Empirical Rule

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. It is also called the 68-95-99.7 rule. The rule applies whenever data follows a bell-shaped distribution, which includes a wide range of natural and financial phenomena.

Think of the standard deviation as a radius around the center of a target: the empirical rule tells you what percentage of shots land within one, two, or three rings from the bullseye.

How the Three Intervals Work

Each interval describes a progressively larger slice of the distribution. Understanding all three together gives you a practical picture of where most data points cluster and where the rare outliers live.

• 68% within one standard deviation: More than two-thirds of all observations fall between the mean minus one standard deviation and the mean plus one standard deviation. This is the core of the distribution where typical values live.
• 95% within two standard deviations: Only 5% of values fall outside this range. In a normal distribution, a result more than two standard deviations from the mean is considered statistically unusual.
• 99.7% within three standard deviations: Only 0.3% of values, or roughly 3 in every 1,000 observations, fall outside this range. Events at three or more standard deviations from the mean are genuinely rare in a normal distribution.

A Concrete Example With Stock Returns

Suppose a mutual fund has an average annual return of 8% and a standard deviation of 12%. The empirical rule tells you:

• In any given year, 68% of returns will fall between -4% and 20%.
• 95% of annual returns will fall between -16% and 32%.
• 99.7% of annual returns will fall between -28% and 44%.

A return of -28% in a single year is a three-standard-deviation event. Based on the empirical rule applied to a normal distribution, you would expect to see that happen only about 0.15% of the time on each tail, or once every few centuries of returns.

The Empirical Rule in Finance and Risk Management

Risk managers, options traders, and portfolio analysts use the empirical rule as a quick reference for assessing probability without running full statistical calculations. Value at Risk models that assume normality rely on this framework to set confidence intervals. A 95% Value at Risk threshold corresponds directly to the two-standard-deviation boundary in the empirical rule.

The rule's most important practical limitation is also its most commonly ignored one: many financial return series are not normally distributed. Equity returns show excess kurtosis, meaning extreme outcomes are more frequent than a normal distribution predicts. The 2008 financial crisis produced daily stock market moves that were theoretically many-standard-deviation events yet occurred repeatedly over a few months. The empirical rule gives you useful ballpark probabilities but understates tail risk in real markets.

Where the Empirical Rule Applies and Where It Fails

The rule is reliable when the underlying data is symmetric and bell-shaped. Human height, measurement error in laboratory experiments, IQ scores, and certain manufacturing processes all fit this pattern well. The empirical rule performs accurately in those contexts.

It fails when distributions are skewed or have fat tails. Income distributions are right-skewed: most people earn near the median, but a few earn vastly more. Log-normal distributions, such as stock price levels, require adjustments before the empirical rule applies. In all such cases, the rule provides a starting approximation but not a precise probability estimate.

Sources

• https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data
• https://www.cfainstitute.org/en/programs/cfa/curriculum