Deciles are positional averages that give us important information about the distribution of the data. The nine deciles denoted as D_{1}, D_{2} …,D_{9} 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 i
^{th}decile D_{i}can then be calculated using the below formula.

D_{i} = (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.

D_{i} = (iN/10)th term.

D_{2} = (2*7/10)th term = (1.4)th term = 1st term = 3.

D_{7} = (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 i
^{th}decile lies. The i^{th}decile lies in the class interval whose cumulative frequency is just greater than (iN/10). - The i
^{th}decile D_{i}for grouped data is calculated using the below formula.

D_{i} = 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=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.

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.

D_{6} = L + h/f*(iN/10 – c.f).

D_{6} = 40 + 10/9*(24 – 21) = 40 + 3.34 = 43.34.