1.7: Percentiles
- Page ID
- 2066
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives
- Define percentiles
- Use three formulas for computing percentiles
A test score in and of itself is usually difficult to interpret. For example, if you learned that your score on a measure of shyness was \(35\) out of a possible \(50\), you would have little idea how shy you are compared to other people. More relevant is the percentage of people with lower shyness scores than yours. This percentage is called a percentile. If \(65\%\) of the scores were below yours, then your score would be the \(65^{th}\) percentile.
Two Simple Definitions of Percentile
There is no universally accepted definition of a percentile. Using the \(65^{th}\) percentile as an example, the \(65^{th}\) percentile can be defined as the lowest score that is greater than \(65\%\) of the scores. This is the way we defined it above and we will call this "\(\text{Definition 1}\)." The \(65^{th}\) percentile can also be defined as the smallest score that is greater than or equal to \(65\%\) of the scores. This we will call "\(\text{Definition 2}\)." Unfortunately, these two definitions can lead to dramatically different results, especially when there is relatively little data. Moreover, neither of these definitions is explicit about how to handle rounding. For instance, what rank is required to be higher than \(65\%\) of the scores when the total number of scores is \(50\)? This is tricky because \(65\%\) of \(50\) is \(32.5\). How do we find the lowest number that is higher than \(32.5\) of the scores? A third way to compute percentiles (presented below) is a weighted average of the percentiles computed according to the first two definitions. This third definition handles rounding more gracefully than the other two and has the advantage that it allows the median to be defined conveniently as the \(50^{th}\) percentile.
Third Definition
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 \(25^{th}\) percentile for the \(8\) numbers in Table \(\PageIndex{1}\). 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 \(25^{th}\) percentile. This is done using the following formula:
\[R = P/100 \times (N + 1)\]
where \(P\) is the desired percentile (\(25\) in this case) and \(N\) is the number of numbers (\(8\) in this case). Therefore,
\[R = 25/100 \times (8 + 1) = 9/4 = 2.25\]
If \(R\) is an integer, the \(P^{th}\) percentile is the number with rank \(R\). When \(R\) is not an integer, we compute the \(P^{th}\) 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, \(IR=2\).
- Define \(FR\) as the fractional portion of \(R\). For this example, \(FR=0.25\).
- Find the scores with Rank \(IR\) and with Rank \(IR+1\). 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 \((0.25)(7 - 5) + 5 = 5.5\).
Therefore, the \(25^{th}\) percentile is \(5.5\). If we had used the first definition (the smallest score greater than \(25\%\) of the scores), the \(25^{th}\) 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 \(25^{th}\) percentile would have been \(5\).
For a second example, consider the \(20\) quiz scores shown in Table \(\PageIndex{2}\).
| Number | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 9 | 9 | 9 | 10 | 10 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
We will compute the \(25^{th}\) and the \(85^{th}\) percentiles. For the \(25^{th}\),
\[R = 25/100 \times (20 + 1) = 21/4 = 5.25\]
\[IR=5\; and\; FR=0.25\]
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 \(25^{th}\) percentile is \(5\). In terms of the formula:
\[25^{th}\; \text{percentile} = (0.25) \times (5 - 5) + 5 = 5\]
For the \(85^{th}\) percentile,
\[R = 85/100 \times (20 + 1) = 17.85.\]
\[IR = 17\; and\; FR = 0.85\]
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 \(85^{th}\) percentile is:
\[(0.85)(10 - 9) + 9 = 9.85\]
Consider the \(50^{th}\) percentile of the numbers \(2, 3, 5, 9\).
\[R = 50/100 \times (4 + 1) = 2.5\]
\[IR=2\; and\; FR=0.5\]
The score with a rank of \(IR\) is \(3\) and the score with a rank of \(IR+1\) is \(5\). Therefore, the \(50^{th}\) percentile is:
\[(0.5)(5 - 3) + 3 = 4\]
Finally, consider the \(50^{th}\) percentile of the numbers \(2, 3, 5, 9, 11\).
\[R = 50/100 \times (5 + 1) = 3\]
\[IR=3\; and\; FR=0\]
Whenever \(FR=0\), you simply find the number with rank \(IR\). In this case, the third number is equal to \(5\), so the \(50^{th}\) percentile is \(5\). You will also get the right answer if you apply the general formula:
\[50^{th}\; \text{percentile} = (0.00) (9 - 5) + 5 = 5\]
Contributors and Attributions
Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University.
- David M. Lane