What does the 68-95-99.7 rule describe for a normal distribution?

What does the 68-95-99.7 rule describe for a normal distribution?

• A. 68% within ±2σ, 95% within ±3σ, 99.7% within ±4σ
• B. 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ
• C. 50% within ±1σ, 75% within ±2σ, 90% within ±3σ
• D. 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ

Answer: (D)

The main idea here is the empirical rule for a normal distribution. This rule quantifies how data spreads around the mean in terms of standard deviations. About 68% of observations lie within one standard deviation of the mean, roughly 95% lie within two standard deviations, and about 99.7% lie within three standard deviations. These percentages come from the shape of the normal curve and correspond to z-scores of ±1, ±2, and ±3. So the description that matches this pattern is: 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ. The other options mix up these boundaries (for example, claiming 68% within ±2σ or 99.7% within ±4σ) or give unrelated percentages, which don’t fit the empirical rule.

The main idea here is the empirical rule for a normal distribution. This rule quantifies how data spreads around the mean in terms of standard deviations. About 68% of observations lie within one standard deviation of the mean, roughly 95% lie within two standard deviations, and about 99.7% lie within three standard deviations. These percentages come from the shape of the normal curve and correspond to z-scores of ±1, ±2, and ±3.

So the description that matches this pattern is: 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ. The other options mix up these boundaries (for example, claiming 68% within ±2σ or 99.7% within ±4σ) or give unrelated percentages, which don’t fit the empirical rule.