Empirical rule (68-95-99.7)

The empirical rule describes how data spreads out in a normal (bell-shaped) distribution. Roughly 68% of all values lie within one standard deviation of the mean, about 95% lie within two standard deviations, and about 99.7% lie within three. That's why it's also called the 68-95-99.7 rule.

Here's the rule in action. Suppose test scores are normally distributed with a mean of 500 and a standard deviation of 100. Then about 68% of scores fall between 400 and 600, about 95% between 300 and 700, and about 99.7% between 200 and 800. Because the normal curve is symmetric, you can split each band in half: about 34% of values sit between the mean and one standard deviation above it, and about 13.5% sit between one and two standard deviations above.

Those halves convert standard deviations into percentiles. A value exactly one standard deviation above the mean sits near the 84th percentile (50% below the mean plus 34%), while a value two standard deviations above the mean is near the 97.5th percentile. One standard deviation below the mean lands near the 16th percentile. Remember the rule only applies to distributions that are approximately normal.

The GRE quantitative section tests the empirical rule frequently, asking you to estimate the percentage of values in a given range or translate a standard-deviation distance into a percentile — memorizing 68-95-99.7 and the 34/13.5/2.35 half-bands handles nearly all of these questions.