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 kth percentile of a data set is a value that divides the data set into the lower kth percentile and the upper (1 − kth) percentile.
The computation of the kth 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 kth percentile using the formula:
If i is an integer, the ith data value is the kth percentile. If i is not an integer, take the mean of the two values on either side of i to give the kth 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:
- Q1 = 25th percentile
- Q2 = 50th percentile = median
- Q3 = 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 90th percentile?
Solution
The 90th 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 90th 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 90th 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.
