Variance is always independent of the change of

Options

A. Origin but not scale
B. Only scale
C. Both origin and scale
D. none of these

Answer

To solve the question regarding the independence of variance from changes, we will analyze the effects of changing the origin and scale of a dataset.

### Step-by-Step Solution:

1. **Understanding Variance**:
Variance is a measure of how much the values in a dataset differ from the mean of that dataset. It quantifies the dispersion or spread of the data points.

2. **Change of Origin**:
Changing the origin means adding a constant value to all data points in the dataset. For example, if we have a dataset \( x_1, x_2, ..., x_n \) and we add a constant \( c \), the new dataset becomes \( x_1 + c, x_2 + c, ..., x_n + c \).

- **Effect on Variance**:
When we add a constant to all values, the mean of the dataset shifts by that constant, but the distances of the data points from the new mean remain the same. Therefore, the variance does not change.

3. **Change of Scale**:
Changing the scale means multiplying all data points by a constant factor. For example, if we multiply each data point by a constant \( k \), the new dataset becomes \( k \cdot x_1, k \cdot x_2, ..., k \cdot x_n \).

- **Effect on Variance**:
When we multiply the values by a constant, the distances of the data points from the mean are also scaled by that factor. This means that the variance will change, specifically, it will be multiplied by \( k^2 \) (the square of the constant).

4. **Conclusion**:
From the above analysis, we can conclude that variance is independent of changes in origin (adding a constant) but is dependent on changes in scale (multiplying by a constant).

Therefore, the correct answer to the question is: **"Origin but not scale."**