Within 2 standard deviations of the mean in a normal distribution, approximately what percent of data lie?

Within 2 standard deviations of the mean in a normal distribution, approximately what percent of data lie?

• A. 68.26%
• B. 99.73%
• C. 34.13%
• D. 95.45%

Answer: (D)

In a normal distribution, the so-called empirical rule tells us how much of the data falls within a given number of standard deviations from the mean. Within two standard deviations of the mean, about 95.45% of the data lie. This comes from the standard normal distribution: P(-2 ≤ Z ≤ 2) = Φ(2) − Φ(−2) = 2Φ(2) − 1 ≈ 2(0.97725) − 1 = 0.9545. So roughly 95% of the data are between μ − 2σ and μ + 2σ. The other options correspond to different ranges: 68.26% is within ±1σ, 99.73% within ±3σ, and 34.13% is the area between the mean and +1σ (half of 68.26%).

In a normal distribution, the so-called empirical rule tells us how much of the data falls within a given number of standard deviations from the mean. Within two standard deviations of the mean, about 95.45% of the data lie. This comes from the standard normal distribution: P(-2 ≤ Z ≤ 2) = Φ(2) − Φ(−2) = 2Φ(2) − 1 ≈ 2(0.97725) − 1 = 0.9545. So roughly 95% of the data are between μ − 2σ and μ + 2σ. The other options correspond to different ranges: 68.26% is within ±1σ, 99.73% within ±3σ, and 34.13% is the area between the mean and +1σ (half of 68.26%).