What Is the Empirical Rule?
Let me explain the empirical rule directly to you—it's also called the three-sigma or 68-95-99.7 rule, and it predicts how data deviates from the mean in a normal distribution. This rule states that 68% of observations fall within the first standard deviation (µ ± σ), 95% within the first two (µ ± 2σ), and 99.7% within the first three (µ ± 3σ) of the mean. You'll find it essential in areas like quality control and risk analysis.
Key Takeaways
Three-sigma limits from the empirical rule set upper and lower bounds in statistical quality control charts and risk assessments. Under this rule, 68% of data falls within one standard deviation, 95% within two, and 99.7% within three from the mean.
Understanding the Empirical Rule
You can use the empirical rule in statistics to forecast outcomes. After calculating the standard deviation, and even before you have all the data, apply this rule for a rough estimate of what to expect from the data you'll analyze. This comes in handy as a probability tool when collecting full data is time-consuming or impossible. Think about it in quality control reviews or risk exposure evaluations—for example, value-at-risk (VaR) assumes risk events follow a normal distribution.
The rule also tests a distribution's normality roughly. If too many points are outside three standard deviations, the distribution isn't normal—it might be skewed or something else. Remember, it's called the three-sigma rule because it covers data within three standard deviations on a bell curve.
Example of the Empirical Rule
Suppose we have a population of zoo animals with a normal distribution of lifespans: mean of 13.1 years and standard deviation of 1.5 years. If you want the probability an animal lives longer than 14.6 years, use the empirical rule. One standard deviation ranges from 11.6 to 14.6 years, two from 10.1 to 16.1, and three from 8.6 to 17.6.
68% falls within one standard deviation, so 32% is outside—half above 14.6 and half below 11.6. That means a 16% probability of living over 14.6 years.
The Empirical Rule in Investing
Market data often isn't normally distributed, so the full 68-95-99.7 rule doesn't apply directly to investments. Still, analysts use standard deviation to estimate volatility. Calculate it for your portfolio or an index to assess risk.
It's straightforward with a spreadsheet: input returns or prices, compute percent changes, and use the STDEV function. For accuracy, use data over three years or more. To annualize, multiply by the square root of 252 trading days. For instance, the S&P 500's standard deviation from 2015 to 2025 was 15.37%. You can find these figures on sites like Morningstar for various periods.
S&P 500 Standard Deviation Example
- Based on March 2025 closing prices, the daily standard deviation was 1.29%, annualizing to 20.42%.
- Higher standard deviation indicates higher perceived risk.
Explain Like I'm Five
The empirical rule shows how data points cluster around the center, based on standard deviation—which measures spread. In a normal distribution, 68% of data is within one standard deviation from the mean, 95% within two, and 99.7% within three.
How Is the Empirical Rule Used?
Apply it to predict outcomes in normal distributions. For a mean of 10 and standard deviation of 3.1, the first deviation ranges from 6.9 to 13.1, the second from 3.8 to 16.2, and so on.
Benefits of the Empirical Rule
It forecasts data effectively, especially for large or unknown-variable datasets.
The Bottom Line
Analysts use the empirical rule to check data within intervals from the mean. In investing, it helps estimate volatility for investments, portfolios, or funds.
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