CDS Mathematics · Statistics

Properties of the Arithmetic Mean

The mean moves with every shift and scaling of the data, the deviations from it always add to zero, and a combined mean is the frequency-weighted mean of the group means.

All Statistics notes CDS Mathematics strategy

Why this matters

Twelve PYQs, and not one of them needs the data written out. Three properties answer them all: what shifting or scaling does, the zero sum of deviations, and the weighted combined mean.

Concept 1 of 3: Shifting and scaling the data

Adding

k

to every value adds

k

to the total of each, so the mean moves by

k

. Multiplying every value by

c

multiplies the total, and the mean, by

c

Definition

If $y = ax + b$ for every value, then $\bar y = a\bar x + b$ .
$\sum (ax_i + b) = a\sum x_i + nb$ .
Changing units (marks out of $250$ to marks out of $50$ ) scales every value, so differences scale too.

Linear change

y = ax + b \;\Rightarrow\; \bar y = a\bar x + b

Worked example

The mean of

40

values is

12

. Each value is doubled and then

3

is subtracted. Find the new mean and the new total.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (II) 2026 — Elementary Mathematics · Q97Moderate

Example 1 · Statistics · Properties of the Arithmetic Mean

If the mean of 15 observations, namely

x_1, x_2, x_3, \ldots, x_{15}

is 2.5, then what is the value of

\sum_{i=1}^{15} [5(4x_i + 1)]

The added constant is added n times to the total

\sum (4x_i + 1)

over

15

values is

4\sum x_i + 15

, not

4\sum x_i + 1

Drill 3 more on shifting and scaling the data

Concept 2 of 3: Deviations from the mean and from other points

The mean is the balance point: deviations above it exactly cancel those below. Deviations from any other point

A

add up to

n(\bar x - A)

, which gives the mean when two such sums are known.

Definition

$\sum (x_i - \bar x) = 0$ for every data set.
$\sum (x_i - A) = n(\bar x - A)$ , so $\bar x = A + \dfrac{\sum(x_i - A)}{n}$ .
Two such sums, from $A$ and $B$ , give two equations in $n$ and $\sum x_i$ .

Deviations

\sum (x_i - \bar x) = 0, \qquad \sum (x_i - A) = n(\bar x - A)

Diagram · mean = the balance point

Treat each value as equal weight on a beam; the mean is the point where it balances. The pulls on the left (deviations −3, −1) exactly cancel those on the right (0, +4), which is the identity Σ(xᵢ − x̄) = 0. One extreme value drags the balance point toward it — why the mean is sensitive to outliers.

Worked example

The deviations of some numbers from

20

add to

-12

, and from

14

they add to

24

. Find how many numbers there are and their mean.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q63Moderate

Example 2 · Statistics · Properties of the Arithmetic Mean

The sum of deviations of a set of

n

values measured from 50 is

-10

and the sum of deviations of the values measured from 46 is 70. What is the mean of the values ?

The zero sum needs no arithmetic

When asked for the sum of deviations from the mean, the answer is

0

whatever the data. Computing the mean first wastes time.

Drill 4 more on deviations from the mean and from other points

Concept 3 of 3: Combined and weighted means

Pooling groups adds their totals and their counts. So the combined mean is a weighted average of the group means, closer to the larger group, and always between the smallest and largest group mean.

Definition

$\bar x = \dfrac{n_1\bar x_1 + n_2\bar x_2 + \cdots}{n_1 + n_2 + \cdots}$ .
It lies strictly between the smallest and largest group means (with positive weights).
It equals the simple average of two group means only when the groups are the same size.
One large value pulls the mean up, so most values can lie below the mean.

Combined mean

\bar x = \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}

Worked example

A class has

30

boys with mean

62

and

20

girls with mean

70

. Find the class mean.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (II) 2025 — Elementary Mathematics · Q95Easy

Example 3 · Statistics · Properties of the Arithmetic Mean

A distribution consists of 3 components with frequencies 45, 40 and 55 having their means 2, 2.5 and 2 respectively. What is the mean of the combined distribution?

Weight by the counts

The mean of two group means is the combined mean only for equal groups. With

30

and

10

values the combined mean sits three-quarters of the way toward the larger group's mean.

Drill 2 more on combined and weighted means

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Shifting and scaling the data
Linear change
$y = ax + b \;\Rightarrow\; \bar y = a\bar x + b$
Deviations from the mean and from other points
Deviations
$\sum (x_i - \bar x) = 0, \qquad \sum (x_i - A) = n(\bar x - A)$
Combined and weighted means
Combined mean
$\bar x = \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}$

Watch out for (3)

The added constant is added n times to the total
→ Shifting and scaling the data
The zero sum needs no arithmetic
→ Deviations from the mean and from other points
Weight by the counts
→ Combined and weighted means

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