*Deciles (Measures of Positions)*

*Deciles (Measures of Positions)*

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. Deciles are denoted by* D _{1}*,

*D*,

_{2}*D*,…

_{3}*D*, where First decile

_{10}*(D*) is the value of order statistics that exceeds 1/10 of the observations and less than the remaining 9/10 and the

_{1}*D*(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 is equal to median. The deciles determine the values for 10%, 20%… and 90% of the data.

_{9}**Calculating Deciles 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 *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 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 *D _{8 }* can be calculated.

*D*= 52.6.

_{8}**Calculating Deciles for grouped Data**

The *m*th 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 *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 to the class containing *m*th deciles

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

**Solution:** Deciles 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 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 *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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**Deciles (Measures of Positions) 141.95 KB**

**Deciles (Measures of Positions) 141.95 KB**