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Everyday arithmetic

Average: balance the numbers

Understand the mean as a fair share, reconstruct totals, and combine groups without losing their sizes.

BeginnerAbout 12 min3 worked examples
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01 / THE STARTING POINT

Before you begin

What you need: Addition, division and simple equations.

  • Find the arithmetic mean
  • Recover missing values from a total
  • Calculate a weighted combined average
02 / BUILD THE CONCEPT

Understand the idea

Average is an equal redistribution

The arithmetic mean is the value each item would have if the total were shared equally. The individual values may differ, but their sum is unchanged.

Keep track of total and count

The useful identity is total = average × count. If an item is added or removed, change both the total and the count before dividing.

Group sizes matter

A mean of two group averages is valid only when the groups have equal sizes. Otherwise, reconstruct the two totals and divide by the combined count.

03 / A DIFFERENT WAY TO SEE IT

Different values, the same fair share

12
18
24
30
The total is 84. Redistributing equally across four values gives a mean of 21.
04 / FROM IDEA TO ANSWER

A method you can reuse

KEEP THIS HANDY

Mean = sum of all values ÷ number of values

The count must be positive. Combined mean = (n₁m₁ + n₂m₂)/(n₁ + n₂).
  1. Calculate or reconstruct the total.
  2. Adjust the total and count for additions, removals or replacements.
  3. Divide the updated total by the updated count and check the range.
A useful insight

Choose a convenient assumed mean. For 48, 51 and 54, use 50: deviations −2, +1, +4 sum to 3, so the mean is 50 + 3/3 = 51.

05 / WATCH THE METHOD WORK

Worked examples

Read the question first. Try a step yourself, then compare your reasoning.

EXAMPLE 01

Find the mean

Average 12, 18, 24 and 30.

  1. Sum = 12 + 18 + 24 + 30 = 84.
  2. There are four values.
  3. 84 ÷ 4 = 21.
Answer21
EXAMPLE 02

Remove a value

Eight values have mean 25. Remove the value 32. Find the new mean.

  1. Original total = 8 × 25 = 200.
  2. New total = 200 − 32 = 168; new count = 7.
  3. 168 ÷ 7 = 24.
Answer24
EXAMPLE 03

Combine unequal groups

10 students average 60 marks; 20 students average 75. Find the combined mean.

  1. Group totals: 10 × 60 = 600 and 20 × 75 = 1500.
  2. Combined total = 2100; combined count = 30.
  3. 2100 ÷ 30 = 70.
Answer70 marks
06 / YOUR TURN

Check your understanding

Quick questions, with explanations. These are for self-study and do not affect your account score.

0 of 3 questions checked

QUESTION 1The mean of five values is 16. What is their sum?

See the explanation

80. Total = 5 × 16 = 80.

QUESTION 2The mean of 4, 8 and x is 7. Find x.

See the explanation

9. The total is 21. Subtract 4 + 8 to get 9.

QUESTION 3Two values average 10; three other values average 20. What is the combined mean?

See the explanation

16. Combined total = 2 × 10 + 3 × 20 = 80. Divide by 5 to get 16.

07 / TAKE THE NEXT STEP

Put your understanding to work

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EXPLORE THE NUMBERS

Average Calculator

Change the inputs and follow the working.

Open calculator →TEST YOUR KNOWLEDGE

Average Practice Quiz 1

Move on to a timed quiz with detailed results.

View quiz →TEST YOUR KNOWLEDGE

Average Practice Quiz 2

Move on to a timed quiz with detailed results.

View quiz →TEST YOUR KNOWLEDGE

Average Practice Quiz 3

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View quiz →BUILD FLUENCY

Average practice

Try a fresh set of ten questions.

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