# Percentile

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 $2+1.35=3.35$ .