MA121: Introduction to Statistics

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: