A student sent me an email about a practice problem involving finding the middle 80% of a normal distribution. The student was confusing the middle 80% of a bell curve with the 80th percentile. The student then tried to answer the practice problem with the 80th percentile, which of course, did not match the answer key. The student sent me an email asking for an explanation. The question in the email is a teachable moment and deserves a small blog post.

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The following figures show the difference between the middle 80% under a bell curve and the 80th percentile of a bell curve.

The middle 80% under a bell curve (Figure 1) is the middle section of the bell curve that exlcudes the 10% of the area on the left and 10% of the area on the right. The 80th percentile (Figure 2) is the area of a left tail that excludes 20% of the area on the right.

Finding the 80th percentile (or for that matter any other percentile) is easy. Either use software or a standard normal table. If you look up the area 0.8000 in a standard normal table, the corresponding z-score is

To find the 90th percentile, look up the area 0.9000 in the standard normal table. There is no exact match and the closest area to 0.9000 is 0.8997, which has a z-score of

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. Thus the middle 80% of a normal distribution is between
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and
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. Now just convert these to the data in the measurement scale that is relevant in the given practice problem at hand.

Find the middle x% is an important skill in an introductory statistics class. The z-score for the middle x% is called a critical value (or z-critical value). The common critical values are for the middle 90%, middle 95% and middle 99%.

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How about critical values not found in the above table? It will be a good practice to find the z-scores for the middle 85%, 92%, 93%, 94%, 96%, 97%, 98%.