When the experts tell us that average incomes have risen, what does that actually tell us? Not much that’s useful in understanding real world politics and the inequities of economic life. I know that most of us understand averages, now don’t we? What’s there to explain? Well, let’s have a look at average incomes.

Incomes can be looked at using three measures of central tendency: the mean (the average), the median and the mode. For a technical explanation of these terms see: https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php.

Each of these measures has its advantages and disadvantages.  If we take a population and want to figure out their average height, it’s simple: measure each individual in the population, add the heights of all of them together and divide by the number of individuals measured. It’s unlikely that all people in a population would be of the same stature, and it doesn’t tell us anything about the range of heights. It’s always good to know what the range is otherwise there’s no way of judging just how variable the heights are in a population. Same goes for incomes. Scenario A below is of a fictitious population of income earners that each earns $100 thousand dollars each per year. That makes the total incomes earned by the group $1 million dollars and the average is $100 thousand. If you know where such a world exists, please let me know.

Scenario A
Income Earner	Income $	% of total income
Income earner 1	100,000	10
Income earner 2	100,000	10
Income earner 3	100,000	10
Income earner 4	100,000	10
Income earner 5	100,000	10
Income earner 6	100,000	10
Income earner 7	100,000	10
Income earner 8	100,000	10
Income earner 9	100,000	10
Income earner 10	100,000	10
Total Income	1,000,000	100
Average Income	100,000

Now, let’s consider Scenario B below. The total income of all ten earners is still $1 million and the average is still $100 thousand. What’s changed? The distribution. In this scenario, the highest income earner brought in $200 thousand while the lowest earner brought in $45 thousand. That’s a significant difference and means that there is substantial income inequality but if a government wanted to obfuscate rather than clarify the issue, it might want to argue that average incomes haven’t changed. What are you worried about?

Scenario B
Income Earner	Income $	% of total income
Income earner 1	200,000	20.00
Income earner 2	150,000	15.00
Income earner 3	130,000	13.00
Income earner 4	100,000	10.00
Income earner 5	95,000	9.50
Income earner 6	85,000	8.50
Income earner 7	75,000	7.50
Income earner 8	65,000	6.50
Income earner 9	55,000	5.50
Income earner 10	45,000	4.50
Total Income	1,000,000	100.00
Average Income	100,000

Now, consider this third scenario. The total incomes are still $1 million, the average is still $100 thousand. However, in this scenario, the highest income earner is taking in 72.7 of the total income. This scenario pushes income inequality to a much greater degree.

Scenario C
Income Earner	Income $	% of total income
Income earner 1	726,990	72.70
Income earner 2	85,000	8.50
Income earner 3	45,000	4.50
Income earner 4	20,430	2.04
Income earner 5	20,430	2.04
Income earner 6	20,430	2.04
Income earner 7	20,430	2.04
Income earner 8	20,430	2.04
Income earner 9	20,430	2.04
Income earner 10	20,430	2.04
Total Income	1,000,000	100.00
Average Income	100,000

Now, think of a situation where aggregate incomes rise, but the lowest earners retain their share of earnings at 2.04 percent. Now the government can say: What are you worried about? Your income hasn’t changed at all and the country is getting richer. Silly you. Maybe you should just work a little harder.

Calculating median income isn’t much more helpful. The median income is the point where half the incomes are above and half below. That will tell us if a population distribution is changing in broad terms, but the median income in scenario C is $20,430.00. What does that tell you about the distribution of incomes in this population.

Calculating the modal income is interesting. The modal income is the one that appears most often in a distribution. In scenario C, the modal income is clearly $20,430.00. That’s the income most people make. It still doesn’t shed much light on the inequality in a distribution.

So, we always need more than a measure of central tendency to tell us what’s really going on in the world in terms of income inequality. If the government, or the head of the Bank of Canada for example, tells you that average incomes are rising, know that you’re not being told the whole story.

If you’re not concerned about these things, never mind.

Note: this post was inspired by a section of Robert Sapolsky’s book: Why Zebras don’t get ulcers.