Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 2 Measures of Dispersion Ex 2.2 Questions and Answers.

Maharashtra State Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2

Question 1.
Find the variance and S.D. for the following set of numbers.
7, 11, 2, 4, 9, 6, 3, 7, 11, 2, 5, 8, 3, 6, 8, 8, 2, 6
Solution:
Given data: 7, 11, 2, 4, 9, 6, 3, 7, 11, 2, 5, 8, 3, 6, 8, 8, 2, 6
The tabulated form of the above data is as given below:
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q1
We prepare the following table for the calculation of variance and S. D.
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q1.1
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q1.2

Question 2.
Find the variance and S.D. for the following set of numbers.
65, 77, 81, 98, 100, 80, 129
Solution:
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q2
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q2.1

Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2

Question 3.
Compute the variance and standard deviation for the following data:
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q3
Solution:
We prepare the following table for the calculation of variance and S.D.:
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q3.1

Question 4.
Compute the variance and S.D.
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q4
Solution:
We prepare the following table for the calculation of variance and S.D.:
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q4.1

Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2

Question 5.
The following data gives the age of 100 students in a school. Calculate variance and S.D.
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q5
Solution:
We prepare the following table for the calculation of variance and S.D:
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q5.1

Question 6.
The mean and variance of 5 observations are 3 and 2 respectively. If three of the five observations are 1, 3, and 5, find the values of the other two observations.
Solution:
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q6
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q6.1
∴ (x4 – 4)(x4 – 2) = 0
∴ x4 = 4 or x4 = 2
From (i), we get
x5 = 2 or x5 = 4
∴ The two numbers are 2 and 4.

Question 7.
Obtain standard deviation for the following data:
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q7
Solution:
We prepare the following table for the calculation of standard deviation.
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q7.1

Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2

Question 8.
The following distribution was obtained by change of origin and scale of variable X.
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q8
If it is given that mean and variance are 59.5 and 413 respectively, determine actual class intervals.
Solution:
Here, Mean = \(\bar{x}\) = 59.5, and
Var(X) = σ2 = 413
Let xi be a mid value of class and
d = \(\frac{x-a}{h}\), where a is assumed mean and h is class width.
We prepare the following table for calculation of mean and variance of di.
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q8.1
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q8.2
Now, Var(X) = h2. Var(D)
∴ 413 = h2 × 4.13
∴ h2 = 100
∴ h = 10
Substituting h = 10 in (i), we get
-0.1 × 10 + a = 59.5
∴ -1 + a = 59.5
∴ a = 59.5 + 1
∴ a = 60.5
We prepare the following table to determine actual class intervals for corresponding values of di.
Maharashtra Board 11th Commerce Maths Solutions Chapter 2 Measures of Dispersion Ex 2.2 Q8.3
∴ The actual class intervals are 15.5 – 25.5, 25.5 – 35.5, …….., 95.5 – 105.5

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