Averages

An average (or arithmetic mean) tells you the "typical" value of a group of numbers. Averages are easy to learn and are asked in almost every placement paper, often mixed with ages, marks, runs and weights. One formula does most of the work, and a few shortcuts save time.

The basic formula

Average = Sum of values / Number of values.

The same formula rearranged gives the two forms you will use constantly:

• Sum = Average × Number of values.
• Number of values = Sum / Average.

Example: the average of 12, 15, 18 and 23 is (12 + 15 + 18 + 23) / 4 = 68 / 4 = 17.

Think in totals

Almost every average problem becomes easy if you turn each average into a total first. "The average of 6 numbers is 15" really means "the total of the 6 numbers is 90". Work with the totals, add or subtract what changes, and only divide again at the end.

Adding, removing or replacing a value

• Adding a value: new total = old total + the new value, and the count goes up by one.
• Removing a value: new total = old total − the removed value, and the count goes down by one.
• Replacing a value: the count stays the same, and the total changes by (new − old).

Example: the average of 5 numbers is 30. One number, 42, is replaced by 22. The total changes by −20, so the average falls by 20 / 5 = 4, and the new average is 26.

Combining two groups (weighted average)

When two groups with different sizes are combined, you cannot simply average the two averages. Use totals:

Combined average = (n₁ × a₁ + n₂ × a₂) / (n₁ + n₂).

Example: 40 students average 55 marks and 10 students average 80 marks. The total is 40 × 55 + 10 × 80 = 2,200 + 800 = 3,000 for 50 students, so the combined average is 60. It is closer to 55 because that group is larger.

Shortcuts

• Consecutive numbers (or any evenly spaced numbers): the average is the middle value, or (first + last) / 2. The average of 1 to 10 is 5.5. The average of the first n natural numbers is (n + 1) / 2.
• Odd count of consecutive numbers: the average is the middle number. For 7 consecutive numbers with average 20, the numbers run from 17 to 23.
• The change-in-average trick: when a new value is added and the average rises by d, the new value equals the new average plus d × (old count).

Average with a changing score

A batsman has an average of 36 after 10 innings. In the 11th innings he scores 58. The total before was 360, so the new total is 418, and the new average is 418 / 11 = 38, a rise of 2. Notice the shortcut: his score of 58 is 20 above the new average of 38, and 20 = 2 × 10, which is the rise multiplied by the old number of innings. In words: the new score exceeds the new average by (rise × old count). You can use this in either direction, so practise both the total method and the shortcut.

Worked examples

Example 1. The average age of 4 friends is 22. A fifth friend, aged 32, joins. The total age was 88, now 120 for 5 people, so the new average is 24.

Example 2. The average of 3 numbers is 14. The average of the first two is 11. The third number = 3 × 14 − 2 × 11 = 42 − 22 = 20.

Example 3. Marks in four tests are 62, 70, 68 and 80. What mark in the fifth test gives an average of 72? Total needed = 5 × 72 = 360. Marks so far = 280. So the fifth mark is 80.

Common mistakes

• Averaging two averages without weighting by the group sizes.
• Forgetting that the number of values changes when something is added or removed.
• Dividing by the wrong count after a replacement (the count does not change when you replace).
• Using the middle-value shortcut for numbers that are not evenly spaced.

Exam tips

1. Write each average as a total (average × count) on the first line.
2. If the average goes up by d when a new value is added, the new value equals the new average plus d × (old count).
3. Sense-check: a combined average always lies between the two group averages.

Now try the quiz. If you get one wrong, rewrite it using totals and see where the count changed.