Averages, and the three ways they mislead you
An average with no spread beside it is half a fact.
Three different things are called the average, they usually disagree, and which one gets quoted is often the most important decision in a piece of statistics.
The three
- Mean - add everything, divide by how many. Uses every value, and is dragged around by extremes.
- Median - sort them, take the middle one. Ignores how extreme the extremes are.
- Mode - the most common value. The only one that works for things that are not numbers.
The owner's 400k drags the mean above every ordinary salary. 'Average salary: 60k' would be arithmetically correct and deeply misleading. This is why income is almost always reported as a median, and why a mean quoted for something with a long tail deserves suspicion.
Which to use
- Mean when values cluster and extremes are genuine - heights, exam marks, repeated measurements.
- Median when there is a long tail or outliers - income, house prices, response times.
- Mode for categories - most common shoe size, most frequent answer.
A useful tell: if mean and median differ a lot, the data is skewed, and whichever one someone chose to quote tells you something about what they wanted you to conclude.
Spread is the missing half
Two classes both average 60%. In one, everyone scored between 55 and 65. In the other, half scored 30 and half scored 90. Same average, completely different situations, and completely different things to do about it.
- Range - largest minus smallest. Simple, and destroyed by one outlier.
- Interquartile range - the span of the middle half. Ignores the extremes on purpose.
- Standard deviation - roughly, the typical distance from the mean. The usual choice for clustered data.
An average quoted without any measure of spread is half a fact, and often the less useful half.
How averages get used badly
Averaging an average
Class A averages 70% with 10 students; class B averages 50% with 30 students. The overall average is not 60%. It is (70x10 + 50x30) / 40 = 55%. You must weight by size, and this error is extremely common in reported figures.
Comparing groups chosen after the fact
'Students who did the revision course averaged higher.' Possibly the course worked. Possibly the students who sign up for extra revision were already the more motivated ones. The average is real; the causal claim attached to it is not established by it.
Simpson's paradox
A trend can appear in every subgroup and reverse when the groups are combined. A treatment can have a better recovery rate in mild cases AND in severe cases, yet a worse rate overall - if it was given mostly to severe cases. The arithmetic is not wrong; the aggregate is answering a different question from the parts.
Ask three questions of any average you are shown. Which average is it? What is the spread? How many things were averaged? An average of four items and an average of four thousand are not the same kind of claim, and the sample size is the detail most often left out.
Why it belongs here
Every one of these mistakes is a reasoning error rather than an arithmetic one - the sums are fine and the conclusion does not follow. That is the same skill the puzzles train: checking whether the thing you have actually established is the thing you are claiming.