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
The given data is:
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
First calculate the standard deviation of group A. The midpoint of the intervals is calculated by adding first and last terms and dividing it by 2.
The formula to calculate the new values of the given data is,
y i = x i − A h
Here, A is the assumed mean and h is the height between the classes.
Substitute 45 for A and 10 for h in the above formula and make a table by using these values.
Marks Group A, f i Midpoint, x i y i = x i − A h ( y i ) 2 f i y i f i ( y i ) 2 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 ∑ i = 1 n f i = 150 ∑ i = 1 n y i f i = − 6 ∑ i = 1 n f i ( y i ) 2 = 342
The formula to calculate the mean is,
x ¯ = A + ∑ i = 1 n y i f i ∑ i = 1 n f i × h x ¯ = A + ∑ i = 1 n y i f i N × h (1)
Where, N is the sum of frequency.
Substitute 45 for A, 10 for h, 150 for N and − 6 for ∑ i = 1 n y i f i in equation (1).
x ¯ = 45 + ( − 6 ) 150 × 10 = 45 − 0.4 = 44.6
Therefore, the mean of the given data is 44.6.
The formula to calculate the variance is,
σ 2 = h 2 N 2 [ N ∑ i = 1 n f i ( y i ) 2 − ( ∑ i = 1 n f i y i ) 2 ] (2)
Substitute 150 for N, 10 for h, − 6 for ∑ i = 1 n y i f i and 342 for ∑ i = 1 n f i ( y i ) 2 in equation (2).
σ 2 = ( 10 ) 2 ( 150 ) 2 [ 150 ( 342 ) − ( − 6 ) 2 ] = 100 22500 ( 51264 ) = 1 225 ( 19275 ) = 227.84
Thus, the variance of the given data is 227.84.
The formula to calculate the standard deviation is,
S .D . = σ 2 (3)
Substitute 227.84 for σ 2 in equation (3).
S .D . = 227.84 = 15.09
Thus, standard deviation of the given data is 15.09.
Now, the standard deviation of group B is being calculated. The midpoint of the intervals is calculated by adding first and last terms and dividing it by 2.
The formula to calculate the new values of the given data is,
y i = x i − A h
Here, A is the assumed mean and h is the height between the classes.
Substitute 45 for A and 10 for h in the above formula and make a table by using these values.
Marks Group A, f i Midpoint, x i y i = x i − A h ( y i ) 2 f i y i f i ( y i ) 2 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 ∑ i = 1 n f i = 150 ∑ i = 1 n y i f i = − 6 ∑ i = 1 n f i ( y i ) 2 = 366
The formula to calculate the mean is,
x ¯ = A + ∑ i = 1 n y i f i ∑ i = 1 n f i × h x ¯ = A + ∑ i = 1 n y i f i N × h (1)
Where, N is the sum of frequency.
Substitute 45 for A, 10 for h, 150 for N and − 6 for ∑ i = 1 n y i f i in equation (1).
x ¯ = 45 + ( − 6 ) 150 × 10 = 45 − 0.4 = 44.6
Therefore, the mean of the given data is 44.6.
The formula to calculate the variance is,
σ 2 = h 2 N 2 [ N ∑ i = 1 n f i ( y i ) 2 − ( ∑ i = 1 n f i y i ) 2 ] (2)
Substitute 150 for N, 10 for h, − 6 for ∑ i = 1 n y i f i and 366 for ∑ i = 1 n f i ( y i ) 2 in equation (2).
σ 2 = ( 10 ) 2 ( 150 ) 2 [ 150 ( 366 ) − ( − 6 ) 2 ] = 100 22500 ( 54864 ) = 1 225 ( 54864 ) = 243.84
Thus, the variance of the given data is 243.84.
The formula to calculate the standard deviation is,
S .D . = σ 2 (3)
Substitute 243.84 for σ 2 in equation (3).
S .D . = 243.84 = 15.61
Thus, standard deviation of the given data is 15.61.
Since the mean of both the groups are the same, the variability is calculated by standard deviation. As standard deviation of group B is more than the standard deviation of group A, therefore, group B has more variability.
Thus, group B has more variability than group A.
The given data is:
| 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 |
First calculate the standard deviation of group A. The midpoint of the intervals is calculated by adding first and last terms and dividing it by 2.
The formula to calculate the new values of the given data is,
Here, A is the assumed mean and h is the height between the classes.
Substitute 45 for A and 10 for h in the above formula and make a table by using these values.
| Marks | Group A, | Midpoint, | | | | |
| 10-20 | 9 | 15 | | 9 | | 81 |
| 20-30 | 17 | 25 | | 4 | | 68 |
| 30-40 | 32 | 35 | | 1 | | 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 |
| | | |
The formula to calculate the mean is,
Where, N is the sum of frequency.
Substitute 45 for A, 10 for h, 150 for N and
Therefore, the mean of the given data is 44.6.
The formula to calculate the variance is,
Substitute 150 for N, 10 for h,
Thus, the variance of the given data is 227.84.
The formula to calculate the standard deviation is,
Substitute 227.84 for
Thus, standard deviation of the given data is 15.09.
Now, the standard deviation of group B is being calculated. The midpoint of the intervals is calculated by adding first and last terms and dividing it by 2.
The formula to calculate the new values of the given data is,
Here, A is the assumed mean and h is the height between the classes.
Substitute 45 for A and 10 for h in the above formula and make a table by using these values.
| Marks | Group A, | Midpoint, | | | | |
| 10-20 | 10 | 15 | | 9 | | 90 |
| 20-30 | 20 | 25 | | 4 | | 80 |
| 30-40 | 30 | 35 | | 1 | | 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 |
| | | |
The formula to calculate the mean is,
Where, N is the sum of frequency.
Substitute 45 for A, 10 for h, 150 for N and
Therefore, the mean of the given data is 44.6.
The formula to calculate the variance is,
Substitute 150 for N, 10 for h,
Thus, the variance of the given data is 243.84.
The formula to calculate the standard deviation is,
Substitute 243.84 for
Thus, standard deviation of the given data is 15.61.
Since the mean of both the groups are the same, the variability is calculated by standard deviation. As standard deviation of group B is more than the standard deviation of group A, therefore, group B has more variability.
Thus, group B has more variability than group A.
