# Deciles for Grouped & Ungrouped Data – Calculation | Examples

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Deciles are positional averages that give us important information about the distribution of the data. The nine deciles denoted as D_1,D_2,\ldots,D_9 divide the given data set into ten equal parts.

For example, the third decile D_3 divides the given data set into two parts such that 30% of the data values lie below it and the remaining 70% of the data lie above that value. Similarly, the eight decile D_8 divides the data into two parts with 80% of the values lying below it and 20% of the values lying above it. We now explain how to calculate the deciles for ungrouped and grouped data sets.

## Deciles for Ungrouped Data:

1. In order to calculate the deciles we must first arrange the given raw data in either ascending or descending order.
2. We then count the number of values (=N) in the given data set.
3. The ith decile D_i can then be calculated using the below formula.

### Decile Formula for Ungrouped Data:

The ith decile D_i for the given data set is calculated using the formula, D_i=\left (\frac{iN}{10}\right )^{th}\text{ term}.

#### Example:

Calculate the second and seventh deciles for the given data values: 4, 3, 8, 6, 9, 11, 10.

Solution: We first arrange the given data set in ascending order as follows, 3, 4, 6, 8, 9, 10, 11. The number of terms is equal to N = 7. We now obtain the required values. The ith decile D_i for the given data set is calculated using the formula, \begin{align*}D_2 &=\left (\frac{2N}{10}\right )^{th}\text{ term} \\ &= \left (\frac{2\times 7}{10}\right )^{th}\text{ term} \\ &= (1.4)^{th}\text{ term} = 1^{st}\text{ term} = 3 \end{align*}

\begin{align*}D_7 &=\left (\frac{7N}{10}\right )^{th}\text{ term} \\ &= \left (\frac{7\times 7}{10}\right )^{th}\text{ term} \\ &= (4.9)^{th}\text{ term} = 5^{th}\text{ term} = 9 \end{align*}

## Deciles for Grouped Data:

1. In order to calculate the deciles for grouped data we first calculate the cumulative frequencies.
2. We add up all frequencies and denote the sum as N.
3. The formula (iN/10) tells us in what class interval the ith decile lies. The ith decile lies in the class interval whose cumulative frequency is just greater than (iN/10).
4. We can then obtain the value of the ith decile by the formula below.

### Decile Formula for Grouped Data:

The ith decile D_i for the given data set is calculated using the formula, D_i = L + \frac{h}{f}\left (\frac{iN}{10} - c.f \right )

where, L = lower limit of the class interval containing the given decile.

h = class size of the class interval.

f = frequency of class interval.

c.f = cumulative frequency of the preceding class interval.

#### Example:

Calculate the sixth decile for the given data set.

Solution: We obtain the cumulative frequencies as follows,

We want to calculate the i=6th decile. Now (iN/10) = (6*40/10) = 24. Since 30 is just greater than 24 in the cumulative frequency column we conclude that the 6th decile lies in the class interval 40-50.

Here, L=40, h=10, f=9 and c.f = 21. Applying the formula we get, D_6 = L + \frac{h}{f}\left (\frac{6\times N}{10} - c.f \right )

D_6 = 40 + \frac{10}{9} (24 - 21) = 40 + 3.34 = 43.34

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