The Normal Distribution

One of the most important distributions for random variables is the Normal Distribution (also known as the Gaussian Distribution). The Normal Distribution looks like the classic “bell curve,” rounded in the middle with long tails, and symmetric around the mean (which equals the median).

The GRE tests on distributions that are both normal and approximately normal. These distributions have the following characteristics:

1. The mean and median are equal, or almost exactly equal.
2. The data is exactly, or almost exactly, symmetric around the mean/median.
3. Roughly two-thirds of the sample will fall within 1 standard deviation of the mean. That means that roughly one-third of the sample falls within 1 standard deviation below the mean and roughly one-third of the sample falls within 1 standard deviation above the mean. Thus, in the previous example, a value of 6 is at roughly , or the 17^th percentile. A value of 14 is at roughly , or the 83^rd percentile.
4. Roughly 96% of the sample will fall within 2 standard deviations of the mean. In other words, roughly 48 percent of the sample falls between the mean and 2 standard deviations below the mean; roughly 48 percent of the sample falls between the mean and 2 standard deviations above the mean. Thus, in the prior example, a value of 2 will be at , or the 2^nd percentile. A value of 18 will be at , or the 98^th percentile.
5. Only about of the curve is 3 or more standard deviations below the mean; the same is true above the mean.

The GRE typically will only test these concepts in a general way, and it will not distinguish between random variables that are normally distributed versus ones that are nearly normally distributed. However, it is important to note that distributions that are not normal or nearly normal do not necessarily share the characteristics above. It is possible, for example, to construct distributions where the mean and median are substantially different, or where 100 percent of the observations fall within 2 standard deviations, or where more than 1 percent of the observations fall more than 3 standard deviations from the mean.

Check Your Skills

For questions #12–15, variable X is nearly normally distributed, with a mean of 6 and a standard deviation of 2.

12. Approximately what percent of the observations in X will be smaller than 4?

13. Approximately what percent of the observations in X will be greater than 12?

14. For variable X, approximately what percentile corresponds to a value of 2?

15. Would the answers to questions #12–14, be the same if variable X were not nearly normally distributed?