3
Calculating the mean and seeing what changes it
Discover Mathematics P6: Term 1, Unit 7, Lessons 1–2
GoalFind a missing value for a given mean and tell how one extra value changes the mean.
New words
missing value · a data value we do not know but can find from the meantarget mean · the mean we want to reachtotal · mean × number of values
Explanation
The mean works backwards too. Since mean = total ÷ number of values, we get total = mean × number of values. If the mean and the number of values are known, we can find the total, and then the missing value is the total minus the values we already have. An extra value changes the mean: a value above the mean raises it, a value below the mean lowers it, and a value equal to the mean leaves it as it is. A far-away value changes it most.
ƒFormula
total = mean × number of values
Worked examples
Marwan’s first three scores are 14, 17 and 12. His four scores must have the mean 15. The total must be 15 × 4 = 60. The first three give 14 + 17 + 12 = 43, so the fourth score is 60 − 43 = 17. Check: (14 + 17 + 12 + 17) ÷ 4 = 60 ÷ 4 = 15.
| New number | New total | New mean |
|---|---|---|
| 24 | 84 | 14 |
| 12 | 72 | 12 |
| 6 | 66 | 11 |
A value far from the others pulls the mean towards it. In the data 2, 2, 3, 3, 10 the mean is 20 ÷ 5 = 4, which is higher than the peak of the dot plot (2 and 3). The mean is larger than most of the values because of the 10.
Class activity
What score do I need? Write down your last three marks from a test (or invent them). Choose a mean you would like to reach with a fourth test. Work out the total you need and the mark that you must get.
Practice
1
Hazem has the marks 18, 20 and 15 out of 25. What must he get in the fourth test so that his mean is 18?
The total must be 18 × 4 = 72. So far 18 + 20 + 15 = 53. He needs 72 − 53 = 19.
2
Six numbers have the mean 10. A seventh number, 24, is added. What is the new mean?
The total was 60. Now it is 84, so the mean is 84 ÷ 7 = 12.
3
Four numbers have the mean 15. One of them is 21. What is the mean of the other three?
The total is 60. Without 21 it is 39, so the mean is 39 ÷ 3 = 13.
4
Hana says: “If I add a number equal to the mean, the mean changes.” Is she right? Try 4, 5, 6 and add 5.
No. The mean of 4, 5, 6 is 5. With 5 added the total is 20 and 20 ÷ 4 = 5. The mean stays the same.
5
In a skewed-right dot plot, is the mean to the left or to the right of the peak? Why?
To the right. The few large values in the tail pull the balance point towards them.