Appendix 1.A Calculating Percentiles Under the Third Definition
Let’s begin with an example. Consider the 25th percentile for the 8 numbers in Table 1.A.15. Notice the numbers are given ranks ranging from 1 for the lowest number to 8 for the highest number.
| Number | 3 | 5 | 7 | 8 | 9 | 11 | 13 | 15 |
|---|---|---|---|---|---|---|---|---|
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
The first step is to compute the rank (\(R\)) of the 25th percentile. This is done using the following formula:
\begin{equation*}
R = P/100 \times (N + 1)
\end{equation*}
where \(P\) is the desired percentile (25 in this case) and \(N\) is the number of numbers (8 in this case). Therefore,
\begin{equation*}
R = 25/100 \times (8 + 1) = 9/4 = 2.25.
\end{equation*}
If \(R\) is an integer, the \(P^\text{th}\) percentile is the number with rank \(R\text{.}\) When \(R\) is not an integer, we compute the \(P^\text{th}\) percentile by interpolation as follows:
-
Define \(IR\) as the integer portion of \(R\) (the number to the left of the decimal point). For this example, \(IR = 2\text{.}\)
-
Find the scores with Rank \(IR\) and with Rank \(IR + 1\text{.}\) For this example, this means the score with Rank 2 and the score with Rank 3. The scores are 5 and 7.
-
Interpolate by multiplying the difference between the scores by \(FR\) and add the result to the lower score. For these data, this is \((0.25)(7 - 5) + 5 = 5.5\text{.}\)
Therefore, the 25th percentile is 5.5. If we had used the first definition (the smallest score greater than 25% of the scores), the 25th percentile would have been 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 5.
For a second example, consider the 20 quiz scores shown in Table 1.A.16.
| Score | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|---|---|
| Score | 7 | 7 | 8 | 8 | 9 | 9 | 9 | 10 | 10 | 10 |
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Rank | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
We will compute the 25th and the 85th percentiles. For the 25th,
\begin{equation*}
R = 25/100 \times (20 + 1) = 21/4 = 5.25.
\end{equation*}
\begin{equation*}
IR = 5 \text{ and } FR = 0.25.
\end{equation*}
Since the score with a rank of \(IR\) (which is 5) and the score with a rank of \(IR + 1\) (which is 6) are both equal to 5, the 25th percentile is 5. In terms of the formula:
\begin{equation*}
\text{25th percentile} = (.25) \times (5 - 5) + 5 = 5.
\end{equation*}
For the 85th percentile,
\begin{equation*}
R = 85/100 \times (20 + 1) = 17.85.
\end{equation*}
\begin{equation*}
IR = 17 \text{ and } FR = 0.85
\end{equation*}
Caution: \(FR\) does not generally equal the percentile to be computed as it does here.
The score with a rank of 17 is 9 and the score with a rank of 18 is 10. Therefore, the 85th percentile is:
\begin{equation*}
(0.85)(10 - 9) + 9 = 9.85
\end{equation*}
Consider the 50th percentile of the numbers 2, 3, 5, 9.
\begin{equation*}
R = 50/100 \times (4 + 1) = 2.5.
\end{equation*}
\begin{equation*}
IR = 2 \text{ and } FR = 0.5.
\end{equation*}
The score with a rank of \(IR\) is 3 and the score with a rank of \(IR + 1\) is 5. Therefore, the 50th percentile is:
\begin{equation*}
(0.5)(5 - 3) + 3 = 4.
\end{equation*}
Finally, consider the 50th percentile of the numbers 2, 3, 5, 9, 11.
\begin{equation*}
R = 50/100 \times (5 + 1) = 3.
\end{equation*}
\begin{equation*}
IR = 3 \text{ and } FR = 0.
\end{equation*}
Whenever \(FR = 0\text{,}\) you simply find the number with rank \(IR\text{.}\) In this case, the third number is equal to 5, so the 50th percentile is 5. You will also get the right answer if you apply the general formula:
\begin{equation*}
\text{50th percentile} = (0.00)(9 - 5) + 5 = 5.
\end{equation*}
