## Percentile Ranks

In the previous post I described a percentile point as a score below which a certain percentage of the other scores in a data set lie. In finding a percentile point, we started with a desired percentage and found the corresponding score. Percentile ranks are similar in that we start with a score to find the corresponding percent. A percentile rank tells us the position of a score relative to the others in the data set, and answers questions such as “What percentage of scores are below mine?”

Suppose your son scores a 19 out of 20 on his Spanish test, and you’d like to know how he did compared to the rest of the class. This, like a percentile point, can be found using either interpolation or a formula. Using the following information, I will demonstrate:

Scores *f* Cum*f* Cum %

20 – 21 4 24 100.00

**18 –** **19** 8 20 **83.33**

16 – 17 3 12** ** **50.00**

14 – 15 7 9 37.50

12 – 13 1 2 8.33

10 – 11 1 1 4.16

n = 24

Again, we have two scales (percent and scores) and we want to use information that we know about one scale to find what we want to know on the other one. In this case, we use what we know about the position of his score to figure out the position on the percentage scale.

First, its easy to see that the distance covered on the score interval is 2 (18 & 19), and we can also see that the distance between the percentage of people scoring in that range and the one below it is 83.33-50 = 33.33.

Now, we need to know how far down on the scale is the score of 19. Since the true upper limit is 19.5 and the true lower limit is 17.5, the score of 19 is .50 down the scale (19.5 – 19). This means that your son’s score of 19 is .25 or ¼ of the way down this scale (.50/2 = .25).

To find what percentage goes with his score of 19, we need to find the percentage that is ¼ of the way down between 83.33% and 50% (on the percent scale). No problem, right? When we subtract we find that the distance between them is 33.33, and ¼ of this is .25 x 33.33 = 8.33%.

Then, when we subtract our 8.33% from the upper percentage on the scale (83.33-8.33), we get 75%. From this, we now know that your son’s score of 19 has a percentile rank of 75%. This means that he scored better than 75% of the others who took the same test.

Sound familiar? It should, because in the previous blog post we found that the score that lies at the 75^{th} percentile was 19. Now you know… Percentile points and percentile ranks are mirror images of each other. Oh, and if you aren’t formula-phobic, you could use this formula to arrive at the same answer:

PR = [(cum*fl + (fi / i)(*X*-X _{L})) N*] (100)

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