This one rule answers nearly every normal-distribution question the GRE writes. In a normal distribution, about 68% of the data lies within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Because the curve is symmetric you can slice these bands in half: 34% sits between the mean and +1 SD, another 34% between the mean and −1 SD, and so on. From there, subtraction gives everything else — for instance the slice between +1 SD and +2 SD holds (95% − 68%)/2 = 13.5% of the data. The tail beyond +2 SD holds (100% − 95%)/2 = 2.5%. The reliable method is to draw the bell, mark the mean and the ±1/±2/±3 gridlines, write 34 / 13.5 / 2.35 into the segments, and simply add the pieces the question asks for. No calculator, no z-table.
✅ Solved examples
1. IQ scores are normal with mean 100 and SD 15. What percent of people score between 85 and 115?
85 = mean − 15 = −1 SD and 115 = mean + 15 = +1 SD. Within ±1 SD is about 68%.
2. Same IQ scores (mean 100, SD 15). What percent score above 130?
130 = mean + 30 = +2 SD. Within ±2 SD is 95%, so the two outer tails hold 5% total; by symmetry the upper tail alone is 5%/2 = 2.5%.
3. Test scores are normal with mean 500 and SD 100. What percent score between 500 and 700?
700 = mean + 2 SD. From the mean out to +2 SD is half of 95% = 47.5%.
4. Quantitative Comparison. Scores are normal with mean 70 and SD 8. Quantity A: the percent scoring between 54 and 70. Quantity B: the percent scoring above 78. Which is greater?
54 = mean − 2 SD, so from −2 SD to the mean is half of 95% = 47.5% (Quantity A). 78 = mean + 1 SD, so above +1 SD is (100% − 68%)/2 = 16% (Quantity B). 47.5% > 16%, so Quantity A is greater.
✏️ Practice — try these, take hints as needed
1. Normal, mean 50, SD 5. What percent lies between 45 and 55?
45 and 55 are ±1 SD.
Within one SD.
The 68 band.
≈ 68%
2. Normal, mean 50, SD 5. What percent lies between 40 and 60?
40 and 60 are ±2 SD.
Within two SD.
The 95 band.
≈ 95%
3. Normal, mean 200, SD 20. What percent is below 160?
160 = mean − 2 SD.
The lower tail beyond −2 SD.
(100 − 95)/2.
≈ 2.5%
4. Normal, mean 12, SD 2. What percent lies between 12 and 16?
16 = mean + 2 SD.
Mean to +2 SD is half of 95%.
95/2.
≈ 47.5%
5. Normal, mean 100, SD 10. What percent lies between 110 and 120 (i.e. +1 SD to +2 SD)?
Within ±1 SD is 68%, within ±2 SD is 95%.
The +1-to-+2 slice is (95 − 68)/2.
27/2.
≈ 13.5%
📝 Topic test — 8 questions
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