Q.Coefficient of variation $= \dfrac{\ldots}{\text{Mean}} \times 100$

The coefficient of variation (CV) is a standardized measure of dispersion that expresses variability relative to the mean, making it useful for comparing datasets with different units or scales.

The CV is defined as the ratio of the standard deviation to the mean, expressed as a percentage. This normalization allows us to compare the relative spread of different distributions regardless of their absolute magnitudes.

The formula is:

$\text{CV} = \frac{\sigma}{\bar{x}} \times 100$ …

The coefficient of variation (CV) is a dimensionless measure that tells us how spread out a dataset is relative to its central value. Unlike the standard deviation, which is an absolute measure (in the same units as the data), the CV allows us to compare variability across datasets with different units or vastly different means.

The logic is straightforward: if two datasets have the same standard deviation but different means, the one with the smaller mean is actually more variable in relative terms. For instance, a standard deviation of 5 kg in weights averaging 50 kg represents much more variability than the same 5 kg spread around a mean of 500 kg. The CV captures this by scaling the standard deviation against the mean.

Why standard deviation?

The numerator must be a measure of dispersion. The three common candidates are:

• Range: Too crude; ignores all intermediate values.
• Mean deviation: Rarely used in practice; lacks the mathematical properties needed for inference.
• Standard deviation: The gold standard; it has desirable statistical properties (relates to variance, appears in the normal distribution, etc.) and is universally adopted.

The formula becomes:

The factor of 100 converts the ratio to a percentage, making it easier to interpret. A CV of 25% means the standard deviation is one-quarter of the mean. …