The Normal Curve and Percentiles

As shown in the histograms in Figure 2, the normal curve is a bell-shaped graph that represents a hypothetical frequency distribution in which the frequency of scores is greatest near the mean and progressively decreases toward the extremes. In essence, the normal curve is a smooth frequency polygon based on an infinite number of scores. The mean, median, and mode of a normal curve all have the same value. This falls at the center of this symmetric distribution and splits it in half, such that 50% of the observations are above and 50% of the observations are below it. Many physical or psychological characteristics, such as height, weight, and scores on many standardized intelligence tests fall on a normal curve.

One useful characteristic of a normal curve is that certain percentages of scores fall at certain distances (measured in standard deviation units) from its mean. A special statistical table makes it a simple matter to determine the percentage of scores that fall above or below a particular score or between two scores on the curve. For example, about 68 percent of scores fall between plus and minus one standard deviation from the mean; about 95 percent fall between plus and minus two standard deviations from the mean; and about 99 percent fall between plus and minus three standard deviations from the mean. Figure 4 contains a normal distribution with the percentage of scores at +/- 1, 2, and 3 standard deviations from the mean shown.

Figure 4.

Consider an IQ test, with a mean of 100 and a standard deviation of 15. What percent of people score above 115? Because intelligence scores fall on a normal curve, about 34 percent of the scores fall between the mean and one standard deviation (in this case 15 points) above the mean. We also know that for a normal distribution 50 percent of the scores fall above the mean and 50 percent fall below the mean. Thus, about 84 percent (50 percent below the mean and 34 percent between the mean and a score of 115) of the scores fall below 115. If 84 percent fall below 115, then 16 percent (100 percent - 84 percent) must fall above a score of 115.

Learning Check #12:
An introductory psychology teacher who has taught for years has developed a comprehensive final exam that is normally distributed with a mean of 200 points and a standard deviation of 25 points. (a) What percentage of the students score above 200 points? (b) What percentage of the students score below 175 points? (c) What percentage of the students score more than 250 points?

Scores along the abscissa of the normal curve also represent percentiles--the scores at or below which particular percentages of scores fall. Percentiles are frequently used, as they give us a quick idea of how a score compares to the rest of the data set. If a score is equal to the 10th percentile, then you know that 10 percent of the scores fell at or below that value and 90 percent of the scores were above that value. With respect to the IQ test mentioned above, a score of 115 would have a percentile rank of 84.

Learning Check #13:
What are the percentile ranks for the three scores listed in Learning Check #12:
200, 175, and 250?

Learning Check #14:
Suppose Emily is taken to the doctor for a well-baby check and it is determined that she is in the 5th percentile for height and 7th percentile for weight. What do you now know about Emily, as compared to other children her age?