Dispersion: range and coefficient of range
Dispersion tells how spread out data are. The simplest measure is the range, L − S. The coefficient of range, (L − S)/(L + S), has no unit.
What dispersion means
Two classes can have the same average mark but very different spreads of marks. Dispersion tells how far the values are spread out from the centre (the mean). The measures of dispersion are range, mean deviation, quartile deviation, standard deviation, variance and coefficient of variation. Range is the simplest. Because it uses only the largest and smallest values, one unusual value can change it a lot.
Range and coefficient of range
| Measure | Formula | Notes |
|---|---|---|
| Range | R = L − S | L = largest value, S = smallest value. Same unit as the data. |
| Coefficient of range | (L − S)/(L + S) | A pure number with no unit, so it can compare data sets. |
| Grouped data (classes) | L = upper limit of the highest class; S = lower limit of the lowest class | Leave out a class whose frequency is 0, since no value lies in it. |
Example 1: weights of students
The weights (in kg) of 7 students are 45, 52, 40, 60, 48, 55, 50. Find the range and the coefficient of range.
- Largest value L = 60 kg, smallest value S = 40 kg.
- Range R = L − S = 60 − 40 = 20 kg
- Coefficient of range = (L − S)/(L + S)
- = 20/(60 + 40) = 20/100 = 0.2
Range = 20 kg, coefficient of range = 0.2
Example 2: working backwards
The range of a data set is 18 and its smallest value is 27. Find the largest value and the coefficient of range.
- R = L − S, so L = R + S = 18 + 27 = 45.
- Coefficient of range = (L − S)/(L + S) = 18/(45 + 27)
- = 18/72 = 0.25
Largest value = 45, coefficient of range = 0.25
Example 3: grouped data
The heights (in cm) of 30 students are grouped as 140-145: 3, 145-150: 8, 150-155: 12, 155-160: 6, 160-165: 1. Find the range and the coefficient of range.
- L = upper limit of the highest class = 165 cm.
- S = lower limit of the lowest class = 140 cm.
- Range = 165 − 140 = 25 cm
- Coefficient of range = 25/(165 + 140) = 25/305 ≈ 0.08
Range = 25 cm, coefficient of range ≈ 0.08
A weather report gives the day's highest and lowest temperatures; the gap between them is the day's range. A small range means the temperature stayed steady all day.
1 marks: “The largest and smallest values of a data set are 40 and 10. Find the coefficient of range.” (40 − 10)/(40 + 10) = 30/50. Answer: 0.6.
2 marks: “Find the range and coefficient of range of 52, 47, 61, 38, 70, 55.” L = 70, S = 38. Range = 70 − 38 = 32. Coefficient = 32/108 ≈ 0.30.
5 marks: “The heights (in cm) of plants in two plots are: Plot A: 42, 50, 38, 55, 47, 60; Plot B: 30, 41, 28, 45, 52, 36. Find the range and coefficient of range of each. Which plot's heights are more spread out compared with their size?” A: range 60 − 38 = 22 cm, coefficient 22/98 ≈ 0.22. B: range 52 − 28 = 24 cm, coefficient 24/80 = 0.30. Plot B is more spread out.
Range is Large minus Small, L − S. For the coefficient, put that same difference over the sum, L + S.