Why spread matters and the range
A measure of centre tells you where the data are located. It does not tell you whether the values are tightly packed or far apart. A measure of variation describes that spread.
Same mean does not mean same spread
A: range 8 · B: range 48
Range uses only the two end values
A measure of variation is a number that describes how spread out the data are. For the range, subtract the smallest value from the largest.
Symbols
- = largest value (data unit)
- = smallest value (data unit)
- The range is quick to calculate and easy to interpret.
- It uses only two values, so one extreme value can change it greatly.
- It has the same unit as the original data: seconds, kilograms, marks, and so on.
Worked example
Range from raw values
The times 14, 9, 12, 17, 11 and 10 seconds are not ordered.
The minimum is 9 and the maximum is 17. The other values do not enter the calculation.
Rounded measurements need boundaries
If a measurement is recorded to the nearest unit, its true value lies in a half-unit interval. There are two possible range questions, and the direction of the boundaries matters.
- Least possible range: make the largest reading as small as it can be and the smallest reading as large as it can be.
- Greatest possible range: make the largest reading as large as it can be and the smallest reading as small as it can be.
Worked example
Least and greatest possible ranges
The shortest recorded height is 150 cm and the tallest is 169 cm, each correct to the nearest centimetre.
The true shortest height lies from 149.5 cm up to, but not including, 150.5 cm. The true tallest height lies from 168.5 cm up to, but not including, 169.5 cm.
Least possible:
Greatest possible:
Common mistake
Know what the range can and cannot say
The range measures the full width of the data, but it ignores every value between the minimum and maximum.
Temperatures 18, 20, 20, 21, 23, 24 and 31 °C are recorded. Find the range and explain why it may give a misleading picture of the spread.
Show worked answer
The range depends on the extreme value 31 °C and ignores the five values packed between 20 °C and 24 °C.