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:**

- 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 i
^{th}decile D_i can then be calculated using the below formula.

**Decile Formula for Ungrouped Data:**

The i^{th} 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 i

^{th}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*}

**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 i
^{th}decile lies. The i^{th}decile lies in the class interval whose cumulative frequency is just greater than (iN/10). - We can then obtain the value of the i
^{th}decile by the formula below.

**Decile Formula for Grouped Data:**

The i^{th} 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.

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=6^{th} decile. Now (iN/10) = (6*40/10) = 24. Since 30 is just greater than 24 in the cumulative frequency column we conclude that the 6^{th} 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