual Checks
Before you apply the 68-95-99.7 rule, verify that your data is approximately normally distributed. Check the shape of the distribution using a histogram or a box plot. A normal distribution is unimodal, symmetric, and bell-shaped. In a box plot, the median should be roughly centred in the box, and the whiskers should be about equal length. The mean and median should be close to each other, if they differ by more than about 0.2 times the standard deviation, the distribution is likely skewed. Calculate the skewness: a value between -1 and 1 is roughly symmetric; outside that range, the data is skewed. For small samples (n < 30), the empirical rule is unreliable because the sample itself may not reflect the population shape.
Fallback Strategies For Non-Normal Data
What to do if the data is not normal: report the median and IQR instead of the mean and standard deviation. The median is resistant to outliers, and the IQR is a robust measure of spread. For a quick estimate of coverage, use Chebyshev's inequality. Do not force a normal model onto non-normal data just because you want to use the empirical rule.
Common Questions
Does the empirical rule apply to all data sets?
No. The empirical rule applies only to data that is approximately normally distributed. For non-normal data, use Chebyshev's inequality, which guarantees at least 75% of data within 2 standard deviations and at least 89% within 3 standard deviations.
What is the difference between the empirical rule and Chebyshev's theorem?
The empirical rule gives exact approximate percentages (68, 95, 99.7) but only for normal distributions. Chebyshev's theorem gives weaker but universal bounds that apply to any distribution. For k = 2, Chebyshev guarantees at least 75% of data within 2 standard deviations; the empirical rule says about 95% for a normal distribution.
How do I use the empirical rule to find a percentile?
The 16th percentile is approximately one standard deviation below the mean, because 68% of data lies within one standard deviation, leaving 16% in the left tail. The 2.5th percentile is about two standard deviations below the mean, and the 0.15th percentile is about three standard deviations below.
Can I use the empirical rule if my sample size is small?
Not reliably. With n < 30, the sample may not represent the population shape well enough to assume normality. Focus on the median and IQR instead. If you must use the rule, verify normality with a histogram and a box plot first.
What is the common mistake when using the empirical rule?
Assuming it applies to all data. The most frequent failure is applying the 68-95-99.7 rule to skewed or multimodal data, which overstates the percentage of data within 1, 2, or 3 standard deviations. Always check for normality first.