NCERT solution class 11 chapter 15 Statistics exercise 15.3 mathematics

EXERCISE 15.3

Question 1:

From the data given below state which group is more variable, A or B?

 Marks 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Group A 9 17 32 33 40 10 9 Group B 10 20 30 25 43 15 7

Firstly, the standard deviation of group A is calculated as follows.

 Marks Group A fi Mid-point xi yi2 fiyi fiyi2 10-20 9 15 –3 9 –27 81 20-30 17 25 –2 4 –34 68 30-40 32 35 –1 1 –32 32 40-50 33 45 0 0 0 0 50-60 40 55 1 1 40 40 60-70 10 65 2 4 20 40 70-80 9 75 3 9 27 81 150 –6 342

Here, = 10, N = 150, A = 45

The standard deviation of group B is calculated as follows.

 Marks Group Bfi Mid-pointxi yi2 fiyi fiyi2 10-20 10 15 –3 9 –30 90 20-30 20 25 –2 4 –40 80 30-40 30 35 –1 1 –30 30 40-50 25 45 0 0 0 0 50-60 43 55 1 1 43 43 60-70 15 65 2 4 30 60 70-80 7 75 3 9 21 63 150 –6 366

Since the mean of both the groups is same, the group with greater standard deviation will be more variable.

Thus, group B has more variability in the marks.

Question 1:

From the data given below state which group is more variable, A or B?

 Marks 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Group A 9 17 32 33 40 10 9 Group B 10 20 30 25 43 15 7

Firstly, the standard deviation of group A is calculated as follows.

 Marks Group Afi Mid-point xi yi2 fiyi fiyi2 10-20 9 15 â€“3 9 â€“27 81 20-30 17 25 â€“2 4 â€“34 68 30-40 32 35 â€“1 1 â€“32 32 40-50 33 45 0 0 0 0 50-60 40 55 1 1 40 40 60-70 10 65 2 4 20 40 70-80 9 75 3 9 27 81 Î£ 150 â€“6 342

Here, = 10, N = 150, A = 45

The standard deviation of group B is calculated as follows.

 Marks Group Bfi Mid-point xi yi2 fiyi fiyi2 10-20 10 15 â€“3 9 â€“30 90 20-30 20 25 â€“2 4 â€“40 80 30-40 30 35 â€“1 1 â€“30 30 40-50 25 45 0 0 0 0 50-60 43 55 1 1 43 43 60-70 15 65 2 4 30 60 70-80 7 75 3 9 21 63 Î£ 150 â€“6 366

Since the mean of both the groups is same, the group with greater standard deviation will be more variable.

Thus, group B has more variability in the marks.

Question 2:

From the prices of shares X and Y below, find out which is more stable in value:

 X 35 54 52 53 56 58 52 50 51 49 Y 108 107 105 105 106 107 104 103 104 101

The prices of the shares X are

35, 54, 52, 53, 56, 58, 52, 50, 51, 49

Here, the number of observations, N = 10

The following table is obtained corresponding to shares X.

 xi 35 –16 256 54 3 9 52 1 1 53 2 4 56 5 25 58 7 49 52 1 1 50 –1 1 51 0 0 49 –2 4 350

The prices of share Y are

108, 107, 105, 105, 106, 107, 104, 103, 104, 101

The following table is obtained corresponding to shares Y.

 yi 108 3 9 107 2 4 105 0 0 105 0 0 106 1 1 107 2 4 104 –1 1 103 –2 4 104 –1 1 101 –4 16 40

C.V. of prices of shares X is greater than the C.V. of prices of shares Y.

Thus, the prices of shares Y are more stable than the prices of shares X.

Question 3:

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

 Firm A Firm B No. of wage earners 586 648 Mean of monthly wages Rs 5253 Rs 5253 Variance of the distribution of wages 100 121

(i) Which firm A or B pays larger amount as monthly wages?

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

(i) Monthly wages of firm A = Rs 5253

Number of wage earners in firm A = 586

∴Total amount paid = Rs 5253 × 586

Monthly wages of firm B = Rs 5253

Number of wage earners in firm B = 648

∴Total amount paid = Rs 5253 × 648

Thus, firm B pays the larger amount as monthly wages as the number of wage earners in firm B are more than the number of wage earners in firm A.

(ii) Variance of the distribution of wages in firm A = 100

∴ Standard deviation of the distribution of wages in firm

A ((σ1) =

Variance of the distribution of wages in firm = 121

∴ Standard deviation of the distribution of wages in firm

The mean of monthly wages of both the firms is same i.e., 5253. Therefore, the firm with greater standard deviation will have more variability.

Thus, firm B has greater variability in the individual wages.

Question 4:

The following is the record of goals scored by team A in a football session:

 No. of goals scored 0 1 2 3 4 No. of matches 1 9 7 5 3

For the team B, mean number of goals scored per match was 2 with a standard

deviation 1.25 goals. Find which team may be considered more consistent?

The mean and the standard deviation of goals scored by team A are calculated as follows.

 No. of goals scored No. of matches fixi xi2 fixi2 0 1 0 0 0 1 9 9 1 9 2 7 14 4 28 3 5 15 9 45 4 3 12 16 48 25 50 130

Thus, the mean of both the teams is same.

The standard deviation of team B is 1.25 goals.

The average number of goals scored by both the teams is same i.e., 2. Therefore, the team with lower standard deviation will be more consistent.

Thus, team A is more consistent than team B.

Question 5:

The sum and sum of squares corresponding to length (in cm) and weight y

(in gm) of 50 plant products are given below:

Which is more varying, the length or weight?