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 must 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 Decile 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: Computing Decile for Grouped Data

Calculate the first and third decile(s) of the following 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*}

Importance of Deciles in Statistics and Data Analysis

Deciles are a valuable statistical measure used to divide a dataset into ten equal parts, each representing 10% of the data distribution. They help in understanding the spread, variability, and central tendencies within a dataset. Below are key reasons why deciles are important:

1. Data Distribution Analysis: Deciles provide a clear breakdown of how data is distributed across different segments. They help identify whether data is skewed, uniform, or concentrated in certain ranges.
2. Comparison of Data Sets: By comparing deciles across different datasets, analysts can assess differences in distributions (e.g., income levels, test scores, or sales performance). Useful in benchmarking (e.g., comparing a company’s performance against industry deciles).
3. Identifying Outliers and Extremes: The 1st decile ($D_1$) and 9th decile ($D_9$) help detect unusually low or high values. Useful in finance (e.g., risk assessment) and healthcare (e.g., identifying extreme patient results).
4. Economic and Social Research: Used by Governments and economists to analyze income/wealth inequality (e.g., the top 10% vs. the bottom 10%). Helps in policy-making (e.g., tax brackets, welfare programs).
5. Business and Marketing Applications: Businesses categorize customers into deciles based on spending habits. Helps in targeted marketing (e.g., focusing on the top 10% of high-value customers).
6. Educational and Performance Assessment: Used by Schools and universities to rank student performance (e.g., standardized test scores). Helps identify students needing extra support or advanced programs.
7. Investment and Portfolio Management: Investors analyze stock or fund performance using decile rankings. Helps in risk management by comparing high-risk vs. low-risk assets.
8. Robust Alternative to Percentiles and Quartiles: While quartiles divide data into four parts, they provide finer granularity (10 parts). More detailed than quintiles (5 parts) but less complex than percentiles (100 parts).

Conclusion

Deciles are a simple yet powerful tool for understanding data distributions, making comparisons, and supporting decision-making in fields like economics, business, education, and finance. They offer a balanced approach between simplicity (like quartiles) and extreme detail (like percentiles), making them widely useful in statistical analysis.

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