Consider the following statements
1. Variance is unaffected by change of origin and change of scale
2 Coefficient of variance is independent of the unit of observations Which of the statements given above is are corect?

Options

A. 1only
B. 2only
C. Both I and II
D. Neither I nor II

Answer

To determine the correctness of the given statements regarding variance and coefficient of variance, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement 1:** Variance is unaffected by change of origin and change of scale.

- **Change of Origin:** When we add a constant (let's say 'c') to all data points, the mean shifts by 'c', but the spread of the data (variance) remains unchanged. This is because variance measures the average squared deviation from the mean, and adding a constant does not affect these deviations.

- **Change of Scale:** If we multiply all data points by a constant (let's say 'k'), the mean will also change, but the variance will be affected. Specifically, if the original variance is 'σ²', after scaling, the new variance will be 'k² * σ²'. This means that variance is affected by a change of scale.

**Conclusion for Statement 1:** Since variance is affected by change of scale, the statement is incorrect.

### Step 2: Analyze the second statement
**Statement 2:** Coefficient of variance is independent of the unit of observations.

- The Coefficient of Variance (CV) is defined as the ratio of the standard deviation (σ) to the mean (μ), often expressed as a percentage:

\[ CV = \left( \frac{\sigma}{\mu} \right) \times 100 \]

- Both the standard deviation and the mean are measured in the same units. When we take the ratio of standard deviation to mean, the units cancel out. Therefore, the CV is a dimensionless quantity and does not depend on the units of measurement used for the data.

**Conclusion for Statement 2:** The statement is correct as the coefficient of variance is indeed independent of the unit of observations.

### Final Conclusion
- **Statement 1:** Incorrect
- **Statement 2:** Correct

Thus, the correct answer is that only the second statement is correct.