**Solve the following :**

Calculate of Q_{3}, D_{5}, and P_{35} for the given data

Income (in lakhs ₹) |
1 | 2 | 3 | 4 | 5 | 6 | |

No. of family |
2 | 5 | 20 | 25 | 15 | 12 |

#### Solution

Arrange the data of the income in ascending order.

(i.e. from lowest number to highest as shown below) and the out the cumulative frequency.

Income (in lakhs RS) |
no. of family |
cumulative frequency(cf) |

1 | 2 | 2 |

2 | 5 | 7 |

3 | 20 | 27 |

4 | 25 | 52 |

5 | 15 | 67 |

6 | 12 | 79 |

N = 79 |

Q_{3} = Size of 3`((n +1)/4)^"th observation"`

= size of 3`((79+1)/4)^"th observation"`

= size of 3`(80/4)^"th observation"`

= size of 3 `(20)^"th observation"`

= size of 60^{th observation} lies in cf 67

Hence, quartile value = Rs .5 lakhs Q_{3 }= Rs. 5 Lakhs

D_{5 }= size of 5`((n+1)/10)^"th observation"`

=size of 5`((79+1)/10)^"th observation"`

= size of 5`(80/10)^"th observation"`

= size of 5`(8)^"th observation"`

= size of `(40)^"th observation"`

Size of 40^{th observation} lies in cf 52 , hence decile value =Rs. 4 Lakhs

D_{5 }=Rs. 4 Lakhs

P_{35} =Size of 35`((n +1)/100)^"th observation"`

P_{35 }=Size of 35`((79 +1)/100)^"th observation"`

P_{35 }=Size of 35`(80/100)^"th observation"`

=Size of 35 `(0.80)^"th observation"`

=Size of 28^{th observation} lies in cf 52

**Hence, percentile value = Rs.4 lakhs P _{35} = Rs. 4 lakhs.**