The Difference between Means and Medians

Wonderful World

Don't know much about algebra,

Don't know what a slide rule is for

If you don’t understand math, then you may get talked into supporting some decisions that are not in your own best interest.

Teen Talk Barbie is right “Math is Hard”. But that doesn’t mean that you shouldn’t try to understand math. If you don’t understand some basic concepts of math, then you can get some unexpected results. One of those is that increasing the average, also known as the mean, does not make things better for the typical person. The average, mean, household income is total income divided by total households. The median household income, the income of the middle, is the income at which 50% of the households have incomes that are higher, and 50% of the households have incomes that are lower. When the median and the mean ( as well as another statistic called the mode) are the same, the name of that is a normal distribution. When their difference becomes greater, should that be called abnormal? To illustrate this, consider the Town of Duckburg, home of Uncle Scrooge McDuck.

In the town of Duckburg, 1,000 households have an income of $50,000 per year while Scrooge McDuck has an income of $5 million per year. The mean, average, income of all households is almost $54,945 per year, while the median, 50th percentile, income of the households is $50,000. The town is going to receive new income of $10 million per year but they have to decide how to divide this new income among the households in the town.

Scrooge says that his income is over 9100% of the mean income, so he should get most of that new income. However, he says that he wants to be generous, and suggests that he should only get 50% of that new income and the other $5 million should be shared among the rest of the households. This increases Scrooge McDuck’s income from $5 million to $10 million per year, while the other duck households increase from $50,000 to $55,000 per year. However, while the mean income increased by almost $10,000 to $64,935 , the median income only increased by only $5,000 to $55,000. The problem is that while Scrooge’s income was over 9100% of the mean income, it was only 9.9% of the total income of all households in Duckburg. To keep the income distribution the same, he and every other household should each only get a 9.9% increase in income. Not a 100% increase for Scrooge and a 10% increase for all other households. Since total income increases by 9.9%, if every household’s income had gotten an increase of 9.9%, including Scrooge’s, then the gap between mean and median income would not have increased.

Might giving most of the new income to Scrooge McDuck have been a good idea? Would he be more likely than most households to use that income to increase the economy, as supply side economics believes, where you accept becoming less equitable but possibly more productive? Maybe, but fans of Uncle Scrooge know the most likely outcome would be that Scrooge would only increase the amount of money in his vault, in which he will swim.