## The Middle 80% Is Not 80th Percentile

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.

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 $z=0.84$. There is no exact match for 0.8000 and the closest is 0.7995, which produces the z-score of $z=0.84$. Once this is found, the z-score can then be converted to the measurement scale that is relevant in the practice problem at hand.

On the other hand, to find the middle 80%, you need to find the 90th percentile. The reason being that the standard normal table only provides the areas of the left tails. The middle area of 80% plus 10% on the left is the area of the left tail of size 90% (or 0.9000). Figure 3 below makes this clear.

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 $z=1.28$. Thus the middle 80% of a normal distribution is between $z=-1.28$ and $z=1.28$. 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%.

$\displaystyle \text{Some common z-critical values} \ \ \ \ \begin{bmatrix} \text{Middle x}\%&\text{ }&\text{Critical Value} \\\text{Or Confidence Level }&\text{ }&\text{ } \\\text{ }&\text{ }&\text{ } \\ 90\%&\text{ }&\text{z=1.645} \\ 95\%&\text{ }&\text{z=1.960} \\ 99\%&\text{ }&\text{z=2.575} \end{bmatrix}$

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%.