Find the mean and the standard deviation for the following data: Number of children 0 - 2 2 - 4 4 - 6 6 - 8 8 - 10 10 - 12 Number of families 272 328 205 120 15 10

Number of children x i Class mark f i d i = x i - A A = 5 u i = d i C C = 2 f i u i f i u i 2 0 - 2 1 272 - 4 - 2 - 544 1088 2 - 4 3 328 - 2 - 1 - 328 328 4 - 6 5 205 0 0 0 0 6 - 8 7 120 2 1 120 120 8 - 10 9 15 4 2 30 60 10 - 12 11 10 6 3 30 90 Σ f i = 950 Σ f i u i = - 692 Σ f i u i 2 = 1686 By step deviation method mean x ¯ = A + ∑ f i u l Σ f i × C = 5 − 692 950 × 2 = 3 .54 And, S . D = σ = ∑ f i u i 2 Σ f i − Σ f i u i Σ f i 2 × C = 1686 950 − − 692 950 2 × 2 = 1 .77474 − 0 .5306 × 2 = 2 .23

Find the mean and variance of the following data by using step deviation method: Classes 0 - 5 5 - 10 10 - 15 15 - 20 20 - 25 25 - 30 30 - 35 35 - 40 40 - 45 Frequency 20 24 32 28 20 11 26 15 24

Classes Class Mark x i f i d i = x i - A A = 22 . 5 u i = d i C C = 5 f i u i f i u i 2 0 - 5 2 . 5 20 - 20 - 4 - 80 320 5 - 10 7 . 5 24 - 15 - 3 - 72 216 10 - 15 12 . 5 32 - 10 - 2 - 64 128 15 - 20 17 . 5 28 - 5 - 1 - 28 28 20 - 25 22 . 5 20 - 0 0 0 0 25 - 30 27 . 5 11 5 1 11 11 30 - 35 32 . 5 26 10 2 52 104 35 - 40 37 . 5 15 15 3 45 135 40 - 45 42 . 5 24 20 4 96 384 Σ f i = 200 Σ f i u i = - 40 Σ f i u i 2 = 1326 By step deviation method, mean = A + Σ f i u i Σ f i × C = 22 .5 − 40 200 × 5 = 22 .5 − 1 = 21 .5 And, S.D ( σ ) = ∑ f i u i 2 Σ f i − Σ f i u i Σ f i 2 × C = 1326 200 − − 40 200 2 × 5 = 6 .63 − 0 .04 × 5 = 12 .83549 So, variance, σ 2 = 164 .75

Find the mean, variance and standard deviation using shortcut method: Height (in cm ) 70 - 75 75 - 80 80 - 85 85 - 90 90 - 95 95 - 100 100 - 105 105 - 110 110 - 115 Number of children 3 4 7 7 15 9 6 6 3

Height (in cm ) x i d i = x i - A A = 92 . 5 f i f i d i f i d i 2 70 - 75 72 . 5 - 20 3 - 60 1200 75 - 80 77 . 5 - 15 4 - 60 900 80 - 85 82 . 5 - 10 7 - 70 700 85 - 90 87 . 5 - 5 7 - 35 175 90 - 95 92 . 5 0 15 0 0 95 - 100 97 . 5 5 9 45 225 100 - 105 102 . 5 10 6 60 600 105 - 110 107 . 5 15 6 90 1350 110 - 115 112 . 5 20 3 60 1200 Σ f i = 60 Σ f i d i = 30 Σ f j d i 2 = 6350 By short cut method, mean = A + Σ f i d i Σ f i = 92 .5 + 30 60 = 92 .5 + 0 .5 = 93 And, variance σ 2 = Σ f i d i 2 Σ f i − Σ f i d i Σ f i 2 = 6350 60 − 30 60 2 = 105 .833 − 0 .25 = 105 .583 And, σ = S . D , = Variance = 105 .583 = 10 .28

From the following, find which series is more consistent: Variables 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 Series A 10 16 30 40 26 18 Series B 22 18 32 34 18 16

Classes Class Marks x i d i = x i - A ' A = 35 u i = d i C C = 10 f i Frequency g i f i u i g i u i f i u i 2 g i u i 2 10 - 20 15 - 20 - 2 10 22 - 20 - 44 40 88 20 - 30 25 - 10 - 1 16 18 - 16 - 18 16 18 30 - 40 35 0 0 30 32 0 0 0 0 40 - 50 45 10 1 40 34 40 34 40 34 50 - 60 55 20 2 26 18 52 36 104 72 60 - 70 65 30 3 18 16 54 48 162 144 Σ f i = 140 Σ g i = 140 Σ f i u i = 110 Σ g i u i = 56 Σ f i u i 2 = 362 Σ g i u i 2 = 356 For series A , x A ¯ = A ' + ∑ f i u i ∑ f i × 10 = 35 + 110 × 10 140 = 42 .857 and σ A = ∑ f i u i 2 Σ f i − Σ f i u i Σ f i 2 × C = 362 140 − 110 140 2 × 10 = 2 .5857 − 0 .61735 × 10 = 14 .0298 So, C . V for series A = σ A x A ¯ × 100 = 14 .0298 42 .857 × 100 = 32 .73 For series B , x ¯ B = A ′ + Σ g i u i Σ g i × 10 = 35 + 56 × 10 140 = 39 And, σ B = Σ g i u i 2 Σ g i − Σ g i u i Σ g i 2 × C = 356 140 − 56 140 2 × 10 = 2 .54286 − 0 .16 × 10 = 15 .4365 So, C . V for series B = σ B x B ¯ × 100 = 15 .4365 × 100 39 = 39 .58 Since C . V for series B > C . V for series A . So, series A is more consistent .

M L Aggarwal Solutions for Chapter: Statistics, Exercise 7: CHAPTER TEST

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Statistics, Exercise 7: CHAPTER TEST

Attempt the practice questions on Chapter 14: Statistics, Exercise 7: CHAPTER TEST with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 11 Volume 2 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Chapter: Statistics, Exercise 7: CHAPTER TEST with Hints & Solutions

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 14 - Statistics>CHAPTER TEST>Q 14

Determine the mean and standard deviation for the following distribution: (Write the answer upto two decimal places.)

Marks	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$	$16$
Frequency	$1$	$6$	$6$	$8$	$8$	$2$	$2$	$3$	$0$	$2$	$1$	$0$	$0$	$0$	$1$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 14 - Statistics>CHAPTER TEST>Q 15

Find the mean and the standard deviation for the following data:

Number of children	$0 - 2$	$2 - 4$	$4 - 6$	$6 - 8$	$8 - 10$	$= \sqrt{\frac{Σ f_{i} x_{i}^{2}}{Σ f_{i}} - {(\frac{Σ f_{i} x_{i}}{Σ f_{i}})}^{2}}$
Number of families	$272$	$328$	$205$	$120$	$15$	$10$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 14 - Statistics>CHAPTER TEST>Q 16

Find the mean and variance of the following data by using step deviation method:

Classes	$0 - 5$	$5 - 10$	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$40 - 45$
Frequency	$20$	$24$	$32$	$28$	$20$	$11$	$26$	$15$	$24$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 14 - Statistics>CHAPTER TEST>Q 17

Find the mean, variance and standard deviation using shortcut method:

Height (in $cm$ )	$70 - 75$	$75 - 80$	$80 - 85$	$85 - 90$	$90 - 95$	$95 - 100$	$100 - 105$	$105 - 110$	$110 - 115$
Number of children	$3$	$4$	$7$	$7$	$15$	$9$	$6$	$6$	$3$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 14 - Statistics>CHAPTER TEST>Q 18

Calculate the standard deviation from the following data: (Write the answer up to two decimal places.)

Mid-point	$5$	$15$	$25$	$35$	$45$	$55$	$65$	$75$
Frequency	$5$	$8$	$7$	$12$	$28$	$20$	$10$	$10$

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 14 - Statistics>CHAPTER TEST>Q 20

The mean and variance of $8$ observations are $9$ and $9.25$ respectively. If the six observations are $6, 7, 10, 12, 12 and 13,$ find the remaining two observations.

EASY

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 14 - Statistics>CHAPTER TEST>Q 21

Weekly average wages of workers in a factory increase from $₹ 80 to ₹ 120$ and standard deviation increases from $₹ 10 to ₹ 30$ . Have the wages become less uniform now?

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 14 - Statistics>CHAPTER TEST>Q 23

From the following, find which series is more consistent:

Variables	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 70$
Series $A$	$10$	$16$	$30$	$40$	$26$	$18$
Series $B$	$22$	$18$	$32$	$34$	$18$	$16$

M L Aggarwal Solutions for Chapter: Statistics, Exercise 7: CHAPTER TEST

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Statistics, Exercise 7: CHAPTER TEST

Questions from M L Aggarwal Solutions for Chapter: Statistics, Exercise 7: CHAPTER TEST with Hints & Solutions

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