Key Takeaways
– The empirical rule (aka the 68–95–99.7 rule or three‑sigma rule) describes how data cluster around the mean in a normal (bell‑shaped) distribution: about 68% within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean.
– It’s a quick, intuitive tool for estimating probabilities, checking approximate normality, and communicating volatility, but it depends on the normality assumption and can be misleading for skewed or heavy‑tailed data (as financial returns often are).
– Practical use requires computing the mean and standard deviation, checking whether the normal approximation is reasonable, and applying the rule cautiously (or using alternatives such as Chebyshev’s inequality, bootstrapping, or empirical quantiles).
Understanding the Empirical Rule
– What it states: For a variable that follows a normal distribution with mean µ and standard deviation σ:
– ≈68% of observations lie between µ − σ and µ + σ.
– ≈95% lie between µ − 2σ and µ + 2σ.
– ≈99.7% lie between µ − 3σ and µ + 3σ.
– Why it matters: It links a measure of central tendency (mean) and dispersion (standard deviation) to probabilities, giving a simple way to estimate likely ranges of outcomes without full distribution tables.
Simple numeric example
– Suppose mean µ = 13.1 years and standard deviation σ = 1.5 years.
– µ + 1σ = 14.6 years. By the empirical rule, about 68% of lifespans fall between 11.6 and 14.6 years, so about 32% fall outside that interval. Half of that (16%) is above 14.6. Thus, using the rule, P(lifespan > 14.6) ≈ 16%.
How the Empirical Rule Is Used
– Quick probability estimates: When data are approximately normal you can rapidly translate a standard deviation distance into a probability (e.g., “within 2σ is about 95%”).
– Normality check: If far more than 0.3% of observations are outside ±3σ, the distribution likely deviates substantially from normal (skewness, heavy tails).
– Risk and quality control:
– Quality control and process monitoring commonly use three‑sigma control limits.
– In finance, standard deviation derived from returns is often used as a volatility measure and feeds into risk metrics such as value‑at‑risk (VaR) under a normality assumption.
Benefits of the Empirical Rule
– Simplicity: Provides immediate, memorable probability benchmarks without complicated calculations.
– Communication: Standardized shorthand for describing variability and unusual events (e.g., “a three‑sigma event”).
– Useful baseline: Good starting point for large samples where normality is plausible.
Limitations and cautions
– Normality assumption: The rule strictly applies only to normal distributions. Financial returns frequently exhibit skewness and fat tails (more extreme outcomes than normal), so the rule will understate extreme risk in those cases.
– Outliers and nonstationarity: Presence of outliers or changing volatility over time (heteroskedasticity) undermines accuracy.
– Sample size and estimation error: With small samples, estimated σ is noisy, and inferences based on the rule can be unreliable.
Practical steps — How to apply the Empirical Rule (step‑by‑step)
1. Define the variable and collect data
– Decide whether you will analyze raw values (e.g., heights, lifespans) or returns (percent changes). For financial returns use a consistent return type (simple or log returns) across observations.
2. Compute the mean and standard deviation
– In a spreadsheet with values in column A:
– Mean: =AVERAGE(A:A)
– Sample standard deviation: =STDEV.S(A:A) (or STDEV.P for full population)
– Note whether you need population (P) or sample (S) formula.
3. Apply the empirical rule ranges
– Compute intervals:
– 1σ range: µ − σ to µ + σ
– 2σ range: µ − 2σ to µ + 2σ
– 3σ range: µ − 3σ to µ + 3σ
– Interpret probabilities per the 68–95–99.7 guideline only if the data are approximately normal.
4. Check approximate normality
– Visual checks: histogram and a Q–Q plot (quantile–quantile plot). If points on a Q–Q plot lie close to the diagonal line, normality is plausible.
– Numeric checks: compute skewness and kurtosis; perform tests such as Jarque–Bera or Shapiro–Wilk. These are diagnostics, not absolute rules—look for substantive deviation rather than purely statistical significance for large samples.
5. For financial time series: compute returns and annualize volatility if needed
– Compute periodic returns (e.g., daily) r_t = (P_t − P_{t−1})/P_{t−1} or log returns ln(P_t/P_{t−1}).
– Periodic standard deviation σ_period = STDEV.S(returns).
– Annualize: σ_annual ≈ σ_period × sqrt(N), where N is periods per year (e.g., N ≈ 252 trading days, 12 months). Be consistent in period and return definition.
6. Use alternatives when normality is suspect
– Chebyshev’s inequality: gives conservative bounds for any distribution (e.g., at least 75% within 2σ).
– Empirical percentiles: use the data’s observed quantiles (e.g., 99th percentile) or bootstrapping to estimate tail probabilities.
– Fit heavy‑tailed distributions (t‑distribution, generalized Pareto) if appropriate.
Practical spreadsheet example (daily returns → annual volatility)
– Suppose you have daily returns in column B for one year:
– Daily standard deviation: =STDEV.S(B:B)
– Annualized volatility: =STDEV.S(B:B) * SQRT(252)
– Interpret: if annualized σ = 20%, then roughly 68% of annual returns would lie within ±20% of the mean return assuming normality.
Explain Like I’m Five (ELI5)
– Imagine most kids’ heights cluster near a middle height. The empirical rule says most kids (about 7 out of 10) are within a small step of that middle, nearly all (19 out of 20) are within two steps, and almost everyone (999 out of 1000) is within three steps. It only works when heights make a nice bell‑shaped pattern.
The Bottom Line
– The empirical rule is a convenient, intuitive summary of how dispersion (σ) maps into probabilities under a normal distribution. It is widely used for quick estimates, communication, and as a baseline in risk or quality contexts. However, its usefulness hinges on the normality assumption; when data are skewed or heavy‑tailed (common in finance), rely on diagnostic checks and consider robust or distribution‑free alternatives before drawing conclusions.
Further reading and sources
– Investopedia, “Empirical Rule” (overview and examples): https://www.investopedia.com/terms/e/empirical_rule.asp
– For normality tests and QQ plots: any standard statistics text or software documentation (e.g., R’s qqnorm/qqplot, Python’s SciPy stats).
Related topics
– Standard deviation and variance
– Z‑scores and standardization
– Chebyshev’s inequality
– Value‑at‑Risk (VaR) and heavy‑tailed distributions
– Bootstrapping and empirical quantiles
If you’d like, I can:
– Walk through a worked spreadsheet example with sample data,
– Show how to test normality with concrete steps in Excel/R/Python, or
– Demonstrate an empirical vs. theoretical tail comparison for a real financial series (e.g., S&P 500 returns).