Averages and Weighted Averages: Shortcuts That Save Time

Averages are simple, but exam questions dress them up in many ways. A few tricks handle most of them.

The formula

Average = Sum ÷ Number of items, so Sum = Average × Number.

Example: the average of 12, 15, 18, 20, 25 is 90 ÷ 5 = 18.

Handy results

• Average of the first n natural numbers = (n + 1) ÷ 2.
• Sum of the first n natural numbers = n(n + 1) ÷ 2.
• Average of consecutive numbers = the middle number (or the mean of the two middle ones).
• Average of an arithmetic progression = (first + last) ÷ 2.

Example: average of 1 to 50 = 51 ÷ 2 = 25.5.

When a member is replaced

If the average of a group changes when one member is replaced, use the change in total.

Example: The average of 5 people is 30. One person aged 40 leaves and a new person joins, and the average rises to 32. The total rose by 5 × 2 = 10, so the new person is 40 + 10 = 50.

Adding a new member

Example: The average of 9 numbers is 20. A tenth number is added and the average becomes 22. The tenth number = 10 × 22 − 9 × 20 = 220 − 180 = 40.

Weighted average

When groups have different sizes, weight each average by its size.

Example: A class of 20 students averages 60, and a class of 30 averages 70.

Combined average = (20 × 60 + 30 × 70) ÷ 50 = (1200 + 2100) ÷ 50 = 66.

Average speed

The average speed for a journey is total distance ÷ total time, not the average of the speeds. See our time, speed and distance guide.

Deviation method

To average numbers close together, pick a guess and average the differences.

Example: 102, 98, 105, 95. Guess 100: differences +2, −2, +5, −5 sum to 0. Average = 100.

Common traps

1. Averaging averages without weights.
2. Forgetting to update the count when adding or removing members.
3. Confusing sum and average.

Tips

• Convert to totals first.
• Work with changes rather than full values.
• Sense-check: the average always lies between the smallest and largest values.

Practice

Short daily sets keep these tricks fresh.

Practise with Online Exam Question & Answer →