TACHS: Mean questions: work with the total
Turn a mean into a total to find a missing value, add or remove a value, correct a mistake, or combine two groups.
How do you find a missing score, or a new mean, when you only know the average?
Short answer
Multiply the mean by the count to get the total, change the total as the question describes, and divide by the new count.
Know first: Mean = total ÷ count, Adding a list of whole numbers
Step 1 of 6
A mean hides a total
If six games have a mean of 14 points, the six scores add up to 6 × 14 = 84. You may not know each score, but you know the total, and the total is what you can work with.
Every question in this family follows three moves: turn each mean into a total, change the total, then divide by the new count.
| The question says | Change the total by | Then divide by |
|---|---|---|
| One value is missing | Total − the known values | Nothing; that is the value |
| A value is added | Adding it | The count + 1 |
| A value is removed | Subtracting it | The count − 1 |
| A value was recorded wrong | The difference, new − wrong | The same count |
| Two groups are combined | Adding both group totals | The combined count |
Means cannot be added or averaged directly. Totals can.
Step 2 of 6
Total minus known values
Worked example: The sixth game
A player's points in six games have a mean of 14. Five of the games were 10, 18, 12, 15 and 11 points. How many points did she score in the sixth game?
- A14
- B18
- C66
- D84
- 1
Turn the mean into a total
6 × 14 = 84 points in all.
- 2
Add the known values
10 + 18 + 12 + 15 + 11 = 66.
- 3
Subtract
84 − 66 = 18 points.
- 4
Check
66 + 18 = 84, and 84 ÷ 6 = 14.
- 5
See why the other choices are there
14 assumes the missing score equals the mean. 66 is the sum of the known games. 84 is the total for all six.
Answer
She scored 18 points in the sixth game.
Wrong: Four students average 8 and six students average 13, so all ten average (8 + 13) ÷ 2 = 10.5.
Right: Totals: 4 × 8 = 32 and 6 × 13 = 78. Combined: 110 ÷ 10 = 11.
The groups are different sizes. Six students at 13 pull the mean toward 13 more than four students at 8 pull it toward 8.
Step 3 of 6
Adding, removing or fixing a value
Change the total, then divide by the new count.
Five values with mean 9 total 45. Add 21 and the new mean is 66 ÷ 6 = 11. Eight values with mean 10 total 80. Remove 24 and the new mean is 56 ÷ 7 = 8.
For a correction, the count stays the same. Four values with mean 12 total 48. If a 7 should have been 15, the total grows by 8 to 56, and the mean is 56 ÷ 4 = 14.
Step 4 of 6
Totals first
Check yourself · Question 1
Four quiz scores have a mean of 9. Three of the scores are 6, 11 and 8. What is the fourth score?
Answer: B
Total: 4 × 9 = 36. Known scores: 6 + 11 + 8 = 25. Fourth score: 36 − 25 = 11.
Trap answer: 9
Assuming the missing value equals the mean
- Why it looks right
- The mean is the typical score, so it seems like a safe guess for one more.
- Why it's wrong
- The known scores total 25, which is 2 short of 3 × 9 = 27. The fourth score must make up that gap, so it is above the mean.
- The right answer
- 11, because 36 − 25 = 11.
Check yourself · Question 2
Six players average 12 points. Four other players average 7 points. What is the mean for all ten players?
Answer: B
Totals: 6 × 12 = 72 and 4 × 7 = 28. Combined: 72 + 28 = 100. Mean: 100 ÷ 10 = 10.
Check yourself · Question 3
Five values have a mean of 20. One value was recorded as 32 but should have been 17. What is the corrected mean?
Answer: B
Total: 5 × 20 = 100. Take out 32 and put in 17: 100 − 32 + 17 = 85. Mean: 85 ÷ 5 = 17.
Common mistake
"I removed a value from the total, then divided by the same count as before."
- Why it's tempting
- The count is easy to forget because the question gives it only once, at the start.
- Do this instead
- After you add or remove a value, recount. Five values minus one is four, so divide the new total by 4.
Step 5 of 6
What to remember
Remember
- Turn every mean into a total: mean × count.
- Add, remove or correct values in the total, and combine groups by adding their totals.
- Divide by the new count, and never average two means from groups of different sizes.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what is the total of all five scores, and which wrong choice is that total?