A student sent out me one email around a practice difficulty involving detect the center 80% of a common distribution. The student was confuse the center 80% of a bell curve through the 80th percentile. The student then tried come answer the practice difficulty with the 80th percentile, which of course, did not match the answer key. The student sent me an e-mail asking for an explanation. The concern in the email is a teachable moment and deserves a tiny blog post.

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The adhering to figures present the difference in between the middle 80% under a bell curve and the 80th percentile of a bell curve.

The center 80% under a bell curve (Figure 1) is the middle section the the bell curve the exlcudes the 10% of the area on the left and 10% that the area ~ above the right. The 80th percentile (Figure 2) is the area of a left tail that excludes 20% the the area ~ above 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 increase the area 0.8000 in a traditional normal table, the equivalent z-score is

To uncover the 90th percentile, look increase the area 0.9000 in the traditional normal table. Over there is no precise match and also the the next area come 0.9000 is 0.8997, which has a z-score that

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. Therefore the middle 80% the a normal distribution is between
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and
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. Currently just transform these come the data in the measurement range that is appropriate in the offered practice difficulty at hand.

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

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