Relative mean deviation is the average distance of everyone’s income from the mean, expressed as a proportion of the mean.

0 20000 40000 60000 80000 100000 120000
0.010 0.060 0.110 0.160 0.210 0.260 0.310 0.360 0.410 0.460 0.510 0.560 0.610 0.660 0.710 0.760 0.810 0.860 0.910 0.960

Income Average A B

a. Calculate the areas A (bounded by the parade curve and the average income) and B (bounded by the parade curve, the average income line, and the vertical line where the proportion of the population is 1).

b. Calculate the relative mean deviation.

c. Reallocate income so that everyone with income below the mean has the same income, say R15,000. Now, calculate the relative mean deviation. How does the answer in (c) compare to that in (b)?

The areas A and B can be calculated by dividing the area into small rectangles and summing their areas. The relative mean deviation can be calculated by finding the average distance of each income value from the mean and dividing it by the mean. After reallocating income so that everyone below the mean has the same income, the relative mean deviation can be calculated again. The answer in (c) can be compared to that in (b) to see if reallocating income has reduced the spread of income values.

Explanation:

To calculate the areas A and B, we need to plot the given income values and their corresponding proportions on a graph. The income values are plotted on the x-axis, and the proportions are plotted on the y-axis.

After plotting the points, we can connect them to form a curve, which is called the parade curve. The average income line is a horizontal line at the mean income value.

To calculate area A, we need to find the area between the parade curve and the average income line. This can be done by dividing the area into small rectangles and summing their areas.

To calculate area B, we need to find the area between the parade curve, the average income line, and the vertical line where the proportion of the population is 1. Again, we can divide the area into small rectangles and sum their areas.

The relative mean deviation can be calculated by finding the average distance of each income value from the mean and dividing it by the mean.

To reallocate income so that everyone below the mean has the same income, we need to set their income to a specific value, in this case, R15,000. After the reallocation, we can calculate the relative mean deviation using the same formula as before.

The answer in (c) can be compared to that in (b) by calculating the difference between the two relative mean deviations. If the answer in (c) is lower than that in (b), it means that reallocating income has reduced the spread of income values.

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