Percentiles

Unless otherwise specified, when we refer to "percentile," we will be referring to this third definition of percentiles. Let's begin with an example. Consider the 25th percentile for the 8 numbers in Table 1. Notice the numbers are given ranks ranging from $\mathrm{1}$ for the lowest number to $\mathrm{8}$ for the highest number.

The first step is to compute the rank ($\mathrm{R}$) of the 25th percentile. This is done using the following formula:

where $\mathrm{P}$ is the desired percentile ($\mathrm{25}$ in this case) and $\mathrm{N}$ is the number of numbers ($\mathrm{8}$ in this case). Therefore,

If $\mathrm{R}$ is an integer, the $\mathrm{Pth}$ percentile is the number with rank $\mathrm{R}$. When $\mathrm{R}$ is not an integer, we compute the $\mathrm{Pth}$ percentile by interpolation as follows:

1. Define $\mathrm{IR}$ as the integer portion of $\mathrm{R}$ (the number to the left of the decimal point). For this example, $\mathrm{IR=2}$.

2. Define $\mathrm{FR}$ as the fractional portion of $\mathrm{R}$. For this example, $\mathrm{FR}=0.25$.

3. Find the scores with Rank $\mathrm{IR}$ and with Rank $\mathrm{IR \, + \, 1}$. For this example, this means the score with Rank 2 and the score with Rank 3. The scores are $\mathrm{5}$ and $\mathrm{7}$.

4. Interpolate by multiplying the difference between the scores by $\mathrm{F}_{\mathrm{R}}$ and add the result to the lower score. For these data, this is $(0.25)(7-5)+5=5.5$.

Therefore, the 25th percentile is $\mathrm{5.5}$. If we had used the first definition (the smallest score greater than 25% of the scores), the 25th percentile would have been $\mathrm{7}$. If we had used the second definition (the smallest score greater than or equal to 25% of the scores), the 25th percentile would have been $\mathrm{5}$.

For a second example, consider the 20 quiz scores shown in Table 2.

We will compute the 25th and the 85th percentiles. For the 25th,

Caution: $\mathrm{FR}$  does not generally equal the percentile to be computed as it does here.

The score with a rank of $\mathrm{17}$ is $\mathrm{9}$ and the score with a rank of $\mathrm{18}$ is $\mathrm{10}$. Therefore, the 85th percentile is:

Consider the 50th percentile of the numbers $\mathrm{2, \, 3, \, 5, \, 9}$.

The score with a rank of IR is $\mathrm{3}$ and the score with a rank of $\mathrm{IR + 1}$ is $\mathrm{5}$. Therefore, the 50th percentile is:

Finally, consider the 50th percentile of the numbers $\mathrm{2,  \, 3,  \, 5,  \, 9, \, 11}$.

Whenever $\mathrm{FR \, = \, 0}$, you simply find the number with rank $\mathrm{IR}$. In this case, the third number is equal to $\mathrm{5}$, so the 50th percentile is $\mathrm{5}$. You will also get the right answer if you apply the general formula: