Question

Consider the data of scores obtained by students in an examination. If the score of every student is increased by 2 marks, then which of the following statements is TRUE?

• A. The mean deviation about the mean does not change.
• B. The mean deviation about the mean is increased by 2.
• C. The mean deviation about the median is increased by 2.
• D. The variance is increased by 2.

Hint:

Measures of dispersion such as Range, Mean Deviation, Standard Deviation, and Variance are independent of the change of origin (i.e., adding or subtracting a constant to/from each observation).
They only change when there is a change of scale (i.e., multiplying or dividing by a constant).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Measures of dispersion (like Range, Mean Deviation, Variance) measure the "spread" of data.

Step 2: Detailed Explanation:
$\bullet$ Let original scores be $x_{i}$. The new scores are $y_{i} = x_{i} + 2$.
$\bullet$ The new mean is $\bar{y} = \bar{x} + 2$.
$\bullet$ The distance of any point from the mean is $|y_{i} - \bar{y}| = |(x_{i} + 2) - (\bar{x} + 2)| = |x_{i} - \bar{x}|$.
$\bullet$ Since the individual deviations from the mean are identical to the original ones, the average of these deviations (Mean Deviation) will remain unchanged.
$\bullet$ Similarly, Variance depends on $(x_{i} - \bar{x})^{2}$, which also remains unchanged. Dispersion measures are independent of the change of origin.

Step 3: Final Answer:
The mean deviation about the mean does not change.
This matches option (A).