Questions

The following table shows the ages of the patients admitted to a hospital during a year:

Age (in years) | 5-15 | 15-25 | 25-35 | 35-45 | 45-55 | 55-65 |

Number of patients | 6 | 11 | 21 | 23 | 14 | 5 |

Find the mode and the mean of the data given above. Compare and interpret the two

measures of central tendency.

detailed solution

Correct option is A

For the modal class, let us the consider the class interval with highest frequency

Here, the greatest frequency = 23, so the modal class = 35 – 45,

l = 35,

class width (h) = 10,

f_{m} = 23,

f_{1} = 21 and f_{2} = 14

The formula to find the mode is

Mode *= l+ [*$\frac{({f}_{m}-{f}_{1})}{2{f}_{m}-{f}_{1}-{f}_{2}}$*]×h*

Substitute the values in the formula, we get

Mode = 35+[$\frac{(23-21)}{46-21-14}$]×10

Mode = 35+($\frac{20}{11}$) = 35+1.8

Mode = 36.8 year

So the mode of the given data = 36.8 year

Calculation of Mean:

First find the midpoint using the formula, x_{i }= $\frac{upperlimit+lowerlimit}{2}$

Class Interval | Frequency (f_{i}) | Mid-point (x_{i}) | f_{i}x_{i} |

5-15 | 6 | 10 | 60 |

15-25 | 11 | 20 | 220 |

25-35 | 21 | 30 | 630 |

35-45 | 23 | 40 | 920 |

45-55 | 14 | 50 | 700 |

55-65 | 5 | 60 | 300 |

Sum f_{i} = 80 | Sum f_{i}x_{i} = 2830 |

The mean formula is

Mean = x̄ = $\frac{{\displaystyle \sum _{}}{f}_{i}{x}_{i}}{\sum _{}{f}_{i}}$

= $\frac{2830}{80}$

= 35.37 years

Therefore, the mean of the given data = 35.37 years

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detailed solution

Correct answer is 1

For the modal class, let us the consider the class interval with highest frequency

Here, the greatest frequency = 23, so the modal class = 35 – 45,

l = 35,

class width (h) = 10,

f_{m} = 23,

f_{1} = 21 and f_{2} = 14

The formula to find the mode is

Mode *= l+ [*$\frac{({f}_{m}-{f}_{1})}{2{f}_{m}-{f}_{1}-{f}_{2}}$*]×h*

Substitute the values in the formula, we get

Mode = 35+[$\frac{(23-21)}{46-21-14}$]×10

Mode = 35+($\frac{20}{11}$) = 35+1.8

Mode = 36.8 year

So the mode of the given data = 36.8 year

Calculation of Mean:

First find the midpoint using the formula, x_{i }= $\frac{upperlimit+lowerlimit}{2}$

Class Interval | Frequency (f_{i}) | Mid-point (x_{i}) | f_{i}x_{i} |

5-15 | 6 | 10 | 60 |

15-25 | 11 | 20 | 220 |

25-35 | 21 | 30 | 630 |

35-45 | 23 | 40 | 920 |

45-55 | 14 | 50 | 700 |

55-65 | 5 | 60 | 300 |

Sum f_{i} = 80 | Sum f_{i}x_{i} = 2830 |

The mean formula is

Mean = x̄ = $\frac{{\displaystyle \sum _{}}{f}_{i}{x}_{i}}{\sum _{}{f}_{i}}$

= $\frac{2830}{80}$

= 35.37 years

Therefore, the mean of the given data = 35.37 years

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