In the frequency distribution of families given below, the number of families corresponding to expenditure group 2000 - 4000 is missing from the table. However value of 25^{th} percentile is 2880. Find the missing frequency.

Weekly Expenditure (₹1000) |
0 – 2 | 2 – 4 | 4 – 6 | 6 – 8 | 8 – 10 |

No. of families |
14 | ? | 39 | 7 | 15 |

#### Solution

Let x be the missing frequency of expenditure group 2000 – 4000.

We construct the less than cumulative frequency table as given below:

Weekly Expenditure |
No. of families (f) |
Less than cumulative frequency (c.f.) |

0 – 2000 | 14 | 14 |

2000 – 4000 | x | 14 + x ← P_{25} |

4000 – 6000 | 39 | 53 + x |

6000 – 8000 | 7 | 60 + x |

8000 – 10000 | 15 | 75 + x |

Total |
75 + x |

Here, N = 75 + x

Given, P_{25} = 2880

∴ P_{25} lies in the class 2000 – 4000.

∴ L = 2000, h = 2000, f = x, c.f. = 14

∴ P_{25} = `"L"+"h"/"f"((25"N")/100-"c.f.")`

∴ 2880 = `2000+2000/"x"((75+"x")/4-14)`

∴ 2880 – 2000 =`2000/"x"((75+"x"-56)/4)`

∴ 880x = 500(x + 19)

∴ 880x = 500x + 9500

∴ 880x – 500x = 9500

∴ 380x = 9500

∴ x = `9500/380` = 25

∴ 25 is the missing frequency of the expenditure group 2000 – 4000.