The deciles: Measure of Position

The deciles are the values (nine in numbers) of the variable that divide an ordered (sorted, arranged) data set into ten equal parts so that each part represents 1/10 of the sample or population, and are denoted by $D_1, D_2, \cdots D_9$, where First decile (D₁) is the value of order statistics that exceed 1/10 of the observations and less than the remaining 9/10 and the D₉ (ninth decile) is the value in order statistic that exceeds 9/10 of the observations and is less than 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 for Ungrouped Data

To calculate deciles for the ungrouped data, first order the all observation according to the magnitudes of the values, then use the following formula for mth decile.

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

Example: Calculate 2nd and 8th deciles of 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 have to more 0.2 of the distance between the sixth and seventh values. i.e. the value of 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 D₈  can be calculated. D₈ = 52.6.

Calculating for Grouped Data

The mth decile for grouped data (in ascending order) can be calculated from the following formula.

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

where

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

Example: Calculate the first and third deciles of the following grouped data

Solution: Deciles class for D₁ can be calculated from $\left(\frac{m.n}{10}-c\right) = \frac{1 \times 30}{10} = 3$rd observation. As 3rd observation lie in 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*}

Deciles class for D₇ 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*}