Coefficient of Variation

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Coefficient of Variation (CV)

Definition

The Coefficient of Variation (CV) is a standardized measure of dispersion that expresses variability relative to the mean. It's particularly useful when comparing datasets with different units or widely different means.

Formula

Sample Coefficient of Variation:

CV = \frac{s}{\bar{x}} \times 100\%

Population Coefficient of Variation:

CV = \frac{\sigma}{\mu} \times 100\%

Where:

$s$ = sample standard deviation
$\bar{x}$ = sample mean
$\sigma$ = population standard deviation
$\mu$ = population mean

Example

Consider two datasets:

Dataset A (temperatures in °C): $20, 22, 21, 23, 19$

Dataset B (heights in cm): $160, 165, 162, 168, 155$

Dataset A:

\begin{align*} \text{Mean} &= 21\text{ °C} \\ \text{Standard Deviation} &= 1.58\text{ °C} \\ CV &= \frac{1.58}{21} \times 100\% = 7.52\% \end{align*}

Dataset B:

\begin{align*} \text{Mean} &= 162\text{ cm} \\ \text{Standard Deviation} &= 4.85\text{ cm} \\ CV &= \frac{4.85}{162} \times 100\% = 2.99\% \end{align*}

Although Dataset B has a larger standard deviation, its CV is smaller, indicating less relative variability.

Key Points

Only applicable to ratio scale data with positive values and non-zero mean
Useful for comparing variability between datasets with different units or means
Generally expressed as a percentage

Common Interpretations

CV < 10%: Very low variability relative to the mean

10% ≤ CV < 20%: Low variability relative to the mean

20% ≤ CV < 30%: Moderate variability relative to the mean

CV ≥ 30%: High variability relative to the mean

Limitations

Not meaningful for interval or ordinal scale data
Cannot be calculated for data with zero mean or negative values
Sensitive to outliers, like other measures based on mean and standard deviation

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