Why Is This Skyline Normal?

Skyline Photograph The skyline pictured here follows the shape of the normal distribution. It is, by far, the most widely used probability distribution. It is so popular that, when a statistician is unsure of the appropriate probabilities, a normal distribution will be usually assumed.

Here are some examples of the normal curve in action:

Grading By The Curve. Some professors grade by absolute standards; for example, 90 percent correct is an A score, 80 percent a B, 70 percent a C, 60 percent a D, and below 60 percent is an F. With absolute grading standards such as these, it is possible for every student in the class to get an A or for every student to get an F.

A very different approach is to judge each student ’s performance relative to the others in the class, for example, “grading by the normal curve.” Each student is given a numerical test score, perhaps the number of questions right or the number right minus a fraction of the number wrong. The professor then uses the mean and standard deviation of these scores to calculate a standardized score, Z, for each student. A value Z = 0 is an average score. Those who did above average have positive Zs, while those who scored below average have negative Zs. If the scores are approximately normally distributed, they will be symmetrically scattered about Z = 0, in the bell shape shown by the skyline above. This particular grading scale gives 7 percent As, 24 percent Bs, 38 percent Cs, 24 percent Ds, and 7 percent Fs.

IQ Tests. There are a number of tests designed to measure a person’s IQ (Intelligence Quotient). These tests are descended from 54 mental “stunts” that two psychologists (Alfred Binet and Theodore Simon) conducted in 1904 to weed dull students out of the Paris school system. The intention is to test a person’s general intelligence, including an accurate memory and the ability to reason clearly and logically.

Scores on IQ tests are approximately normally distributed for people of the same age. The standard scaling gives a mean IQ of 100 with a standard deviation of 16. Thus, a score of 100 indicates that a person’s intelligence is average for people of that age. About half of the people tested will score above 100, while half will score below 100.

Smith, Gary. Statistical Reasoning. 3rd edition. United States: McGraw-Hill, Inc., 1994.

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