Coefficient of variation
The coefficient of variation is (σ/x̄) × 100%. It compares spread across data sets: the set with the smaller C.V. is more consistent.
Why we need C.V.
Standard deviation has the same unit as the data, and it grows with the size of the values. So σ alone cannot fairly compare, for example, the heights of children with the heights of adults. The coefficient of variation divides σ by the mean and writes it as a percentage. The data set with the smaller C.V. is more consistent: more stable, with less variation. The one with the larger C.V. is less consistent.
Standard deviation vs coefficient of variation
Example 1: finding C.V. and finding σ
(a) The mean of a data set is 25 and its standard deviation is 5. Find the C.V. (b) The C.V. of another data set is 15% and its mean is 40. Find its standard deviation.
- Formula: C.V. = (σ/x̄) × 100%.
- (a) C.V. = (5/25) × 100% = 20%
- (b) 15 = (σ/40) × 100
- σ = 15 × 40 ÷ 100 = 6
(a) C.V. = 20% (b) σ = 6
Example 2: which batsman is more consistent?
Over a season, batsman A has an average score of 50 runs with standard deviation 15 runs. Batsman B has an average of 40 runs with standard deviation 8 runs. Who is more consistent?
- C.V. of A = (15/50) × 100% = 30%
- C.V. of B = (8/40) × 100% = 20%
- 20% is less than 30%, so B's scores vary less compared with his average.
B is more consistent, although A has the higher average.
Points to write in a C.V. answer
- Find x̄ and σ for each data set first.
- Write each C.V. as a percentage, rounded to two decimal places.
- State the conclusion in words: the set with the smaller C.V. is more consistent.
- A larger average does not mean more consistent. Compare the C.V., not the means.
Selectors compare cricket players using C.V.: the player with the lower C.V. scores more consistently, even if another player has a higher average.
1 marks: “The mean of a data set is 10 and its C.V. is 20%. Find its standard deviation.” σ = C.V. × x̄ ÷ 100 = 20 × 10 ÷ 100 = 2.
2 marks: “The standard deviation of a data set is 4.2 and its mean is 35. Find the coefficient of variation.” C.V. = (4.2/35) × 100% = 12%.
5 marks: “The runs scored by two batsmen in 5 matches are A: 40, 50, 60, 70, 80 and B: 45, 50, 50, 55, 50. Find the coefficient of variation for each. Who is more consistent?” A: x̄ = 60, σ = √200 ≈ 14.14, C.V. ≈ 23.57%. B: x̄ = 50, σ = √10 ≈ 3.16, C.V. ≈ 6.32%. B is more consistent because his C.V. is smaller.
C.V. = SD over mean, times 100. Small C.V., steady player.