Add a fifth game
Four games have mean 12, so their total is 48. A score of 20 gives total 68 over five games: 68 ÷ 5 = 13.6.
Before this page: calculate a mean as total ÷ count, and use the inverse operation to reverse a division.
The total is mean times count
A mean is the total shared across the count. Reverse that sharing by multiplying the mean by the number of values.
Total = mean × number of values
- Use the same count that produced the stated mean.
- Keep the units of the original values in the total.
Four game scores have mean 12
Recover the player's total score across the four games.
- 12 × 4 = 48The player scored 48 points in total.
- 48 ÷ 4 = 12The inverse check returns the stated mean.
Subtract known values to find a missing value
Find the full total from the mean and count, then subtract the values that are already known.
A fifth game score is missing
A player averages 18 points over five games. Four scores are 12, 17, 21 and 20. Find the fifth score.
- 18 × 5 = 90All five scores total 90.
- 12 + 17 + 21 + 20 = 70The known scores total 70.
- 90 − 70 = 20The missing score is 20.
Checking (12 + 17 + 21 + 20 + 20) ÷ 5 = 18 confirms the missing value.
Recalculate the total when a value is added or removed
Changing the count changes the divisor, so find the new total and the new count before calculating the new mean.
Remove one game
Five games have mean 18, so their total is 90. Removing a score of 20 leaves total 70 over four games: 70 ÷ 4 = 17.5.
Check the original mean
Substitute a missing value into the full list, add all the values, and divide by the original count. The result must equal the stated mean.
Check the missing score
12 + 17 + 21 + 20 + 20 = 90, and 90 ÷ 5 = 18. The missing score gives the required mean.
Common mistakes
- Dividing the mean by the count Four scores with mean 12 have total 12 × 4 = 48, not 12 ÷ 4 = 3.
- Using the known-value total as the full total For the scores 12, 17, 21 and 20 with mean 18 over five scores, 70 is the known total. The full total is 90, so the missing score is 20.
- Keeping the old count after adding a value Adding 20 to four values with total 48 gives 68 across five values, so the new mean is 13.6, not 68 ÷ 4 = 17.
Summary
- Recover a total by multiplying the mean by the count.
- Subtract known values from the full total to find a missing value.
- After adding or removing a value, change both the total and the count.
- Check a recovered value by recalculating the original mean.