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# Percentile

A percentile may be defined as the comparison score between an individual score and the scores earned by the rest of a group. It reveals the percentage of scores that a given score surpassed. For example, if you earned a 75 on your last math test and ranked in the 85th percentile, that means that your 75 was higher than 85% of the other scores. Tough math test!

## How to calculate the percentile of any score

You can calculate the percentile rank of any score using the following formula:

$R=\frac{P}{100}\left(N\right)$

In this formula, R represents the percentile rank, P is the desired percentile, and N is the total number of data points. Dividing by 100 comes from the word percentile, as "percent" literally means "per 100." It's just like how there are 100 cents in a dollar.

If R is an integer, the Pth percentile would be the score with rank R when the data points are arranged in ascending order. For example, consider the following scores on a math test: 20, 30, 15, and 75. If we arrange them in ascending order (lowest to highest), we get 15, 20, 30, and 75. Next, we assign each a rank such as in the table below:

 Number 15 20 30 75 Rank 1 2 3 4

Now, we can sub in the rank of any of these numbers for R in the formula above to calculate its percentile rank. Let's use 30 as our example. We have a total of 4 data points and 30 earned rank 3, giving us the following equation:

$3=\frac{P}{100}\left(4\right)$

Then, we simply solve for P:

$3=\frac{P}{25}$

$75=P$

A score of 30 represents the 75th percentile for this particular math test. Remember that each data set is entirely independent of other data sets, so you could later get a problem where 30 is the 20th, 40th, 90th, or 100th percentile.

## Calculating percentile scores when R Is not an integer

Let's assume that our data set is 7, 3, 12, 15, 14, 4, and 20. The first step is listing them all in ascending order and assigning each a rank:

 Number 3 4 7 12 14 15 20 Rank 1 2 3 4 5 6 7

Next, let's say that we're looking for the 35th percentile of these scores. We know that P is 35 and N is 7 since we have a total of seven data points, There are actually a number of different ways of calculating percentiles, we present the same formula used by most spreadsheet software such as Excel or Google Sheets; be sure to check with your teacher or textbook if they use a different definition:

$R=\frac{P}{100}\left(n-1\right)+1$

$R=\frac{35}{100}\left(6\right)+1=3.1$

Now, we simply solve for R and get 3.1.

Since 3.1 is between 3 and 4 we need to find the "weighted average" of the values assigned to rank 3 and 4, and we want to go "0.1 of the way" between them using the formula ${x}_{i}×\left(1-d\right)+{x}_{\mathrm{i+1}}×d$ , where d is the truncated decimal part and i is the integer part.

In this case, $i=3$ , so we look at rank 3 for ${x}^{3}$ and rank 4 for ${x}^{4}$ , and $d=0.1$

$7×\left(1-0.1\right)+12×\left(0.1\right)=7.5$

So the 35th percentile of the dataset is 7.5.

## Practice Questions

a. Given the data set 1, 7, 5, 6, 9 what is the percentile rank of 6?

50th percentile

b. Given the following table, what is the percentile rank of the value 27?

 # 3 6 9 13 17 18 25 27 34 42 Rank 1 2 3 4 5 6 7 8 9 10

80th percentile

c. Given the data set 2, 4, 6, 8 what score would represent the 45th percentile?

4.7

Percent

## Flashcards covering the Percentile

Statistics Flashcards

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