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 or interquartile 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 and interquartile ranges rather than means and ranges

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
Exam tip

Put the data in order before finding a median or a quartile. In a frequency table the mode is the value itself, never the count beside it. "Estimate of the mean" always signals grouped data: use the midpoints, add the extra columns and show both totals, because the method marks sit there rather than in the final division.

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