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Topic 3 of 7 · Form 3

Measures of average and dispersion

Mean, median and mode for ungrouped and grouped data (estimated mean using mid-points; median by interpolation or cumulative frequency; modal class). Choosing an appropriate average; effect of extreme values.

Notes for this topic 1 file, in order

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Averages

Mean, median and mode for ungrouped and grouped data (estimated mean using mid-points; median by interpolation or cumulative frequency; modal class). Choosing an appropriate average; effect of extreme values.

Spread

Range, interquartile range, percentiles, variance and standard deviation.

  • Standard deviation s = √(Σ(x − x̄)² ÷ n) = √(Σx²/n − x̄²); for frequency data use Σf.

Worked example: Find the mean and standard deviation of 3, 5, 7, 9, 11.

Mean = 35 ÷ 5 = 7; Σ(x − x̄)² = 16 + 4 + 0 + 4 + 16 = 40 → s = √(40 ÷ 5) = √8 = 2.83

Coding

If y = (x − a)/b, then ȳ = (x̄ − a)/b and s_y = s_x ÷ b.

Comparing distributions

Compare averages and spreads in context; skewness from box plots and the relationship of mean, median and mode.

Exam tip: Always compare both an average AND a measure of spread, in the context of the question.

Check yourself

  1. Which average is unaffected by an extreme value?
  2. Find the range and IQR given minimum 12, Q1 20, median 26, Q3 35, maximum 50.
  3. Data x have mean 40 and standard deviation 6. Find the mean and SD of y = (x − 40) ÷ 2.
Show answers
  1. The median (and the mode).
  2. Range 38; IQR 15
  3. Mean 0; SD 3
Next topicIndex numbers, weighted averages and vital statistics

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