Question

Which of the following has no units?

• A. variance
• B. inter-quartile range
• C. standard deviation
• D. coefficient of variation

Hint:

Relative measures of dispersion, such as the Coefficient of Variation or the Coefficient of Quartile Deviation, are always dimensionless and have no units.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Measures of dispersion can be absolute (having the same units as the original data) or relative (dimensionless ratios expressed as percentages).
Key Formula or Approach:
Let us evaluate the formulas and units for each measure:
- Variance:
\[ \sigma^2 = \frac{\sum (X_i - \bar{X})^2}{N} \quad (\text{Unit: } [\text{original unit}]^2) \]
- Inter-quartile range:
\[ \text{IQR} = Q_3 - Q_1 \quad (\text{Unit: } [\text{original unit}]) \]
- Standard deviation:
\[ \sigma = \sqrt{\sigma^2} \quad (\text{Unit: } [\text{original unit}]) \]
- Coefficient of variation (CV):
\[ \text{CV} = \left( \frac{\sigma}{\mu} \right) \times 100\% \]

Step 2: Detailed Explanation:
In the formula for the Coefficient of Variation, both standard deviation ($\sigma$) and mean ($\mu$) possess the exact same units.
When dividing $\sigma$ by $\mu$, the units cancel out completely, yielding a dimensionless, unitless ratio.
All other options (variance, standard deviation, and inter-quartile range) have units derived from the original scale of measurement.
Therefore, the coefficient of variation has no units.

Step 3: Final Answer
The correct option is (D).