Arithmetic averages

The arithmetic mean#

The arithmetic mean, often called the average, is found by adding all values and dividing by how many values there are.

For 6, 8, 10, and 12:

1. Add the values: 6 + 8 + 10 + 12 = 36.
2. Count the values: 4.
3. Divide: 36 ÷ 4 = 9.

The mean is 9.

Use the total-count relationship#

Three quantities are connected:

• average = total ÷ count;
• total = average × count;
• count = total ÷ average, when the values and context allow a meaningful count.

This relationship is more useful than memorising the first formula alone.

Find a missing value#

The average of five numbers is 18. Four of the numbers are 12, 15, 20, and 21. Find the fifth number.

1. Required total: 18 × 5 = 90.
2. Known total: 12 + 15 + 20 + 21 = 68.
3. Missing value: 90 − 68 = 22.

Effect of adding a value#

Suppose four scores have an average of 70. Their total is 70 × 4 = 280. If a fifth score of 80 is added, the new total is 360 and the new average is:

360 ÷ 5 = 72.

Do not average the old average and the new score directly unless both represent groups of equal size.

Combined averages#

One group of 10 learners has an average score of 64, and another group of 15 learners has an average of 72.

1. First total: 10 × 64 = 640.
2. Second total: 15 × 72 = 1,080.
3. Combined total: 1,720.
4. Combined count: 25.
5. Combined average: 1,720 ÷ 25 = 68.8.

The simple average of 64 and 72 would be wrong because the groups have different sizes.

Common mistakes#

Dividing by the wrong count

Count the data values, not the number of additions written in the expression.

Forgetting to update both total and count

When a value is added or removed, both quantities may change.

Assuming the mean must appear in the data

The average of 4 and 7 is 5.5, even though 5.5 is not one of the original values.

Reasonableness check#

For ordinary non-weighted data, the mean should lie between the smallest and largest values. A result outside that range signals an error in the total or count.