Sum

The median of the following incomplete table is 92. Find the missing frequencies:

C.I. |
30 – 50 | 50 – 70 | 70 – 90 | 90 – 110 | 110 – 130 | 130 – 150 | Total |

f |
6 | ? | 18 | 20 | ? | 10 | 80 |

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#### Solution

Let a and b be the missing frequencies of the class 50 – 70 and class 110 – 130 respectively.

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

C.I. |
f |
Less than cumulative frequency (c.f.) |

30 – 50 | 6 | 6 |

50 – 70 | a | 6 + a |

70 – 90 | 18 | 24 + a |

90 – 110 | 20 | 44 + a ← Q_{2} |

110 – 130 | B | 44 + a + b |

130 – 150 | 10 | 54 + a + b |

Total |
N = 80 |

Here, N = 54 + a + b

Since, N = 80

∴ 54 + a + b = 80

∴ a + b = 26 .........(i)

Given, Median = Q_{2 }= 92

∴ Q_{2} lies in the class 90 – 110.

∴ L = 90, h = 20, f = 20, `(2"N")/4`=`(2xx80)/4` = 40, c.f. = 24 + a

∴ Q_{2} = `"L"+"h"/"f"((2"N")/4-"c.f.")`

∴ 92 = `90 + 20/20[40 - (24 + "a")]`

∴ 92 – 90 = 40 – 24 – a

∴ 2 = 16 – a

∴ a = 14

Substituting the value of a in equation (i), we get

14 + b = 26

∴ b = 26 – 14 = 12

∴ 14 and 12 are the missing frequencies of the class 50 – 70 and class 110 – 130 respectively.

Concept: Concept of Median

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