Statistics From Zero · Lesson 2 of 22
The Average and Its Evil Twins
In one sentence: There are three different 'averages' (mean, median and mode), and picking the wrong one can make a pub full of normal people look like millionaires.
The idea
"Average" sounds like one thing. It's actually three different things wearing the same name tag:
- The mean: add everything up and divide by how many there are. This is what most people mean by "the average".
- The median: line everything up from smallest to biggest and take the one in the middle.
- The mode: the value that turns up most often.
Most of the time they're close to each other and nobody cares which one you use. The interesting part is when they split apart, because that's exactly when someone can mislead you by picking the convenient one.
A billionaire walks into a pub
Uncle Gary's local, on a quiet Tuesday. Nine regulars, each earning $30,000 a year. The mean income in the pub is $30,000. So is the median. So is the mode. Everyone agrees on what's typical.
Then a billionaire walks in for a pint.
The mean income in the pub jumps to about $100 million. The median is still $30,000. Gary is now, on average, a multi-millionaire. Sadly, the barman still won't let him run a tab.
One extreme value, an outlier, dragged the mean a hundred million dollars away from anything real. The median didn't even blink, because the person in the middle of the line is still a regular earning $30,000.
The pub after the billionaire walks in: yearly income
So which one should I use?
| Use | When | Example |
|---|---|---|
| Mean | Numbers are fairly balanced, no extreme values | Heights, exam marks, daily temperatures |
| Median | A few extreme values pull one way (skewed data) | Incomes, house prices, waiting times |
| Mode | Categories, or "what's most common?" | Most popular drink at the bar, most sold shoe size |
The mode has a superpower the others don't: it works for things that aren't numbers. You can't calculate the mean of "lager, lager, cider, gin and tonic", but you can say the most common order is lager. And a shoe shop has no use for "the mean shoe size is 7.43". It wants to know which size to stock most of.
Try it yourself
Gary's pub quiz team scored this over seven weeks: 4, 5, 5, 6, 7, 5 and 31. (In week six, by pure coincidence, every question was about 1980s darts.)
Work out the mean, the median and the mode. Which one best describes a typical week for the team?
Show the answer
Mean: 4 + 5 + 5 + 6 + 7 + 5 + 31 = 63, divided by 7 = 9.
Median: in order, 4, 5, 5, 5, 6, 7, 31. The middle (4th) value is 5.
Mode: 5 appears three times, more than anything else.
The median and mode (5) describe a typical week. The mean (9) is higher than six of the seven actual scores, thanks to one week of darts trivia. If Gary tells you the team "averages 9 points", he's technically right and practically lying.
The words your lecturer uses
| Lecturer says | It means |
|---|---|
| Measures of central tendency | The different kinds of "average": mean, median, mode |
| Arithmetic mean | Add up and divide (the usual average) |
| Median | The middle value when the data is in order |
| Mode | The most frequent value |
| Outlier | A value far away from the rest (the billionaire) |
| Skewed distribution | Data with a long tail on one side, so mean and median split apart |
| Robust statistic | One that outliers barely affect, like the median |
| Bimodal | Data with two peaks, so it has two modes |
Three things to remember
- "Average" can mean the mean, the median or the mode. Always ask which.
- The mean is pulled by outliers. The median isn't. For incomes, prices and anything lopsided, trust the median.
- When the mean and the median are far apart, the data is skewed, and someone may be choosing the number that suits them.
Part of Statistics From Zero, a free beginner course by WrongStat. Not a substitute for your lecturer, but hopefully a good translator.