Variance is independent of change of :

Question

Accepted Answer

Variance is a statistical measure that quantifies the spread or dispersion of a set of data points. It is calculated as the average of the squared deviations from the mean. Understanding how variance behaves under transformations of data is crucial in statistics.

1. **Explanation for Option 2 (Origin only):**

Variance is independent of changes in the origin. This means that if you add or subtract a constant from every data point in a dataset, the variance remains unchanged. This is because variance measures the spread of the data points around the mean, and shifting all data points by a constant does not affect their relative distances from each other. Mathematically, if you have a dataset $X = \{x_1, x_2, ..., x_n\}$ and you create a new dataset $Y = \{x_1 + c, x_2 + c, ..., x_n + c\}$, where $c$ is a constant, the variance of $Y$ is the same as the variance of $X$.

2. **Explanation for why other options are incorrect:**

- **Option 1 (Scale only):** Variance is not independent of changes in scale. If you multiply every data point by a constant, the variance is affected. Specifically, if each data point is multiplied by a constant $k$, the variance is multiplied by $k^2$. This is because scaling affects the spread of the data points. For example, if you double each data point, the variance increases by a factor of four.

- **Option 3 (Origin and scale both):** As explained, variance is independent of changes in origin but not in scale. Therefore, this option is incorrect because it suggests that variance is independent of both, which is not true.

- **Option 4 (Not in origin and not in scale):** This option is incorrect because it implies that variance is dependent on changes in both origin and scale. As established, variance is independent of changes in origin, so this option contradicts the correct understanding of variance.

In conclusion, the correct answer is Option 2 because variance remains unchanged when there is a shift in the origin of the data, but it is affected by changes in scale.