## Percentile Points

Percentile points and ranks are used for comparison within a group, and are commonly used in education. For example, if your daughter comes home with her state exam grade for math showing she is at the 90^{th} percentile for math, it means that she is scoring better than 90% of the others who took the same test. In other words, her percentile rank is 90%. Percentile points give us similar information, but are used a little differently.

A percentile point is the score below which a certain percentage of the other scores fall. When we calculate a percentile point we will answer the question “What score in the data set is at the 75^{th} percentile?” An educator might want to know this if he plans to implement remedial help to all of his students who are scoring in the bottom 75%. In this case, he begins with a given percent with the goal of finding the corresponding score. This can easily be done using a frequency table and either interpolation or a formula. The interpolation method is quite simple, really, and may be more acceptable to those who fear mathematical formulas. Interpolation involves using knowledge of one scale of measurement (such as percentage known, in this case 75%) and applying it in the same way to the other scale of measurement (such as students’ scores). Here is how it works. Suppose we have the following frequency table:

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

We have two scales here: scores and percentiles (cum %). So we need to find the fraction for the one we know (percent) and apply it to the score scale.

It’s easy to see that the 75% percentile is somewhere between 83.33% and 50% (see in bold). This tells us that the score the teacher is looking for is somewhere in the interval of 18-19. We must remember that the true limits of this interval, however, are 17.5 and 19.5, with the width being 2 (this will be important in a jif!).

The top percentage is 83.33 and the one below it is 50, so that is a difference of 33.33 percentage points. The one we want is 75%, which is 8.33 points from the 83.33, so the fraction will be 8.33/33.33, which translates to .25 or 25% or 1/4.

Now we know that the score we are looking for is 1/4 of the way down the interval of 18-19. And the true limits of this interval are 17.5 and 19.5, with a width of 2. Therefore:

2 x .25 = .50

The value of 2 is the interval width, and by multiplying it by our .25, we find that .50 is the distance we need to go down from the true upper limit of the score range of 18-19. So we do this by 19.5 –.50 = 19. Now we find that the score at the 75^{th} percentile is 19. So, this fictitious teacher would place everyone who scored 19 and below to the remedial lessons.

If you aren’t one of the masses who is afraid of formulas, you can also arrive at the same answer with this:

PP = *X _{L}* + (

*i*/

*fi*)(cum

*fp*– cum

*f*)

_{L}Next time, I’ll show you how to arrive at any percentile rank you desire.

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