Class 11 RD Sharma Solutions - Chapter 32 Statistics - Exercise 32.7

Question 1. Two plants A and B of a factory show the following results about the number of workers and the wages paid to them

In which plant A or B is there greater variability in individual wages?

Solution:

Variation of the distribution of wages in plant A (σ² =18)

So, Standard deviation of the distribution A (σ – 9)

Similarly, the Variation of the distribution of wages in plant B (σ² =100)

So, Standard deviation of the distribution B (σ – 10)

And, Average monthly wages in both the plants is 2500,

Since, the plant with a greater value of SD will have more variability in salary.

∴ Plant B has more variability in individual wages than plant A

Question 2. The means and standard deviations of heights and weights of 50 students in a class are as follows:

Which shows more variability, heights or weights?

Solution:

We observe that the average weights and height for the 50 students is same i.e. 63.2.

Therefore, the parameter with greater variance will have more variability.

Thus, height has greater variability

Question 3. The coefficient of variation of two distribution are 60% and 70%, and their standard deviations are 21 and 16 respectively. What is their arithmetic means?

Solution:

Coefficient of variation = 

So, we have:

∴ Means are 35 and 22.85

Question 4. Calculate coefficient of variation from the following data:

Solution:

Class	Fi	xi	fiui	fiui2
1000 - 1700	12	1350	-2	-24	48
1700 - 2400	18	2050	-1	-18	18
2400 - 3100	20	2750	0	0	0
3100 - 3800	25	3450	1	25	25
3800 - 4500	35	4150	2	70	140
4500 - 5200	10	4850	3	30	90

Now,

N = 120, 

Mean, 

Variance = 1076332.64

Standard Deviation, 

Coefficient of variation = 

= 32.08

∴ The coefficient variation is 32.08

Question 5. An analysis of the weekly wages paid to workers in two firms A and B, belonging to the same industry gives the following results:

(i) Which firm A or B pays out the larger amount as weekly wages?

(ii) Which firm A or B has greater variability in individual wages?

Solution:

(i) Average weekly wages = 

Total weekly wages = (Average weekly wages) × (No. of workers)

Total weekly wages of Firm A = 52.5 × 586 = Rs 30765

Total weekly wages of Firm B = 47.5 × 648 = Rs 30780

Firm B pays a larger amount as Firm A

(ii) Here, 

S.D (Firm A) = 10 and S.D (Firm B) = 11

Coefficient variance (Firm A) = 

= 19.04

Coefficient variance (Firm B) = 

= 23.15

∴ Coefficient variance of Firm B is greater than that of Firm A, Firm B has greater variability in individual wages.

Question 6. The following are some particulars of the distribution of weights of boys and girls in a class:

Which of the distributions is more variable?

Solution:

Given:

S.D (Boys) is 3 and S.D (Girls) is 2

Coefficient variance (Boys) = 

= 5

Coefficient variance (Girls) = 

= 4.4

∴ Coefficient variance of Boys is greater than Coefficient variance of girls, and then the distribution of weights of boys is more variable than that of girls.

Question 7. The mean and standard deviation of marks obtained by 50 students of a class in three subjects, mathematics, physics and chemistry are given below:

Solution:

In order to compare the variability of marks in Math, Physics and Chemistry.

We have to calculate their coefficient of variation.

Let σ₁, σ₂ and σ₃ denote the standard deviation of marks in Math, Physics and Chemistry respectively. Further, Let  be the mean scores in Math, Physics and Chemistry respectively.

We have 

⇒ σ₁  = 12       σ₂  = 15         σ₃  = 20

Now,

Coefficient of variation in Maths = 

Coefficient of variation in Physics = 

Coefficient of variation in Chemistry = 

Clearly, coefficient of variation in marks is greatest in Chemistry and lowest in Math.

So, marks in chemistry show highest variability and marks in maths show lowest variability.

Question 8. From the data given below state which group is more variable, G₁ or G₂?

Solution:

Let's first find the coefficient of variable for group G₁

CI              f            10 - 20         9     20 - 30        17	x       u=(x - A)/h           15               -3      25               -2	fu         u2          -27         9     -34         4	fu2          81      68
30 - 40        32      40 - 50        33	35               -1      45                0	-32         1       0          0	32       0
50 - 60        40	55                1	40          1	40
60 - 70        10     70 - 80         9	65                2      75                3	20          4      27          9	40      81
150	-6	342

Here, N = 150, A = 45,   and h = 10

∴ Mean = 

Coefficient of variation = 

Now, lets find the coefficient of variable for group G₂

CI                   f               10 - 20              10      20 - 30              20	x                 u=(x - A)/h              15                        -3       25                        -2	fu              u2            -30               9      -40               4	fu2             90        80
30 - 40              30      40 - 50              25	35                        -1       45                         0	-30               1         0                0	30         0
50 - 60              43	55                         1	43               1	43
60 - 70              15      70 - 80               7	65                         2       75                         3	30               4        21               9	60        63
150	-6	366

Here, N = 150, A = 45,  and h = 10

∴ Mean = 

Coefficient of variation = 

Group G₂ is more variable

Question 9. Find the coefficient of variation for the following data:

Solution:

CI                 f                  x               10 - 15            2                12.5      15 - 20            8                17.5	u=(x - A)/h               fu                  u2                           -2                       -4                   4                 -1                       -8                   1	fu2              8            8
20 - 25           20               22.5      25 - 30           35               27.5	0                        0                   0                   1                       35                  1	0        35
30 - 35           20               32.5	2                       40                  4	80
35 - 40           15               37.5	3                       45                  9	135
100	108	266

Here, N = 100, A = 22.5,  and h = 5

∴ Mean = 

Coefficient of variation = 

Question 10. From the prices of shares X and Y given below: find out which is more stable in value:

Solution:

x                  d = (x - Mean)                 35                           -13         24                           -24	d2            169      576
52                              4         53                              5	16       25
56                              8	64
58                             10         52                              4	100       16
50                              2	4
51                              3	9
49                              1	1
480	980

∴ Mean = 

Coefficient of variation = 

x                  d = (x - Mean)                  35                           -13         24                           -24	d2                   169          576
52                            4          53                            5	16           25
56                            8	64
58                           10          52                            4	100           16
50                            2	4
51                            3	9
49                            1	1
480	980

∴ Mean = 

Coefficient of variation = 

Since the coefficient of variation for share Y is smaller than the coefficient of variation for shares X, they are more stable

Summary

Chapter 32 of RD Sharma's Class 11 mathematics textbook covers Statistics, a branch of mathematics dealing with data collection, analysis, interpretation, and presentation. Exercise 32.7 specifically focuses on measures of dispersion, particularly the concept of standard deviation. This exercise teaches students how to calculate and interpret standard deviation for both ungrouped and grouped data, understand its significance in statistical analysis, and apply it to solve real-world problems.