Empirical Rule (68-95-99.7 Rule)
The Empirical Rule (68-95-99.7 Rule) states that for normal distributions, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 — a fundamental statistics concept in enVision Algebra 1 Chapter 11 for Grade 11. For test scores with mean μ = 75 and standard deviation σ = 10: 68% of scores fall between 65 and 85, 95% between 55 and 95, and 99.7% between 45 and 105. This rule allows quick probability estimates without calculations when data is approximately normally distributed.
Key Concepts
For normal distributions, approximately: 68% of data falls within 1 standard deviation of the mean: $\mu \sigma \leq x \leq \mu + \sigma$ 95% of data falls within 2 standard deviations of the mean: $\mu 2\sigma \leq x \leq \mu + 2\sigma$ 99.7% of data falls within 3 standard deviations of the mean: $\mu 3\sigma \leq x \leq \mu + 3\sigma$.
Common Questions
What does the Empirical Rule state?
For normal distributions: about 68% of data falls within 1 standard deviation of the mean (μ ± σ), about 95% within 2 standard deviations (μ ± 2σ), and about 99.7% within 3 standard deviations (μ ± 3σ).
If heights have μ = 68 inches and σ = 3 inches, what range contains 95% of heights?
95% of heights fall within μ ± 2σ = 68 ± 6 = between 62 and 74 inches.
What percentage of data falls beyond 2 standard deviations from the mean?
About 5% (100% - 95% = 5%), split roughly equally on both sides, meaning about 2.5% above and 2.5% below.
For test scores with μ = 75 and σ = 10, what range covers 68% of scores?
68% of scores fall within μ ± σ = 75 ± 10 = between 65 and 85.
Does the Empirical Rule apply to all data sets?
No, only to data that is approximately normally distributed (bell-shaped). Skewed data or data with outliers does not follow the 68-95-99.7 pattern.