# Long Questions

Marks obtained | 60-75 | 65-70 | 70-75 | 75-80 | 80-85 | 85-90 |
---|---|---|---|---|---|---|

No. of students | 7 | 5 | 8 | 4 | 3 | 3 |

Tabulating given data for calculation of Quartile deviation (QD),

Marks obtained | Number of students (f) | Cumulative frequency (c.f.) |
---|---|---|

60-75 | 7 | 7 |

65-70 | 5 | 12 |

70-75 | 8 | 20 |

75-80 | 4 | 24 |

80-85 | 3 | 27 |

85-90 | 3 | 30 |

N = 30 |

Position of first quartile = $(\frac N4 )^{th}$ item = $(\frac{30}{4})^{th}$ item = 7.5^{th} item

In c.f. column, 12 is just greater than 7.5 so its corresponding class is 65-70.

∴ L = 65, f = 5, c.f. = 7, i = 5

Now, we know,
Q_{1} = `L` + $\frac{\frac N4 , - , c.f.}{f}$ × `i`
or, Q_{1} = 65 + $\frac{7.5 , - , 7}{5}$ × 5

∴ Q_{1} = 65.5

Position of third quartile = $(\frac{3N}{4} )^{th}$ item = $(\frac{3 \, \times \, 30}{4})^{th}$ item = 22.5^{th} item

In c.f. column, 24 is just greater than 22.5 so its corresponding class is 75-80.

∴ L = 75, f = 4, c.f. = 20, i = 5

Now, we know,
Q_{3} = `L` + $\frac{\frac{3N}{4} , - , c.f.}{f}$ × `i`
or, Q_{3} = 75 + $\frac{22.5 , - , 20}{5}$ × 5

∴ Q_{3} = 78.125

Finally,

QD = $\frac{Q_3 \, - \, Q_1}{2}$ = $\frac{78.125 \, - \, 65.5}{2}$ = 6.3132

Weight (kg) | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
---|---|---|---|---|---|---|

No. of person | 5 | 15 | 10 | 8 | 6 | 2 |

Tabulating given data for calculation of Quartile deviation (QD),

Weight | Number of person (f) | Cumulative frequency (c.f.) |
---|---|---|

20-30 | 5 | 5 |

30-40 | 15 | 20 |

40-50 | 10 | 30 |

50-60 | 8 | 38 |

60-70 | 6 | 44 |

70-80 | 2 | 46 |

N = 46 |

Position of first quartile = $(\frac N4 )^{th}$ item = $(\frac{46}{4})^{th}$ item = 11.5^{th} item

In c.f. column, 20 is just greater than 11.5 so its corresponding class is 30-40.

∴ L = 30, f = 15, c.f. = 5, i = 10

Now, we know,
Q_{1} = `L` + $\frac{\frac N4 , - , c.f.}{f}$ × `i`
or, Q_{1} = 30 + $\frac{11.5 , - , 5}{15}$ × 10

∴ Q_{1} = 34.33

Position of third quartile = $(\frac{3N}{4} )^{th}$ item = $(\frac{3 \, \times \, 46}{4})^{th}$ item = 26.25^{th} item

In c.f. column, 38 is just greater than 34.5 so its corresponding class is 50-60.

∴ L = 50, f = 8, c.f. = 30, i = 10

Now, we know,
Q_{3} = `L` + $\frac{\frac{3N}{4} , - , c.f.}{f}$ × `i`
or, Q_{3} = 50 + $\frac{34.5 , - , 30}{8}$ × 10

∴ Q_{3} = 55.63

Finally,

QD = $\frac{Q_3 \, - \, Q_1}{2}$ = $\frac{55.63 \, - \, 34.33}{2}$ = 10.65 kg

Class interval | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
---|---|---|---|---|---|

Frequency | 8 | 16 | 4 | 4 | 3 |

Tabulating given data for calculation of Quartile deviation (QD),

Class interval | Frequency (f) | Cumulative frequency (c.f.) |
---|---|---|

20-30 | 8 | 8 |

30-40 | 16 | 24 |

40-50 | 4 | 28 |

50-60 | 4 | 32 |

60-70 | 3 | 35 |

N = 35 |

Position of first quartile = $(\frac N4 )^{th}$ item = $(\frac{35}{4})^{th}$ item = 8.75^{th} item

In c.f. column, 24 is just greater than 8.75 so its corresponding class is 30-40.

∴ L = 30, f = 16, c.f. = 8, i = 10

Now, we know,
Q_{1} = `L` + $\frac{\frac N4 , - , c.f.}{f}$ × `i`
or, Q_{1} = 30 + $\frac{8.75 , - , 8}{16}$ × 10

∴ Q_{1} = 30.47

Position of third quartile = $(\frac{3N}{4} )^{th}$ item = $(\frac{3 \, \times \, 35}{4})^{th}$ item = 26.25^{th} item

In c.f. column, 28 is just greater than 26.25 so its corresponding class is 40-50.

∴ L = 40, f = 4, c.f. = 24, i = 10

Now, we know,
Q_{3} = `L` + $\frac{\frac{3N}{4} , - , c.f.}{f}$ × `i`
or, Q_{3} = 40 + $\frac{26.25 , - , 24}{4}$ × 10

∴ Q_{3} = 45.63

Finally,

QD = $\frac{Q_3 \, - \, Q_1}{2}$ = $\frac{45.63 \, - \, 30.47}{2}$ = 7.58

Size | 4-8 | 8-12 | 12-16 | 16-20 | 20-24 | 24-28 | 28-32 | 32-36 | 36-40 |
---|---|---|---|---|---|---|---|---|---|

Frequency | 6 | 10 | 18 | 30 | 15 | 12 | 10 | 6 | 5 |

Tabulating given data for calculation of Quartile deviation (QD),

Size | Frequency (f) | Cumulative frequency (c.f.) |
---|---|---|

4-8 | 6 | 6 |

8-12 | 10 | 16 |

12-16 | 18 | 34 |

16-20 | 30 | 64 |

20-24 | 15 | 79 |

24-28 | 12 | 91 |

28-32 | 10 | 101 |

32-36 | 6 | 107 |

36-40 | 5 | 112 |

N = 112 |

Position of first quartile = $(\frac N4 )^{th}$ item = $(\frac{112}{4})^{th}$ item = 28^{th} item

In c.f. column, 34 is just greater than 28 so its corresponding class is 12-16.

∴ L = 12, f = 18, c.f. = 16, i = 4

Now, we know,
Q_{1} = `L` + $\frac{\frac N4 , - , c.f.}{f}$ × `i`
or, Q_{1} = 12 + $\frac{28 , - , 16}{18}$ × 4

∴ Q_{1} = 14.67

Position of third quartile = $(\frac{3N}{4} )^{th}$ item = $(\frac{3 \, \times \, 112}{4})^{th}$ item = 84^{th} item

In c.f. column, 91 is just greater than 84 so its corresponding class is 24-28.

∴ L = 24, f = 12, c.f. = 79, i = 4

Now, we know,
Q_{3} = `L` + $\frac{\frac{3N}{4} , - , c.f.}{f}$ × `i`
or, Q_{3} = 24 + $\frac{84 , - , 79}{12}$ × 4

∴ Q_{3} = 25.67

Finally,

QD = $\frac{Q_3 \, - \, Q_1}{2}$ = $\frac{45.67 \, - \, 14.67}{2}$ = 5.5

Height (in inches) | 60-62 | 62-64 | 64-66 | 66-68 | 68-70 | 70-72 |
---|---|---|---|---|---|---|

No. of students | 4 | 6 | 8 | 12 | 7 | 2 |

Tabulating given data for calculation of Quartile deviation (QD),

Height | Number of students (f) | Cumulative frequency (c.f.) |
---|---|---|

60-62 | 4 | 4 |

62-64 | 6 | 10 |

64-66 | 8 | 18 |

66-68 | 12 | 30 |

68-70 | 7 | 37 |

70-72 | 2 | 39 |

N = 39 |

Position of first quartile = $(\frac N4 )^{th}$ item = $(\frac{39}{4})^{th}$ item = 9.75^{th} item

In c.f. column, 10 is just greater than 9.75 so its corresponding class is 62-64.

∴ L = 62, f = 6, c.f. = 4, i = 2

Now, we know,
Q_{1} = `L` + $\frac{\frac N4 , - , c.f.}{f}$ × `i`
or, Q_{1} = 62 + $\frac{9.75 , - , 4}{6}$ × 2

∴ Q_{1} = 63.92

Position of third quartile = $(\frac{3N}{4} )^{th}$ item = $(\frac{3 \, \times \, 39}{4})^{th}$ item = 29.25^{th} item

In c.f. column, 30 is just greater than 29.25 so its corresponding class is 66-68.

∴ L = 66, f = 12, c.f. = 18, i = 2

Now, we know,
Q_{3} = `L` + $\frac{\frac{3N}{4} , - , c.f.}{f}$ × `i`
or, Q_{3} = 66 + $\frac{29.25 , - , 18}{12}$ × 2

∴ Q_{3} = 67.875

Finally,

QD = $\frac{Q_3 \, - \, Q_1}{2}$ = $\frac{67.875 \, - \, 63.92}{2}$ = 1.978 inches

Coefficient of quartile deviation = $\frac{Q_3 \, - \, Q_1}{Q_3 \, + \, Q_1}$ = $\frac{3.955}{131.795}$ = 0.03

Expenditure | 0 ≤ x ≤ 10 | 10 ≤ x ≤ 20 | 20 ≤ x ≤ 30 | 30 ≤ x ≤ 40 | 40 ≤ x ≤ 50 |
---|---|---|---|---|---|

No. of workers | 5 | 15 | 10 | 8 | 6 |

Tabulating given data for calculation of Quartile deviation (QD),

Expenditure | Number of workers (f) | Cumulative frequency (c.f.) |
---|---|---|

0-10 | 5 | 5 |

10-20 | 15 | 20 |

20-30 | 10 | 30 |

30-40 | 8 | 38 |

40-50 | 6 | 44 |

N = 44 |

Position of first quartile = $(\frac N4 )^{th}$ item = $(\frac{44}{4})^{th}$ item = 11^{th} item

In c.f. column, 20 is just greater than 11 so its corresponding class is 10-20.

∴ L = 10, f = 15, c.f. = 5, i = 10

Now, we know,
Q_{1} = `L` + $\frac{\frac N4 , - , c.f.}{f}$ × `i`
or, Q_{1} = 10 + $\frac{11 , - , 5}{15}$ × 10

∴ Q_{1} = 14

Position of third quartile = $(\frac{3N}{4} )^{th}$ item = $(\frac{3 \, \times \, 44}{4})^{th}$ item = 33^{th} item

In c.f. column, 38 is just greater than 33 so its corresponding class is 30-40.

∴ L = 30, f = 8, c.f. = 30, i = 10

Now, we know,
Q_{3} = `L` + $\frac{\frac{3N}{4} , - , c.f.}{f}$ × `i`
or, Q_{3} = 30 + $\frac{33 , - , 30}{8}$ × 10

∴ Q_{3} = 33.75

Finally,

QD = $\frac{Q_3 \, - \, Q_1}{2}$ = $\frac{33.75 \, - \, 14}{2}$ = 9.875

Coefficient of quartile deviation = $\frac{Q_3 \, - \, Q_1}{Q_3 \, + \, Q_1}$ = $\frac{19.75}{47.75}$ = 0.414

The following are the marks obtained by class 9 students in their internal examination. Taking class interval of (10-20) as first class, prepare a frequency distribution table and find the quartile deviation. Also find its coefficient from the following data:

22, 25, 46, 34, 57, 69, 44, 36, 12, 27, 50, 36, 35, 62, 46, 52, 54, 61, 66, 55, 29, 39, 40, 33, 14, 41, 25, 20, 16.

Given data,

22, 25, 46, 34, 57, 69, 44, 36, 12, 27, 50, 36, 35, 62, 46, 52, 54, 61, 66, 55, 29, 39, 40, 33, 14, 41, 25, 20, 16

Tabulating given data starting from class interval of (10 - 20) for calculation of Quartile deviation (QD),

Class interval | Frequency (f) | Cumulative frequency (c.f.) |
---|---|---|

10-20 | 3 | 3 |

20-30 | 6 | 9 |

30-40 | 6 | 15 |

40-50 | 5 | 20 |

50-60 | 5 | 25 |

60-70 | 4 | 29 |

N = 29 |

Position of first quartile = $(\frac N4 )^{th}$ item = $(\frac{29}{4})^{th}$ item = 7.25^{th} item

In c.f. column, 9 is just greater than 7.25 so its corresponding class is 20-30.

∴ L = 20, f = 6, c.f. = 3, i = 10

Now, we know,
Q_{1} = `L` + $\frac{\frac N4 , - , c.f.}{f}$ × `i`
or, Q_{1} = 20 + $\frac{7.25 , - , 3}{6}$ × 10

∴ Q_{1} = 27.083

Position of third quartile = $(\frac{3N}{4} )^{th}$ item = $(\frac{3 \, \times \, 29}{4})^{th}$ item = 21.75^{th} item

In c.f. column, 25 is just greater than 21.75 so its corresponding class is 50-60.

∴ L = 50, f = 5, c.f. = 20, i = 10

Now, we know,
Q_{3} = `L` + $\frac{\frac{3N}{4} , - , c.f.}{f}$ × `i`
or, Q_{3} = 50 + $\frac{21.75 , - , 20}{5}$ × 10

∴ Q_{3} = 53.5

Finally,

QD = $\frac{Q_3 \, - \, Q_1}{2}$ = $\frac{53.5 \, - \, 27.083}{2}$ = 13.21

Coefficient of quartile deviation = $\frac{Q_3 \, - \, Q_1}{Q_3 \, + \, Q_1}$ = $\frac{26.417}{80.58}$ = 0.33

The following are the marks obtained by class 9 students in their internal examination. Taking class interval of (10-20) as first class, prepare a frequency distribution table and find the quartile deviation. Also find its coefficient from the following data:

21, 45, 60, 57, 15, 41, 48, 50, 34, 29, 56, 40, 14, 62, 28, 70, 22, 30, 38, 74, 13, 47, 20, 53, 64, 34, 75, 66.

Given data,

21, 45, 60, 57, 15, 41, 48, 50, 34, 29, 56, 40, 14, 62, 28, 70, 22, 30, 38, 74, 13, 47, 20, 53, 64, 34, 75, 66

Tabulating given data starting from class interval of (10 - 20) for calculation of Quartile deviation (QD),

Class interval | Frequency (f) | Cumulative frequency (c.f.) |
---|---|---|

10-20 | 3 | 3 |

20-30 | 5 | 8 |

30-40 | 4 | 12 |

40-50 | 5 | 17 |

50-60 | 4 | 21 |

60-70 | 4 | 25 |

70-80 | 3 | 28 |

N = 28 |

Position of first quartile = $(\frac N4 )^{th}$ item = $(\frac{28}{4})^{th}$ item = 7^{th} item

In c.f. column, 8 is just greater than 7 so its corresponding class is 20-30.

∴ L = 20, f = 5, c.f. = 3, i = 10

Now, we know,
Q_{1} = `L` + $\frac{\frac N4 , - , c.f.}{f}$ × `i`
or, Q_{1} = 20 + $\frac{7 , - , 3}{5}$ × 10

∴ Q_{1} = 28

Position of third quartile = $(\frac{3N}{4} )^{th}$ item = $(\frac{3 \, \times \, 28}{4})^{th}$ item = 21^{th} item

In c.f. column, corresponding class of 21 is 50-60.

∴ L = 50, f = 4, c.f. = 17, i = 10

Now, we know,
Q_{3} = `L` + $\frac{\frac{3N}{4} , - , c.f.}{f}$ × `i`
or, Q_{3} = 50 + $\frac{21 , - , 17}{4}$ × 10

∴ Q_{3} = 60

Finally,

QD = $\frac{Q_3 \, - \, Q_1}{2}$ = $\frac{60 \, - \, 28}{2}$ = 16

Coefficient of quartile deviation = $\frac{Q_3 \, - \, Q_1}{Q_3 \, + \, Q_1}$ = $\frac{32}{88}$ = 0.36