🌱 A garden where numbers grow
The Average Machine
You know how to work out an average, but what does the number actually mean?
You’ve learned how to find an average.
You add the numbers up, divide, and write down the answer.
But what does that number actually mean?
The set-up
Five friends share a pile of sweets
The mean is what everyone would have if you shared the whole pile out evenly. Once you can picture that, you can also spot when an average gives the wrong idea.
Five friends brought sweets to the cinema: 2, 3, 5, 6 and 9. The dashed line is where they’d all end up if you shared the pile out equally.
Your turn. Drag a tower up or down (or use the buttons), then press share and see where everything ends up.
The dashed line shows the fair share, the height every tower would be if you shared the pile out equally. Change any tower and the line moves straight away. That line is the mean.
You don’t need to memorise the mean as a rule.
Just picture the level a pile reaches when you share it out evenly.
The trap
Does anyone have to be average?
Those five towers have a mean of 5. Answer this before you look: if five numbers average 5, does at least one of them have to be 5?
One number on its own can change the whole average.
If it gets big enough…
…it pulls the average up with it.
The wrecker
One kid can drag the whole average
Five kids get pocket money. Four of them get €5, and the fifth is Jo. Before you touch anything, think about this: if Jo’s money goes up to €100, what will the mean be? Pick a number in your head, then drag Jo’s bar and see.
Here’s a simpler version. You’ve got €5. Your mate has…
Finding a mean takes only two steps.
Pile everything up, then deal it out.
Doing the sums
Two steps: pile up, then deal out
There are two steps. Pile up: add all the numbers to get one total. Deal out: share the total between however many numbers there are. Change the values and follow the working. The tempting wrong move is shown crossed out.
Here’s the smallest example, with two numbers: 4 and 10. Pile up: 4 + 10 = 14. Deal out between 2: 14 ÷ 2 = 7. As towers, you’d slide 3 blocks off the 10 onto the 4, and they’d both be 7. Levelling them out is the same as dividing.
To go backwards, work out mean × how many = total. So 6 × 4 = 24. You can’t tell which four numbers they were (they could be 6, 6, 6, 6 or 1, 2, 9, 12), but the total has to be 24. So if someone tells you their four test scores “average 6 out of 10”, you can work out that their total is 24.
Every average works the same way.
It’s a pile, shared out evenly…
…even if no real day, game or person matches it. Goals per game, screen time and July rainfall all work like this. Once you can picture the share-out, you can tell when an average gives the wrong idea.
One more thing
Next time someone tells you “the average family has 2.4 children”, you can explain it: it’s all the children shared out between all the families. Nobody has 0.4 of a sibling. And when an average sounds surprising, ask this first: what’s the biggest number in that pile? If you can explain to someone why a made-up number can still be really useful, you understand averages properly.