Deciles and Percentiles

Deciles:

The deciles are the partition values which divides the set of observations into tem equal parts. There are nine deciles namely {D_1},{D_2},{D_3}, \ldots ,{D_9}. The first decile is {D_1} 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:

The 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 percentiles. The percentiles are calculated for very large number of observations like workers in factories and the population in provinces or countries. The 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}}itemlies 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}