In computing the median income of any group, some federal agencies omit all members of the group who had no income. Give an example to show that the reported median income of a group can go down even though the group becomes economically better off. Is this true of the mean income?
To understand how the reported median income of a group can go down even though the group becomes economically better off, let's consider an example.
Suppose there is a group of five individuals with the following incomes in thousands of dollars: $20,000, $25,000, $30,000, $35,000, and $1,000,000. If we calculate the median income of this group without omitting any members, the median would be $30,000.
However, let's say a federal agency decides to omit the one person with an income of $1,000,000 from the calculation. In this case, the remaining four individuals have incomes of $20,000, $25,000, $30,000, and $35,000. The new median income, after omitting the $1,000,000 income, is $27,500.
Here's how this happened: by excluding the person with a high income, the median income of the remaining group decreased. Although the group became economically better off by having a millionaire within it, the reported median income decreased due to the omission.
On the other hand, for mean income, the situation is different. The mean is calculated by summing all the incomes and dividing by the total number of individuals. In the example above, the mean income of the group with all five individuals would be $222,000 ($1,110,000 divided by 5). If we omit the $1,000,000 income and calculate the mean with the remaining four individuals, it becomes $27,500 ($110,000 divided by 4).
In this example, even though the group is economically better off, the mean income decreased as well, as it takes into account the values of all individuals, including those with extremely high or low incomes.
In summary, omitting individuals with no income can cause the reported median income of a group to decrease, indicating a decline despite economic improvement. However, this is not necessarily the case for mean income, as it factors in the values of all individuals and can be influenced by extreme values.