Chapter 14: Problem 6

A group of 10 people have the following annual incomes: $24,000, $18,000, $50,000, $100,000, $12,000, $36,000, $80,000, $10,000, $24,000, $16,000. Calculate the share of total income that each quintile receives from this income distribution. Do the top and bottom quintiles in this distribution have a greater or larger share of total income than the top and bottom quintiles of the U.S. income distribution?

Short Answer

Expert verified

In this income distribution, the top quintile receives 48.6% of the total income, slightly less than the 52% share in the U.S. income distribution. The bottom quintile receives 5.9% of the total income, significantly larger than the 3% share in the U.S. income distribution.

Step by step solution

Calculate the total income of the group

Add all individual annual incomes together to find the total income of the group: $Total\,Income = \$24,000 + \$18,000 + \$50,000 + \$100,000 + \$12,000 + \$36,000 + \$80,000 + \$10,000 + \$24,000 + \$16,000 $ $Total\,Income = \$370,000 $

Sort the individual incomes in ascending order

Arrange the individual annual incomes in ascending order to divide them into quintiles later: $ \$10,000, \$12,000, \$16,000, \$18,000, \$24,000, \$24,000, \$36,000, \$50,000, \$80,000, \$100,000 $

Define the quintiles and find the sum of each quintile

There are 10 people in the group, so each quintile will have two people. Divide the sorted incomes into quintiles: - Quintile 1 (Bottom): $ \$10,000 + \$12,000 = \$22,000 $ - Quintile 2: $ \$16,000 + \$18,000 = \$34,000 $ - Quintile 3: $ \$24,000 + \$24,000 = \$48,000 $ - Quintile 4: $ \$36,000 + \$50,000 = \$86,000 $ - Quintile 5 (Top): $ \$80,000 + \$100,000 = \$180,000 $

Calculate the share of total income for each quintile

To find the share of total income for each quintile, divide the sum of each quintile by the total income: - Quintile 1 (Bottom): $ \$22,000/ \$370,000 = 0.059 = 5.9\% $ - Quintile 2: $ \$34,000/ \$370,000 = 0.0919 = 9.19\% $ - Quintile 3: $ \$48,000/ \$370,000 = 0.130 = 13\% $ - Quintile 4: $ \$86,000/ \$370,000 = 0.232 = 23.2\% $ - Quintile 5 (Top): $ \$180,000/ \$370,000 = 0.486 = 48.6\% $

Compare the top and bottom quintiles with the U.S. income distribution

According to the U.S. Census Bureau, the income distribution for the top and bottom quintiles in the U.S. are as follows: - Top Quintile: Approximately 52% - Bottom Quintile: Approximately 3% Comparing these values with our calculated shares: - Top Quintile: 48.6% （ours） vs. 52% （U.S.） - Bottom Quintile: 5.9% （ours） vs. 3% （U.S.） In conclusion, the top quintile in our distribution has a slightly smaller share of total income than the top quintile of the U.S. income distribution, while the bottom quintile in our distribution has a significantly larger share of total income than the bottom quintile of the U.S. income distribution.

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