Understanding Data
Measures of variability
Find the range and standard deviation of data.
Two data sets can have the same mean but be spread out very differently. Measures of variability (or dispersion) show how spread out the values are around the mean.
- Range = maximum − minimum. It is quick to find, but it uses only the two extreme values, so a single outlier can distort it badly.
- Standard deviation (σ) uses all the values. Find how far each value is from the mean, square each difference, take the average of the squares, and then the positive square root. A small standard deviation means the values are close together; a large one means they are widely spread.
For the children's heights in the previous lesson: range = 115 − 85 = 30 cm; the squared differences from the mean add up to about 938, so σ = √(938 ÷ 9) ≈ 10.21 cm.
Which measure to use depends on the question:
- To find the class average: the mean.
- To find the most popular car colour: the mode.
- To study the disparity (inequality) in salaries in a company: the standard deviation.
Range is quick; standard deviation uses every value.
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