**Calculate the range and coefficient of range from the following data.**

**a) The heights of 10 children in cm: 122, 144, 154, 101, 168, 118, 155, 133, 160, 140 **

Solution:

Heights of 10 children in cm: 122, 144, 154, 101, 168, 118, 155,133,

160,140.

Range: H – L = 168 – 101 = 67

Coefficient of Range: ^{H – L}/_{H + L}

= ^{67}/_{168 + 101} = ^{67}/_{269} **= 0.249**

**b) Marks scored by 12 students in a test: 31, 18, 27, 19, 25, 28, 49, 14, 41, 22, 33, 13.**

Solution:

** **Marks scored by 12 students

31, 18, 27, 19, 25, 28, 49, 14, 41, 22, 33, 13.

H = 49; L = 13

Range: H – L = 49 – 13 = 36

Coefficient of Range: ^{H – L}/_{H + L} = ^{36}/_{49 + 13} = ^{36}/_{62} = **0.58. **

**c) ****Number of trees planted in 6 months: 186, 234, 465, 361, 290, 142. **

Solution:

No .of trees planted in 6 months:

186, 234, 465, 361, 290, 142.

H = 465; L = 142

Range: H – L = 465 – 142 = 323

Coefficient of Range: ^{H – L}/_{H + L} = ^{323}/_{465+142} = ^{323}/_{607} = **0.532 **

**State quartile deviation for the following data:**

**a) ****30, 18, 23, 15, 11, 29, 37, 42, 10, 21.**

Solution:

** **Arrange the scores in ascending order:

n=10,

10, 11, 15, 18, 21, 23, 29, 30, 37, 42.

(a) First Q1 = ^{n+1}/_{4}

= ^{10+1}/_{4} = ^{11}/_{4} = 2.75 = 3rd score = **15 **

(ii) Third Quartile

Q3 = ^{3(n+1)}/_{4} = ^{3×11}/_{4} = ^{33}/_{4} = 8.25 = 9th = **37 **

(iii) Quartile Deviation:

= (Q3−Q1)/_{2} = ^{37−15}/_{2} = ^{22}/_{2} = **11**

**b) 3, 5, 8, 10, 12, 7, 5.**

Solution:

3,5,5,7,8,10,12

n = 7

(i) Quartile Q1 = ^{n+1}/_{4} = ^{7+1}/_{4} = ^{8}/_{4} = 2nd score

Q1 = **5 **

(ii) Quartile Q3 = 3^{(}^{n + 1)}/_{4} = 3[(8)/_{4}] = 6th score

Q3 = **10 **

(iii) Quartile Deviation:

= [Q3 − Q1]/_{2} = ^{10 – 5}/_{2} = ^{5}/_{2} = **2.5**

**(c)**

Age | 3 | 6 | 9 | 12 | 15 |

No. of children | 4 | 8 | 11 | 7 | 12 |

Solution:

x | f | commutative frequency |

3 | 4 | 4 |

6 | 8 | 12 |

9 | 11 | 23 |

12 | 7 | 30 |

15 | 12 | 42 |

\ n = 42

Q1 = ^{n}/_{4} ; score = ^{42}/_{4} = 10.5;11th score

From fc ∴Q1 = 6

Q3 = ^{3n}/_{4} = ^{3 ×42}/_{4} = 31.5; 32nd score ∴Q3 = 15

Q.D = [Q3−Q1]/_{2} = ^{15−6}/_{2} = ^{9}/_{2} = **4.5**

**d)**

Marks scored | 10 | 20 | 30 | 40 | 50 | 60 |

No. of students | 12 | 7 | 16 | 08 | 18 | 22 |

Solution:

x | f | fc |

10 | 12 | 12 |

20 | 07 | 19 |

30 | 16 | 35 |

40 | 08 | 43 |

50 | 18 | 61 |

60 | 22 | 83 |

n = 83

Q1= ^{n}/_{4} = ^{83}/_{4} = **20.75; **21st score

∴Q1 = 30

Q3 = ^{3n}/_{4} = ^{3 ×83}/_{4} = 3X20.75 = 62.25; 62nd score

∴Q3 = 60

Q.D = [Q3−Q1]/2 = ^{60 – 30}/_{2} = ^{30}/_{2} = **15 **

∴QD = 15

**Compute quartile deviation for each of the following tables.**

**a)**

Class interval | Frequency | fc |

5 – 15 | 11 | 11 |

15 – 25 | 5 | 16 |

25 – 35 | 15 | 31 |

35 – 45 | 9 | 40 |

45 – 55 | 22 | 62 |

55 – 65 | 8 | 70 |

65 – 75 | 17 | 87 |

Solution:

n = 87

Q1= ^{n}/_{4} = ^{87}/_{4} = **21.75 **

22nd score CI = 25 – 35

∴LRC = 25

fc = 31; i = 10

Q2 = LRL +( [^{N}/_{4}−fc]/_{fm})i

= 25 + [(^{87}/_{4}−16)/_{15}] 10

= 25 + [^{21.75 −16}/_{15}] ×10

**Q****2 ****= 28.83**

Q3 = LRL + [(^{3N}/_{4} – fc)/_{fm}]*i

^{3N}/_{4} = ^{3 × 87}/_{4} = 65.25 class interval 55 – 65

L = 55, fc = 62, fm = 8, CI = 10

LRL = 55 +(^{65.25 – 62})/_{8} ×10

= 55 + 4.06

**LRL = 59.06 **

Quartile Deviation = [Q3 − Q1]/_{2}

= ^{59.06 – 28.83}/_{2}

= ^{30.23}/_{2}

**Q.D = 15.11**

**(b)**

class interval | frequency | fc |

1 – 9 | 4 | 4 |

10 – 19 | 3 | 7 |

20 – 29 | 20 | 27 |

30 – 39 | 12 | 39 |

40 – 49 | 5 | 44 |

50 – 59 | 8 | 52 |

60 – 69 | 14 | 66 |

70 – 79 | 27 | 93 |

80 – 89 | 2 | 95 |

90 – 99 | 5 | 100 |

H = 100

Solution:

n = 100

^{100}/_{4} = 25th Score 20 – 29; LRL = 19.5

Fc = 7; fm = 20

Q1 = LRL+[ (^{N}/_{4}−fc)/_{fm}]*i

= 19.5 + [^{25 – 7}/_{20}]× 10

= 19.5 + ^{18}/_{20}*10

**Q****1 ****= 28.5 **

^{3N}/_{4} = ^{3 }^{× }^{100}/_{4} = 3 × 25 = 75th score cl 70 – 79

LRL = 69.5; fc = 66; fm = 14

Q3 = 69.5 + [^{(}^{75 – 66)}/_{14}] × 10

= 69.5 + 3.33

**Q****3 ****= 72.83**

Quartile Deviation = (Q3 − Q1)/_{2}

= ^{72.83 – 28.5} /_{2}

= ^{44.33}/_{2}

**Q.D = 22.16**

**STATISTICS – EXERCISE 1.5.2 – Class 9**

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