Theorem

Deviations about the mean add to zero

Statistics • Mean

Formula

\sum_{i = 1}^{n} (x_{i} - x ˉ) = 0

For any

n

values

x_{1}, x_{2}, \dots, x_{n}

with mean

x ˉ

, the algebraic sum of the deviations about the mean is zero:

\sum_{i = 1}^{n} (x_{i} - x ˉ) = 0

. It follows at once from

n x ˉ = \sum x_{i}

, it holds whatever the values are, and it is what makes the mean the balancing point of the data.

How to use it

1Asked for the algebraic sum of deviations about the mean, answer $0$ without computing the mean or the deviations at all.
2Given all but one value and the mean, set the deviations to sum to zero — it is usually shorter than forming the total.
3Use it as a check after computing a mean by hand: if the deviations do not cancel, the mean is wrong.

Worked example

The mean of

4

8

and

x

7

. Find

x

from the deviations about the mean.

Step-by-step

1
$(4 - 7) + (8 - 7) + (x - 7) = 0$
Why: The deviations about the mean sum to zero.
2
$- 3 + 1 + x - 7 = 0 \Rightarrow x = 9$

Answer

9

Watch out

Only the signed sum vanishes. The sum of absolute deviations

\sum ∣ x_{i} - x ˉ ∣

does not, and neither does the sum of squared deviations — that one is smallest at the mean but zero only if every value equals it. Deviations taken about some other value

a

add to

n (x ˉ - a)

, not to zero, which is exactly what the assumed-mean method corrects for.

Builds on

Mean of a frequency distribution

Also known as

sum of deviations from the mean is zero, algebraic sum of deviations, deviations cancel about the mean, mean as the balancing point

Appears in past papers

2020 (II)

Q97