Average

high importance~1 Q in Tier 116 formulas⚡ 12 shortcuts4 subtopics

Overview Learn (4)Formulas (16)Examples (5)Practice (60)

Subtopic 1 of 4·Members joining or leaving →

Average & the sum bridge

$\text{Average} = \dfrac{\text{Sum of observations}}{\text{Number of observations}}$ . The useful direction is the reverse: Sum = Average × n — most CGL average questions are solved by converting to sums immediately.

Properties:

If every observation is increased by k, the average increases by k (same for −, ×, ÷).
The average is always between the smallest and largest observation.
If all numbers are equal, average = each number; the sum of deviations from the average is zero.

Standard averages (memorise): first n naturals $= \frac{n+1}{2}$ ; first n odd $= n$ ; first n even $= n+1$ ; squares $= \frac{(n+1)(2n+1)}{6}$ ; cubes $= \frac{(n+1)^2}{4}$ . Consecutive numbers: average = middle term = $\frac{\text{first} + \text{last}}{2}$ .

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 17 practice Q

Detailed notes

What is an average?

An average is one number that stands for a whole group. If five friends have ₹20, ₹30, ₹25, ₹35 and ₹40, the total is ₹150. Shared equally, each would get $150 \div 5 = ₹30$ . That ₹30 is the average (also called the arithmetic mean). $\text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}}$ An "observation" simply means one value in the list.

The sum bridge — the most useful idea

Turn the formula around: $\text{Sum} = \text{Average} \times n$ . Almost every average question is solved by changing averages into sums, doing the adding or subtracting with sums, and only at the end dividing again. Example: the average of 4 numbers is 25 and three of them are 20, 30 and 18. Sum $= 4 \times 25 = 100$ , so the fourth $= 100 - 68 = 32$ .

Properties you should know

The average always lies between the smallest and the largest value.
Add (or subtract) k to every value → the average goes up (or down) by k.
Multiply (or divide) every value by k → the average is multiplied (or divided) by k.
Do the operations in the same order: "×3 then +5" on an average of 15 gives $15 \times 3 + 5 = 50$ .
The deviations (differences) of all values from the average add up to zero.

Standard averages (learn by heart)

Series	Average
First n natural numbers	$\frac{n+1}{2}$
First n odd numbers	$n$
First n even numbers	$n+1$
Squares of first n naturals	$\frac{(n+1)(2n+1)}{6}$
Cubes of first n naturals	$\frac{n(n+1)^2}{4}$
Example: first 20 even numbers (2 to 40) → average 21. First 25 odd numbers → average 25.

Consecutive numbers and equal gaps

When numbers go up by the same step (consecutive, consecutive even, multiples of 7 …) the average is the middle term, which is also $\frac{\text{first} + \text{last}}{2}$ . 7 consecutive even numbers with average 48 → middle (4th) is 48 → numbers 42 to 54 → largest 54. Multiples of 6 from 1 to 100: 6, 12, …, 96 → average $\frac{6 + 96}{2} = 51$ .

Assumed-mean (deviation) method

To average numbers that are close together, pick a round "assumed" value, write each number as a plus or minus gap from it, average the gaps and add back. 284, 291, 296, 302, 307 around 300: gaps $-16, -9, -4, +2, +7$ → total $-20$ → average gap $-4$ → average $296$ . No big addition needed.

Common traps

Average of averages is NOT the overall average when groups are of different sizes (see weighted average).
Consecutive even/odd numbers step by 2, not 1.
Read "each number is multiplied by 3 and then increased by 5" in order.

Quick revision

Sum = average × count; change everything to sums.
Shift by k → average shifts by k; scale by k → average scales by k.
Equal steps → average = middle term = (first + last)/2.
Series table: $\frac{n+1}{2}$ , $n$ , $n+1$ .
Close numbers → assumed-mean method.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Sum from average / missing value from the sumvery common2 practice Q

How to spot it:

An average and a count are given; the total, or one missing value among known values, is asked.

\text{Sum} = \text{Average} \times n

Multiply average by the number of values to get the sum.
Subtract the known values from that sum.
What is left is the missing value (or the asked total).

Why: the average is only the sum shared equally, so the sum carries all the information.

Example: The average of 4 numbers is 25. Three of them are 20, 30 and 18. Find the fourth.

Sum $= 4 \times 25 = 100$ ; known $= 68$ → fourth $= 32$ .

Type 2: Average of standard series (naturals, odd, even, squares, cubes)very common3 practice Q

How to spot it:

'Average of the first n natural / odd / even numbers' or of their squares or cubes.

\frac{n+1}{2},\quad n,\quad n+1,\quad \frac{(n+1)(2n+1)}{6},\quad \frac{n(n+1)^2}{4}

Identify the series and n (count of terms, not the last term).
Read the average straight from the table.
For even numbers up to 2n, n is half the last term.

Why: each table value is the series-sum formula divided by n.

Example: Find the average of the first 25 odd numbers.

Average of first n odd numbers $= n$ → $25$ . (Sum $= 25^2 = 625$ , $625 \div 25 = 25$ .)

Type 3: Consecutive numbers / equal-gap seriesvery common3 practice Q

How to spot it:

'Average of 7 consecutive even numbers is 48 — find the largest', or average of multiples of k in a range.

\text{Average} = \text{middle term} = \frac{\text{first} + \text{last}}{2}

The average is the middle term of the list.
Step out from the middle by the common gap (1 for consecutive, 2 for even/odd, k for multiples).
For a range of multiples, average = (first multiple + last multiple)/2.

Why: equal gaps are balanced on both sides of the middle, so the middle is the mean.

Example: The average of 7 consecutive even numbers is 48. Find the largest.

Middle (4th) term $= 48$ ; three steps of 2 up → $48 + 6 = 54$ .

Type 4: Same operation on every valuecommon2 practice Q

How to spot it:

'Each number is multiplied by 4 / increased by 6 / divided by 2' — the new average is asked.

\text{new average} = k \times A + c \ \ (\text{apply in the stated order})

Apply the same operations to the old average, in the same order.
Addition shifts the average; multiplication scales it.

Why: sum and average respond to a uniform change exactly like each single value does.

Example: The average of 6 numbers is 15. Each is multiplied by 3 and then increased by 5. New average?

$15 \times 3 + 5 = 50$ .

Type 5: Assumed mean (deviation method)common2 practice Q

How to spot it:

A list of 4-8 close numbers (weights, marks, runs) is given and their average is asked.

\text{Average} = \text{assumed value} + \frac{\text{sum of deviations}}{n}

Pick a round number near the middle.
Write each value's gap from it (+ or −) and add the gaps.
Divide the gap total by n and add to the assumed value.

Why: shifting every value by the same amount shifts the average by that amount.

Example: Find the average of 284, 291, 296, 302 and 307.

Gaps from 300: $-16, -9, -4, +2, +7$ → total $-20$ → $\frac{-20}{5} = -4$ → average $296$ .

Formulas

Definition

\bar{x} = \frac{\sum x_i}{n} \iff \sum x_i = n\bar{x}

Shift property

\overline{x_i + k} = \bar{x} + k,\quad \overline{k x_i} = k\bar{x}

First n naturals / odd / even

\frac{n+1}{2},\quad n,\quad n+1

Squares / cubes

\frac{(n+1)(2n+1)}{6},\quad \frac{(n+1)^2}{4}

Consecutive / AP terms

\text{average} = \frac{\text{first} + \text{last}}{2} = \text{middle term}

Shortcut tricks

⚡ Sum = average × n first

Never manipulate averages directly — flip to sums, adjust, divide back.

Example: The average of 8 numbers is 32.5. Find their sum.

Sum = 8 × 32.5 = 260.

⚡ Uniform shift

Adding k to every number just adds k to the average.

Example: The average of 6 numbers is 15. If each number is multiplied by 3 and increased by 5, find the new average.

New average $= 15 \times 3 + 5 = 50$ .

⚡ Middle of consecutive terms

For evenly spaced numbers, the average is the central value — no addition needed.

Example: The average of 5 consecutive numbers is 30. Find the largest.

Middle = 30 ⇒ numbers 28…32 ⇒ largest 32.

Where students lose marks

Reporting a sum where an average is asked (read the units).
Averaging two averages with different group sizes (a 40-60 'average of averages' error).
Using $\frac{n(n+1)}{2}$ -style sums without dividing by n when the average is asked.
For 'each multiplied by 3 and increased by 5', applying only one of the two operations.

Practice sets — 17 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

Suggested time 4 min · wrong answers go to your mistake notebook automatically.