Coefficient of Variation (CV) is a standardised measure of dispersion that expresses standard deviation as a percentage of the mean, enabling comparison of variability across datasets with different units or scales.
## Core concept
The Coefficient of Variation measures relative dispersion rather than absolute dispersion. While standard deviation tells you the absolute spread of data, CV answers: "How much does the data vary *relative to its average*?"
- Used to compare variability between two or more datasets
- Eliminates the effect of different units of measurement
- Particularly useful when means are different
- Expressed as a percentage—higher CV means higher relative variability
- Assumption: Mean ≠ 0 (CV is undefined when mean is zero or near-zero)
## Formula / rule
$$\text{Coefficient of Variation (CV)} = \frac{\text{Standard Deviation}}{\text{Mean}} \times 100\%$$
Or in symbolic form:
$$\text{CV} = \frac{\sigma}{\bar{X}} \times 100$$
Where: - σ = Standard Deviation - $\bar{X}$ = Arithmetic Mean - Result is always expressed as a percentage
Key relationship: - If two datasets have the same CV, they have the same relative variability regardless of their actual values - Lower CV = more consistent/homogeneous data - Higher CV = more scattered/heterogeneous data
## Common exam applications
1. Comparative Analysis (Most Common) When comparing variability of two datasets measured in different units or with different means, always use CV, not SD alone.
Example: Comparing investment returns - Stock A: Mean return = 12%, SD = 2% - Stock B: Mean return = 8%, SD = 1.5%
CV for Stock A = (2 ÷ 12) × 100 = 16.67% CV for Stock B = (1.5 ÷ 8) × 100 = 18.75%
Conclusion: Despite lower absolute SD, Stock B has higher *relative* variability. Stock A is more stable in proportional terms.
2. Consistency Assessment - Manufacturing quality control: Which batch has consistent output? - Student performance: Which class has uniform marks? - Comparing precision across different measurement scales
3. Risk Assessment In finance/statistics: Higher CV indicates higher risk per unit of return.
## Common exam mistakes
| Mistake | Why it's wrong | Correct approach | |---------|---|---| | Using SD alone to compare datasets with different means | Ignores the scale difference; misleading comparison | Always calculate CV when units or means differ | | Forgetting to multiply by 100 | Leaves CV as decimal instead of percentage | Always express as % for exam answers | | Comparing CV when means are very different in magnitude | Can distort interpretation if one mean is near zero | Use CV only when both means are reasonably far from zero | | Confusing CV with Variance or SD | Different measures serve different purposes | CV = relative; SD = absolute; Variance = SD² |
## Quick checklist for exam
- ✓ Calculate mean and standard deviation first
- ✓ Divide SD by mean
- ✓ Multiply by 100 to express as percentage
- ✓ Always write the % symbol
- ✓ Use CV for comparative questions; use SD for absolute spread questions
- ✓ If question asks "which is more consistent/uniform/stable," interpret as: lower CV = more consistent
Typical exam question pattern: "Compute CV for two investments and state which is more stable." Answer format requires both calculations AND interpretation.