A mean is the total divided by the count, so multiplying the mean by the count gives the total. The missing value is that total minus the known values, and then you check that the answer is possible.
This page is part of choosing the right summary of a data set. It follows explaining how an outlier changes the mean and median differently.
What is the method?
- Total = mean × number of values.
- Add the known values.
- Missing value = total − known sum.
- Check that the answer is possible and recompute the mean.
Worked example 1: one test missing
The mean of five test scores is 62. Four scores are 55, 70, 64 and 58. Find the fifth.
Total = 62 × 5 = 310. Known sum = 55 + 70 + 64 + 58 = 247. Missing = 310 − 247 = 63.
Plausibility. 63 lies between 0 and 100 and close to the other scores. Check: (247 + 63) ÷ 5 = 310 ÷ 5 = 62, which matches the given mean.
Worked example 2: a new value added
The mean of 9 scores is 50. After a new student joins, the mean of 10 scores is 52. Find the new score.
Old total = 50 × 9 = 450. New total = 52 × 10 = 520. New score = 520 − 450 = 70. This is a possible mark, and it is above the old mean, which is why the mean rose.
Worked example 3: when the answer is impossible
The mean of four marks is 85, and the maximum mark is 100. Three marks are 50, 60 and 70.
Total = 85 × 4 = 340, and the known sum is 180, so the missing mark is 340 − 180 = 160.
A mark of 160 is impossible when the maximum is 100. The correct conclusion is that the given mean cannot be right for these marks, so the data contains an error. Even the maximum possible fourth mark of 100 gives a mean of 280 ÷ 4 = 70.
The slips to avoid
The first slip is to divide the known sum by the count instead of multiplying the mean by the count. The second is to use the wrong count, such as 9 instead of 10 in the new mean.
The third is to stop at the number. Always ask whether it is a possible value, and write one sentence to show you asked. You can rehearse the arithmetic with the algebra step repair trainer.
Check yourself
The mean of eight numbers is 15, and seven of them add up to 100. Find the eighth. Then state whether it is plausible if each number is a test mark out of 20.
Answer
Total = 15 × 8 = 120. Missing = 120 − 100 = 20.
A mark of 20 out of 20 is possible, because it equals the maximum. The mean of 15 is achievable, since the other seven marks average 100 ÷ 7 ≈ 14.3.
What to study next
Continue with deciding whether two data summaries are genuinely comparable. Test the cluster with the practice set.
For a teacher to build the checking habit with you, see online one-to-one Mathematics tuition.