Deciles are positional averages that give us important information about the distribution of the data. The nine deciles denoted as D1, D2 …,D9 divide the given data set into ten equal parts.
Deciles for Ungrouped Data:
- In order to calculate the deciles we must first arrange the given raw data in either ascending or descending order.
- We then count the number of values (=N) in the given data set.
- The ith decile Di can then be calculated using the below formula.
Di = (iN/10)th 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.
Di = (iN/10)th term.
D2 = (2*7/10)th term = (1.4)th term = 1st term = 3.
D7 = (7*7/10)th term = (4.9)th term = 5th term = 9.
Deciles for Grouped Data:
- In order to calculate the deciles for grouped data we first calculate the cumulative frequencies.
- We add up all frequencies and denote the sum as N.
- 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).
- The ith decile Di for grouped data is calculated using the below formula.
Di = L + h/f*(iN/10 – c.f).
- 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.
| Class Interval | Frequency |
| 0-10 | 3 |
| 10-20 | 7 |
| 20-30 | 4 |
| 30-40 | 7 |
| 40-50 | 9 |
| 50-60 | 2 |
| 60-70 | 3 |
| 70-80 | 5 |
Solution: We obtain the cumulative frequencies as follows,
| Class Interval | Frequency | Cumulative Frequency |
| 0-10 | 3 | 3 |
| 10-20 | 7 | 10 |
| 20-30 | 4 | 14 |
| 30-40 | 7 | 21 |
| 40-50 | 9 | 30 |
| 50-60 | 2 | 32 |
| 60-70 | 3 | 35 |
| 70-80 | 5 | 40 |
| N=40 |
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.
The frequency of the decile class is 9 and the cumulative frequency of the class just before it is 21.
Here, L=40, h=10, f=9 and c.f = 21.
D6 = L + h/f*(iN/10 – c.f).
D6 = 40 + 10/9*(24 – 21) = 40 + 3.34 = 43.34.
