A Normal Curve Puzzle

First, think of a normal (Gaussian) curve. Don’t ponder its formula, origin, or application; instead, just picture it. To be more specific, create a mental image of the famous bell-shaped curve resting on a straight, horizontal “baseline” that’s labeled with z-scores.

In the picture now in your mind, the z-score on the baseline directly under the tallest part of the curve should equal zero, with that value corresponding to the mean. Other z-scores on the baseline, of course, simply indicate the lateral distance from the mean as measured in standard deviation units. The z-scores to the right of zero are positive numbers, corresponding to scores that are above the mean. Conversely, z-scores to the left of the mean are negative.

Clearly, the curved line above the baseline does not have the same degree of “curvature” at all points from far left to far right. When the curve is 10 standard deviations the right (or left) of the mean, the curvature is quite small. When this far out from the mean, the curve is nearly flat. Closer in, the curve has more curvature.

OK. Now that you have access to a mental image of the normal curve sitting on a baseline of z-scores, here comes the puzzle question:

At what z-score value(s) is the normal curve curved the most?

(If you come up with an answer to this little puzzle and would like to see if it’s correct, send your solution in an email message to Sky Huck at shuck@utk.edu

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