# Percentile

A
**
percentile
**
is a comparison score between a particular score and the scores of the rest of a group. It shows the percentage of scores that a particular score surpassed. For example, if you score
$75$
points on a test, and are ranked in the
$85$
^{
th
}
percentile, it means that the score
$75$
is higher than
$85\%$
of the scores.

The percentile rank is calculated using the formula

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

where $P$ is the desired percentile and $N$ is the number of data points.

**
Example 1:
**

If the scores of a set of students in a math test are $20$ , $30$ , $15$ and $75$ what is the percentile rank of the score $30$ ?

Arrange the numbers in ascending order and give the rank ranging from $1$ to the lowest to $4$ to the highest.

$\begin{array}{|ccccc|}\hline \text{Number}& 15& 20& 30& $ 75$\\ \text{Rank}& 1& 2& 3& 4\\ \hline\end{array}$

Use the formula:

$\begin{array}{l}3=\frac{P}{100}\left(4\right)\\ 3=\frac{P}{25}\\ $ 75$=P\end{array}$

Therefore, the score
$30$
has the
$75$
^{
th
}
percentile.

Note that, if the percentile rank
$R$
is an integer, the
$P$
^{
th
}
percentile would be the score with rank
$R$
when the data points are arranged in ascending order.

If
$R$
is not an integer, then the
$P$
^{
th
}
percentile is calculated as shown.

Let
$I$
be the integer part and be the decimal part of
$D$
of
$R$
. Calculate the scores with the ranks
$I$
and
$I+1$
. Multiply the difference of the scores by the decimal part of
$R$
. The
$P$
^{
th
}
percentile is the sum of the product and the score with the rank
$I$
.

**
Example 2:
**

Determine the
$35$
^{
th
}
percentile of the scores
$7,3,12,15,14,4$
and
$20$
.

Arrange the numbers in ascending order and give the rank ranging from $1$ to the lowest to $7$ to the highest.

$\begin{array}{|cccccccc|}\hline \text{Number}& 3& 4& 7& 12& 14& 15& 20\\ \text{Rank}& 1& 2& 3& 4& 5& 6& 7\\ \hline\end{array}$

Use the formula:

$\begin{array}{l}R=\frac{35}{100}\left(7\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2.45\end{array}$

The integer part of $R$ is $2$ , calculate the score corresponding to the ranks $2$ and $3$ . They are $4$ and $7$ . The product of the difference and the decimal part is $0.45\left(7-4\right)=1.35$ .

Therefore, the
$35$
^{
th
}
percentile is
$2+1.35=3.35$
.