A random sample of bonuses (in millions of dollars) paid by large companies to their executives is shown below.
Class boundary
Frequency
0.5----3.5
11
3.5----6.5
12
6.5----9.5
4
9.5----12.5
2
12.5---15.5
1
Find
Mean deviation about the mean
To find the mean deviation about the mean, we first need to calculate the mean of the given data set.
The midpoints of each class interval can be calculated as follows:
Midpoint of 0.5 - 3.5: (0.5 + 3.5) / 2 = 2
Midpoint of 3.5 - 6.5: (3.5 + 6.5) / 2 = 5
Midpoint of 6.5 - 9.5: (6.5 + 9.5) / 2 = 8
Midpoint of 9.5 - 12.5: (9.5 + 12.5) / 2 = 11
Midpoint of 12.5 - 15.5: (12.5 + 15.5) / 2 = 14
Next, we calculate the mean of the data set using the midpoints and frequencies:
Mean = (Midpoint1 * Frequency1 + Midpoint2 * Frequency2 + ... + Midpointn * Frequencyn) / (Total Frequency)
Mean = (2*11 + 5*12 + 8*4 + 11*2 + 14*1) / (11 + 12 + 4 + 2 + 1) = 6.25
The mean of the data set is 6.25.
To calculate the mean deviation about the mean, we find the deviation of each midpoint from the mean, multiply it by the respective frequency, and then sum these values:
Mean Deviation about the Mean = (|Midpoint1 - Mean| * Frequency1 + |Midpoint2 - Mean| * Frequency2 + ... + |Midpointn - Mean| * Frequencyn) / (Total Frequency)
Mean Deviation about the Mean = (|2 - 6.25|*11 + |5 - 6.25|*12 + |8 - 6.25|*4 + |11 - 6.25|*2 + |14 - 6.25|*1) / (11 + 12 + 4 + 2 + 1)
Simplifying this expression, we get:
Mean Deviation about the Mean = (4.25*11 + 1.25*12 + 1.75*4 + 4.75*2 + 7.75*1) / 30
Calculating the values in the numerator:
Mean Deviation about the Mean = (46.75 + 15 + 7 + 9.5 + 7.75) / 30
Calculating the final value:
Mean Deviation about the Mean = 86 / 30
Mean Deviation about the Mean ≈ 2.87
Therefore, the mean deviation about the mean is approximately 2.87.