# Meaning of phrases (median vs mean): Income of the average person in my city vs. The average income of people in my city

Hello Cross Validated,

I have been discussing things like the average income of a person in my city, and have come to think there is a subtle but important difference in the phrases "Income of the average person in my city" and "The average income of people in my city" in that the first would be the median income in the city, while the later would be the mean income.

I would guess that this difference of phrasing would be often ignored and might be confusing (especially to the average person), but is it technically correct?

Thanks!

• How would you classify someone as “average” in order to then find out their income?
– Dave
Jul 29 '20 at 2:18
• My instinct is to line them up and pick the middle one, though I guess that's a bit of a tautology? Jul 29 '20 at 2:27
• Line them up according to what? Height? Last name alphabetical order? Alphabetically by height? (That last one is a joke.)
– Dave
Jul 29 '20 at 2:31
• @Dave Since we are discussing income, I assume we would line them up by income. Are you suggesting it is incorrect to refer to the average person at all? Jul 29 '20 at 3:30

In ordinary (non-technical) English the word average can be used for various concepts relating to a typical value of a list of numbers: maybe the "most common" income, middle income in @Dave's line-up, or the number you get when you add all the incomes and divide by the number of incomes. If two people disagree about "the average," they may need to discuss what each of them really has in mind.

In statistical terminology the first of these is called the mode (most common), the second is called the median (middle entry in a sorted list), and the third is called the (arithmetic) mean. [There are also 'geometric' and 'harmonic' means, but let's leave them for later.]

Sometimes mode, median and mean are all the same number (or nearly the same number). If we have a list of heights of 25 randomly chosen college students (measured to the nearest inch), we might get:

  57 61 62 64 64 65 66 66 66 66 66 66 68
68 68 69 70 70 71 71 71 71 71 71 72


For this sample: the mode is 66, the median is 68, and the mean is 67.2.

However, for the data: 1, 2, 3, 3, 3, 4, 6, 7, 10, 15, 23, the mode is 3, the median is 4 and the mean is 77/11 = 7.

A billing agency might have 300 clerks making \$40,000 a year, 10 supervisors making \$80,000, and a CEO making \$800,000. Then the mode and median are both \$40,000, and the mean is \\$43,730. The mode and median may seem more typical, but only the mean is directly related to the total annual payroll.

The harmonic mean is the reciprocal of the arithmetic mean of reciprocals of the numbers. It can be useful for finding average MPG (miles per gallon) for driving a car.

I live on a hill. If I drive a mile down the hill at 30 MPG, a mile on flat roads at 25 MPG and back home for a mile up hill at 20 MPG, my average gas mileage is not 25 MPG. I use 1/30 gal. downhill, 1/25 gal. on flat roads, and 1/20 gal. uphill, for a total of 0.1233 gal. or an average of 0.1233/3 = 0.0411 gal./mi. So my average MPG is 1/0.0411 = 24.3 MPG. [Where liters and kilometers are used, a customary measure of fuel efficiency is the number of liters to go 100km, and the arithmetic mean works fine for that.]

The geometric mean is used, among other things, to compute average interest rates or stock portfolio returns. I will give links for that: Investopia and Wikipedia.

• Thanks @BruceET, that was helpful. Since there are multiple technical meanings of the word average, then there is not single specific technical meaning of the "average person" or "average income"? Jul 29 '20 at 16:37
• Right. But the default supposition would be that 'average' is 'arithmetic mean', subject to revision with more information or context. Jul 29 '20 at 17:58