Big z-Scores

Dear Students,

Many people incorrectly think that z-scores range from -3.0 to +3.0. In other words, there's a myth floating around that it's impossible for a score to lie more than 3 standard deviations above or below the mean.

To prove that it's possible for a given score to lie more than 3 SDs away from M, consider these 26 IQ scores: 105, 105, 105, 105, 105, 106, 106, 106, 106, 106, 107, 107, 107, 107, 107, 108, 108, 108, 108, 108, 109, 109, 109, 109, 109, and 130. For these scores, the mean is equal to 107.88 while the standard deviation is equal to 4.64. The z-score for the 26th score in the data set is found by dividing the difference between 130 and 107.88 by 4.64, which is equal to 4.77. This z-score clearly is larger than 3!

In the data set just considered, the 26 scores are not normally distributed. Although that's obviously true, there's no requirement that scores form a bell-shaped curve before z-scores can be computed. A z-score can be computed for any score in any group of scores, regardless of what kind of distributional shape exists.

But what if the scores are normally distributed? Do all scores here lie within 3 standard deviations of the mean? The answer is "No!"

In a normal distribution, remember that the "tails" of the curve extend infinitely far up and down the numerical baseline upon which the curve sits. Speaking formally, the curved tails are asymptotic to the baseline. The further out we go from the mean (either direction), the closer the tail of the curve will come to the baseline; however, it will never actually touch down because the normal curve extends out to positive and negative infinity.

Consider now the real distribution of IQ scores of all living adults. Such scores do not form a true normal curve, but they do create a bell-shaped distribution. The mean of these IQ scores would be about 100, and the standard deviation would be about 15. In such a distribution, your z-score would be equal to +1 if you had an IQ of 115, +2 if you had an IQ of 130, and +3 if you had an IQ of 145. But what if you had, as some people do, an IQ of 160?

It's my guess that the myth about z-scores ranging from -3 to +3 comes from the picture of the normal curve that appears in many statistics books. Unfortunately, such pictures often appear with the curve touching down (on the baseline) just beyond the places where you'd be if you were 3 SDs above or below the mean. Accompanying that picture is usually the statement that 99.73 percent of the scores lie within 3 SDs of the mean. The picture and the percentage, I think, mislead people into thinking that no score can lie more than 3 standard deviations from the mean.

If you've read this entire email message and think you understand it's main point, consider yourself to have a z-score of +5!

Sky Huck

Copyright © 2012

Schuyler W. Huck
All rights reserved.

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