# Deciles and Percentiles

Deciles

Deciles are the partition values which divide the set of observations into ten equal parts. There are nine deciles: ${D_1},{D_2},{D_3}, \ldots ,{D_9}$. The first decile is ${D_1}$, which is a point which has 10% of the observations below it.

Deciles for Individual Observations (Ungrouped Data)

$\begin{gathered} {D_1} = {\text{Value of }}{\left( {\frac{{n + 1}}{{10}}} \right)^{th}}{\text{item}} \\ {D_2} = {\text{Value of 2}}{\left( {\frac{{n + 1}}{{10}}} \right)^{th}}{\text{item}} \\ {D_3} = {\text{Value of 3}}{\left( {\frac{{n + 1}}{{10}}} \right)^{th}}{\text{item}} \\ \vdots \\ {D_9} = {\text{Value of 9}}{\left( {\frac{{n + 1}}{{10}}} \right)^{th}}{\text{item}} \\ \end{gathered}$

Quartile for a Frequency Distribution (Discrete Data)

$\begin{gathered} {D_1} = {\text{Value of }}{\left( {\frac{{n + 1}}{{10}}} \right)^{th}}{\text{item}}\left( {n = \sum f} \right) \\ {D_2} = {\text{Value of 2}}{\left( {\frac{{n + 1}}{{10}}} \right)^{th}}{\text{item}} \\ {D_3} = {\text{Value of 3}}{\left( {\frac{{n + 1}}{{10}}} \right)^{th}}{\text{item}} \\ \vdots \\ {D_9} = {\text{Value of 9}}{\left( {\frac{{n + 1}}{{10}}} \right)^{th}}{\text{item}} \\ \end{gathered}$

Quartile for Grouped Frequency Distribution

$\begin{gathered} {D_1}{\text{ = }}l + \frac{h}{f}\left( {\frac{n}{{10}} – c} \right){\text{ }}\left( {n = \sum f} \right) \\ {D_2} = l + \frac{h}{f}\left( {\frac{{2n}}{{10}} – c} \right) \\ {D_3} = l + \frac{h}{f}\left( {\frac{{3n}}{{10}} – c} \right) \\ \vdots \\ {D_9} = l + \frac{h}{f}\left( {\frac{{9n}}{{10}} – c} \right) \\ \end{gathered}$

Percentiles

Percentiles are the points which divide the set of observations into one hundred equal parts. These points are denoted by ${P_1},{P_2},{P_3}, \ldots ,{P_{99}}$, and are called the first, second, third… ninety ninth percentile. The percentiles are calculated for a very large number of observations like workers in factories and the populations in provinces or countries. Percentiles are usually calculated for grouped data. The first percentile denoted by ${P_1}$ is calculated as ${P_1} = {\text{Value of }}{\left( {\frac{n}{{100}}} \right)^{th}}{\text{item}}$. We find the group in which the ${\left( {\frac{n}{{100}}} \right)^{th}}$ item lies and then ${P_1}$ is interpolated from the formula.

$\begin{gathered}{P_1}{\text{ = }}l + \frac{h}{f}\left( {\frac{n}{{100}} – c} \right){\text{ }}\left( {n = \sum f} \right) \\ {P_2} = l + \frac{h}{f}\left( {\frac{{2n}}{{100}} – c} \right) \\ {P_3} = l + \frac{h}{f}\left( {\frac{{3n}}{{100}} – c} \right) \\ \vdots \\ {P_{99}} = l + \frac{h}{f}\left( {\frac{{99n}}{{100}} – c} \right) \\ \end{gathered}$