Mathematics (Core)Exam code: 0580
Averages and Range
Statistics
One average and one spread
- A proper comparison quotes an average and a measure of spread, and interprets both
- A higher mean or median means the values are typically larger
- A smaller range means the values are more consistent or less spread out
- Say what this means in context rather than only stating the numbers
- Where one set contains an extreme value, compare medians rather than means, because one unusual value drags a mean but barely moves a median
Working backwards from a mean
- Knowing a mean and how many values there are gives the total, since total = mean × number of values
- That total is what makes a missing value findable
- If five numbers have a mean of 12, their total is 60, so a missing fifth value is 60 minus the sum of the other four
- The same idea handles an added or removed value, by adjusting both the total and the count
Worked example
Working backwards from a mean
Five numbers have a mean of 12. Four of them are 8, 15, 11 and 9. Find the fifth.
Solution:
- The mean is the total divided by how many values there are, so the total can be recovered from it
- Total = mean × number of values = 12 × 5 = 60
- Add the four known values: 8 + 15 + 11 + 9 = 43
- The fifth value is the part of the total not yet accounted for
- 60 − 43 = 17
- Check by finding the mean of all five: (8 + 15 + 11 + 9 + 17) ÷ 5 = 60 ÷ 5 = 12
- Reconstructing the total is the whole method, and it works for any "find the missing value" version of this question
Worked example
Two missing values from the mean and the range
Seven whole numbers have a mean of 11 and a range of 16. Five of them are 3, 8, 9, 12 and 15. The other two are p and q, where p is greater than q. Find p and q.
Solution:
- Total of all seven = mean × number of values = 11 × 7 = 77
- The five known numbers add to 3 + 8 + 9 + 12 + 15 = 47, so p + q = 77 − 47 = 30
- The known numbers run only from 3 to 15, a spread of 12, which is less than the range of 16, so one of the missing numbers must be the largest or the smallest
- Range = largest − smallest, so largest = smallest + range
- Try the bigger missing number as the largest: p = 3 + 16 = 19
- Then q = 30 − 19 = 11
- Check: 11 lies between 3 and 19, so 3 is still the smallest and 19 the largest, and the range is 19 − 3 = 16
- Check the mean: (3 + 8 + 9 + 11 + 12 + 15 + 19) ÷ 7 = 77 ÷ 7 = 11