The Mean
When your school report shows your average mark, or a commentator quotes a footballer's average
goals per game, that one handy number is doing a big job: it squashes a whole pile of results into
a single fair figure you can compare at a glance.
The mean is what most people call the average. It answers a
simple question: if everyone shared equally, how much would each one get? You find it with two
easy steps:
- Add up all the values to get the total.
- Divide the total by how many values there are.
Suppose three friends picked 4, 6 and
2 apples. Together that is 4 + 6 + 2 = 12
apples, shared between 3 friends:
\text{mean} = \frac{4 + 6 + 2}{3} = \frac{12}{3} = 4 \text{ apples each.}
So even though nobody actually picked exactly four, four is the fair share — the amount
each friend would have if they pooled their apples and split them evenly.
Grown-up books write the same recipe with symbols. The mean of n
values is written \bar{x} (say "x-bar"), and the stretched-out S,
\sum, just means "add them all up":
\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \cdots + x_n}{n}.
Don't let the symbols scare you — it is exactly the two steps above: add, then divide by the count.
See it: levelling out the towers
Here is a picture that makes the mean feel obvious. Build a tower of cubes for each value — a tall
tower for a big number, a short one for a small number. Now imagine taking cubes off the tall towers
and stacking them onto the short ones until every tower is the same height. The
height they all settle at is the mean.
That is why the mean is sometimes called "levelling out" or "sharing equally": the total number of
cubes never changes, you just spread them out flat. Press Play to find the mean
line and level the towers, and Refresh for a brand-new set of towers.
You and two friends empty your pockets onto the table: you have
5 sweets, one friend has 1, and the other
has 6. To be fair, you pool them and share equally.
Altogether that is 5 + 1 + 6 = 12 sweets for
3 people, so each gets 12 / 3 = 4. The
mean is just the fair share — exactly what "share it out equally" means.
A few worked examples
The recipe never changes: add everything up, then divide by how many values there are.
1. Test scores. Sam scored 7, 8
and 9 on three spelling tests. The mean score is
\frac{7 + 8 + 9}{3} = \frac{24}{3} = 8.
2. More values, same idea. Five children counted the goals they scored:
4, 6, 6,
8 and 6. There are five numbers, so divide by
five:
\frac{4 + 6 + 6 + 8 + 6}{5} = \frac{30}{5} = 6.
3. The mean can be a decimal. Three plants are 5,
6 and 8 cm tall. Their mean height is
\frac{5 + 6 + 8}{3} = \frac{19}{3} = 6.33\ldots \text{ cm.}
None of the plants is exactly 6.33 cm tall — and that is perfectly
normal. The mean is a calculated, "fair-share" number; it does not have to match any value in the
list, and it often comes out as a fraction or decimal even when every value is a whole number.
The three commonest mean mistakes:
- Divide by the count, not the biggest value. For
2, 4, 6 you divide by 3 (three values),
not by 6. The mean is
12/3 = 4.
- The mean need not be one of your numbers. It is often a
decimal or fraction — like 6.33 — even when all the data
are whole numbers.
- A zero is still a value — count it. If scores are
0, 4, 8, that is three values:
(0 + 4 + 8)/3 = 4, not 12/2.
Three friends measure how tall they are: 120 cm,
130 cm and 134 cm. The average (mean)
height is
\frac{120 + 130 + 134}{3} = \frac{384}{3} = 128 \text{ cm.}
Notice that nobody is exactly 128 cm tall. The mean
height is the single height you would get if you could "level everyone out" — like the towers of
cubes above — so that the tall ones lend some height to the short ones until everybody matches.
The balance point
There is one more vivid way to picture the mean. Lay the data out as equal weights on a ruler. The
mean is the exact spot where the ruler balances — the values pulling from the left
and the values pulling from the right cancel out. That is why a single far-away value tips the
balance so much: the mean slides toward it to keep the ruler level.
Drag the orange point below. Watch the balance line — the mean — chase it. Push the point to an
extreme and the mean follows, because every value gets an equal vote.
- \bar{x} = \frac{1}{n}\sum_i x_i — the sum of the values divided by their count n.
- It is the data's balance point, and the height the values "level out" to.
- Every value counts equally, so a single extreme value pulls the mean toward it.
See it explained