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Empirical Rule (68-95-99.7 Rule)

The Empirical Rule is a shortcut for normal distributions: approximately 68% of all data falls within 1 standard deviation of the mean, about 95% falls within 2 standard deviations, and about 99.7% falls within 3 standard deviations.

For a dataset that follows a normal (bell-shaped) distribution with mean μ\mu and standard deviation σ\sigma, the Empirical Rule provides three approximate containment intervals. Roughly 68.27% of observations lie in the interval μ±σ\mu \pm \sigma, roughly 95.45% lie in μ±2σ\mu \pm 2\sigma, and roughly 99.73% lie in μ±3σ\mu \pm 3\sigma. These percentages are derived from the properties of the standard normal distribution and are typically rounded to 68%, 95%, and 99.7% for practical use.

Key Formula

P(μ−σ≤X≤μ+σ)≈0.68P(μ−2σ≤X≤μ+2σ)≈0.95P(μ−3σ≤X≤μ+3σ)≈0.997\begin{aligned} P(\mu - \sigma \le X \le \mu + \sigma) &\approx 0.68 \\ P(\mu - 2\sigma \le X \le \mu + 2\sigma) &\approx 0.95 \\ P(\mu - 3\sigma \le X \le \mu + 3\sigma) &\approx 0.997 \end{aligned}
Where:
  • μμ = the mean of the distribution
  • σσ = the standard deviation of the distribution
  • XX = a normally distributed random variable

Worked Example

Problem: The heights of students at a school are normally distributed with a mean of 170 cm and a standard deviation of 6 cm. Use the Empirical Rule to find the range of heights that contains the middle 95% of students, and determine what percentage of students are taller than 182 cm.
Step 1: Identify the mean and standard deviation.
μ=170 cm,σ=6 cm\mu = 170 \text{ cm}, \quad \sigma = 6 \text{ cm}
Step 2: For the middle 95%, apply the 2-standard-deviation interval.
μ±2σ=170±2(6)=170±12\mu \pm 2\sigma = 170 \pm 2(6) = 170 \pm 12
Step 3: Calculate the endpoints of this interval.
[158 cm,  182 cm][158 \text{ cm},\; 182 \text{ cm}]
Step 4: To find the percentage taller than 182 cm, note that 95% of data lies between 158 and 182. The remaining 5% is split equally between the two tails.
100%−95%2=2.5%\frac{100\% - 95\%}{2} = 2.5\%
Answer: The middle 95% of students have heights between 158 cm and 182 cm. Approximately 2.5% of students are taller than 182 cm.

Visualization

Why It Matters

The Empirical Rule gives you a fast way to estimate probabilities without a calculator or z-table. In AP Statistics, it appears frequently in questions about normal distributions, and it helps you quickly judge whether a data point is unusual. Outside the classroom, quality control in manufacturing uses the rule to set tolerance limits — if a measurement falls beyond 3 standard deviations, something has likely gone wrong in the process.

Common Mistakes

Mistake: Applying the Empirical Rule to distributions that are not approximately normal.
Correction: The rule only works for bell-shaped, roughly symmetric distributions. For skewed data or data with outliers, the percentages 68-95-99.7 will not hold. Always check the shape of the distribution first.
Mistake: Forgetting to split the tail percentage evenly on both sides.
Correction: Because the normal distribution is symmetric, the area outside any interval μ±kσ\mu \pm k\sigma is divided equally between the left and right tails. For instance, the 5% outside μ±2σ\mu \pm 2\sigma means 2.5% in each tail, not 5% on one side.

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