Remember that the median of a data set divides the lower 50% of the data from the upper 50%. We say that the median is the 50th percentile of the data set. If a number divides the lower 34% of the data from the upper 66%, that number is the 34th percentile. In general, a *k*th percentile of a data set is a value that divides the data set into the lower *k*th percentile and the upper (1 − *k*th) percentile.

The computation of the *k*th percentile is similar to the one for the median. First arrange the *n* data values in ascending order, and then compute the **location** (or index, *i* ) of the *k*th percentile using the formula:

If *i* is an integer, the *i*th data value is the *k*th percentile. If *i* is not an integer, take the mean of the two values on either side of *i* to give the *k*th percentile.

You’ve already seen the usefulness of the ** quartiles** of a data set (recall the 5-Number Summary and boxplots from Unit 2?). The quartiles are the 25th, 50th, and 75th percentiles:

- Q
_{1}= 25th percentile - Q
_{2}= 50th percentile = median - Q
_{3}= 75th percentile

You can find the quartiles using the same index formula above.

**Example: Finding the z-Value That Represents a Given Percentile**

What *z*-value represents the 90^{th} percentile?

**Solution **

The 90^{th} percentile is the *z*-value for which 90% of the area under the standard normal curve is to the left of *z*. So, we need to find the value of *z* that has an area of 0.9000 to its left.

Looking for 0.9000 (or an area extremely close to it) in the interior of the cumulative normal tables, we find 0.8997, which corresponds to a *z*-value of 1.28. Thus *z* ≈ 1.28 represents the 90^{th} percentile.

Using a TI-83/84 Plus calculator, we can find a value of *z* with a given area to its left. As noted previously, the 90^{th} percentile is the *z*-value that has an area of 0.9000 to its left. Enter invNorm(0.9000), as shown in the screenshot, and press ENTER. The answer is *z* ≈ 1.28.