The Deciles: Measure of Position

The deciles are the values (nine in number) of the variable that divides an ordered (sorted, arranged) data set into ten equal parts so that each part represents $\frac{1}{10}$ of the sample or population and are denoted by $D_1, D_2, \cdots D_9$, where First decile ($D_1$) is the value of order statistics that exceed 1/10 of the observations and less than the remaining $\frac{9}{10}$. The $D_9$ (ninth decile) is the value in order statistic that exceeds $\frac{9}{10}$ of the observations and is less than $\frac{1}{10}$ remaining observations. Note that the fifth deciles are equal to the median. The deciles determine the values for 10%, 20%… and 90% of the data.

Calculating Deciles for Ungrouped Data

To calculate the decile for the ungrouped data, first order all observations according to the magnitudes of the values, then use the following formula for $m$th decile.

\[D_m= m \times \left( \frac{(n+1)}{10} \right) \mbox{th value; } \qquad \mbox{where} m=1,2,\cdots,9\]

Example: Calculate the 2nd and 8th deciles of the following ordered data 13, 13,13, 20, 26, 27, 31, 34, 34, 34, 35, 35, 36, 37, 38, 41, 41, 41, 45, 47, 47, 47, 50, 51, 53, 54, 56, 62, 67, 82.
Solution:

\begin{eqnarray*}
D_m &=&m \times \{\frac{(n+1)}{10} \} \mbox{th value}\\
&=& 2 \times \frac{30+1}{10}=6.2\\
\end{eqnarray*}

We have to locate the sixth value in the ordered array and then move 0.2 of the distance between the sixth and seventh values. i.e. the value of the 2nd decile can be calculated as
\[6 \mbox{th observation} + \{7 \mbox{th observation} – 6 \mbox{th observation} \}\times 0.2\]
as 6th observation is 27 and 7th observation is 31.
The second decile would be $27+\{31-27\} \times 0.2 = 27.8$

Similarly, $D_8$ can be calculated. $D_8=52.6$.

Calculating Deciles for Grouped Data

The following formula can calculate the $m$th decile for grouped data (in ascending order).

\[D_m=l+\frac{h}{f}\left(\frac{m.n}{10}-c\right)\]

where

$l$ = is the lower class boundary of the class containing $m$th deciles
$h$ = is the width of the class containing $m$th deciles
$f$ = is the frequency of the class containing $m$th deciles
$n$ = is the total number of frequencies
$c$ = is the cumulative frequency of the class preceding the class containing $m$th deciles

Example: Calculate the first and third decile(s) of the following grouped data

Deciles for Grouped data

Solution: The Decile class for $D_1$ can be calculated from $\left(\frac{m.n}{10}-c\right) = \frac{1 \times 30}{10} = 3$rd observation. As 3rd observation lies in the first class (first group) so

\begin{eqnarray*}
D_m&=&l+\frac{h}{f}\left(\frac{m.n}{10}-c\right)\\
D_1&=&85.5+\frac{5}{6}\left(\frac{1\times30}{10}-0\right)\\
&=&88\\
\end{eqnarray*}

The Decile class for $D_7$ is 100.5—105.5 as $\frac{7 \times 30}{10}=21$th observation which is in fourth class (group).
\begin{eqnarray*}
D_m&=&l+\frac{h}{f}\left(\frac{m.n}{10}-c\right)\\
D_7&=&100.5+\frac{5}{6}\left(\frac{7\times30}{10}-20\right)\\
&=&101.333\\
\end{eqnarray*}

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