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Sum

Calculate Q. D. for the following data:

X |
24 | 25 | 26 | 27 | 28 | 29 | 30 |

F |
6 | 5 | 3 | 2 | 4 | 7 | 3 |

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#### Solution

X |
F |
c.f. (less than type) |

24 | 6 | 6 |

25 | 5 | 11 ← Q_{1} |

26 | 3 | 14 |

27 | 2 | 16 |

28 | 4 | 20 |

29 | 7 |
27 ← Q |

30 | 3 | 30 |

Total |
30 |

Here, N = 30

Q_{1} = value of `(("N" + 1)/4)^"th"` observation

= value of `((30 + 1)/4)^"th"` observation

= value of (7.75)^{th} observation

Cumulative frequency which is just greater than (or equal to) 7.75 is 11.

∴ Q_{1} = 25

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

= value of `[3((30 + 1)/4)]^"th"` observation

= value of (3 × 7.75)^{th} observation

= value of (23.25)^{th} observation

Cumulative frequency which is just greater than (or equal to) 23.25 is 27.

∴ Q_{3} = 29

∴ Q.D. = `("Q"_3 - "Q"_1)/(2)`

= `(29 - 25)/(2)`

= `(4)/(2)`

∴ Q.D. = 2

Concept: Measures of Dispersion - Quartile Deviation (Semi - Inter Quartile Range)

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