Measures of central tendency · Ages 14–15
Finding an unknown from an average
Sum = mean × how many.
How to do it
Method
Sum = mean × how many
7, 9, x, 12 with mean 10: x = 40 − 28 = 12
Another way
An equation
(7 + 9 + x + 12) ÷ 4 = 10, x = 12
In full
Given the mean, find the sum first: sum = mean × how many. If 7, 9, x, 12 have a mean of 10, their sum is 10 × 4 = 40, so x = 40 − 7 − 9 − 12 = 12. Or set up an equation: (7 + 9 + x + 12) ÷ 4 = 10 gives x = 12. A value put in or taken out: the new sum − the old sum is that value; 5 numbers with mean 8 (sum 40) become 6 with mean 9 (sum 54), so 54 − 40 = 14 was put in. Given the median: once the data are in order, the value in the median's place (or the mean of the two there) must equal it; with a and x the middle two and m the median, (a + x) ÷ 2 = m, so x = 2m − a.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Four numbers have a mean of 18; three of them are 16, 10 and 11, and the fourth is 35.
True or false?
Show the answer and working
Answer: True
The sum is 18 × 4 = 72, so the fourth = 72 − 16 − 10 − 11 = 35.
Example 2
6 data have a mean of 30. One more is put in, and the mean of the 7 becomes 30 + 3. What was put in?
Show the answer and working
Answer: 51
The old sum is 6 × 30 = 180; the new 7 × (30 + 3) = 231; so 231 − 180 = 51 went in.
Example 3
7 data have a mean of 23. One is taken out, and the mean of the other 6 is 23 − 2. What was taken out?
Show the answer and working
Answer: 35
The old sum is 161 and the rest add up to 6 × (23 − 2) = 126, so 161 − 126 = 35 came out.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Four numbers have a mean of 20; three of them are 13, 12 and 11, and the fourth is 43.
True or false?
Answer:
2.The five numbers 11, x, 27, 40 and 33 have a mean of 25. Their sum is , so x = .
Fill in each blank
Answer:
3.4 data have a mean of 25. One more is put in, and the mean of the 5 becomes 25 + 4. What was put in?
Answer:
4.5 data have a mean of 22. One is taken out, and the mean of the other 4 is 22 − 2. What was taken out?
Answer:
5.Six data in order, smallest first, are 15, 16, 18, x, 30, 36, and their median is 20. x =
Answer:
All the answers
- 1. False
- 2. 125, 14
- 3. 45
- 4. 30
- 5. 22
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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