5. Percentiles

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:

EquationIf 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.

Sec03. Percentile 1

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.

Sec03. Percentile 2

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.

Sec03. Percentile 3