Q1. 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 |
Sol:
The standard deviation of group A is calculated as given in the table below.
Marks | ||||||
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 |
N = 150
h = 10
A = 45
Mean,
=
=
= 45 – 0.4
= 44.6
Variance,
=
=
=
=
= 227.84
Standard deviation,
=
= 15.09
The standard deviation of group A is calculated as given in the table below.
Marks | ||||||
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 |
N = 150
h = 10
A = 45
Mean,
=
=
= 45 – 0.4
= 44.6
Variance,
=
=
=
=
= 243.84
Standard deviation,
=
= 15.61
As 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.
Q2. 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 |
Sol:
The prices of the shares X are 35, 54, 52, 53, 56, 58, 52, 50, 51, and 49.
Here, the number of observations, N = 10
Mean,
=
=
= 51
The following table is obtained corresponding to shares X.
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 |
Variance,
=
=
= 35
Standard deviation,
=
= 5.91
C.V.
=
= 11.58
The prices of share Y are 108, 107, 105, 105, 106, 107, 104, 103, 104, and 101.
Mean,
=
=
= 105
The following table is obtained corresponding to shares Y.
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 |
Variance,
=
=
= 4
Standard deviation,
=
= 2
C.V.
=
= 1.9
C.V. of prices of shares X is greater than the C.V. of prices of shares Y.
Therefore, the prices of shares Y are more stable than the prices of shares X.
Q3. 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?
Sol:
(i)
Monthly wages of firm A = Rs. 5253
Number of wage earners in firm A = 586
Therefore,
Total amount paid = Rs. 5253 × 586
Monthly wages of firm B = Rs. 5253
Number of wage earners in firm B = 648
Therefore,
Total amount paid = Rs. 5253 × 648
Hence, 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,
Therefore,
Standard deviation of the distribution of wages in firm A,
Variance of the distribution of wages in firm B,
Therefore,
Standard deviation of the distribution of wages in firm B,
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.
Therefore, firm B has greater variability in the individual wages.
Q4. The following is the record of goals scored by team A in a football session:
Number of goals scored | 0 | 1 | 2 | 3 | 4 |
Number of matches | 1 | 9 | 7 | 5 | 3 |
For the team B, mean number of goals scored per match was 2 with standard deviation 1.25 goals. Find which team may be considered more consistent?
Sol:
The mean and the standard deviation of goals scored by team A are calculated as follows.
No. of goals scored | No. of matches | |||
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 |
23 | 50 | 130 |
Mean,
=
= 2
Therefore, the mean of both the teams is same.
Standard deviation,
=
=
=
=
= 1.09
The standard deviation of team B is 1.25 goals.
The average number of goals scored by both the teams is same that is 2. Therefore, the team with lower standard deviation will be more consistent.
Therefore, team A is more consistent than team B.
Q5. The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are as given below:
Which is more varying, the length or weight?
Sol:
Length
Here, N = 50
Mean,
=
=
= 4.24
Variance,
=
=
=
=
=
=
=
= 0.07
Standard deviation,
=
= 0.26
C.V. (Length)
=
=
= 6.13
Weight
Mean,
=
=
= 5.22
Variance,
=
=
=
=
=
=
=
= 1.89
Standard deviation,
=
= 1.37
C.V. (Weight)
=
=
= 26.24
Therefore, C.V. of weights is greater than the C.V. of lengths. Hence, weights vary more than the lengths.