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?

Let the original scores be $x_1, x_2, \dots, x_n$.

When every score is increased by $2$, the new scores are $y_i = x_i + 2$.

The new mean is $\bar{y} = \bar{x} + 2$.

The mean deviation about the mean for the new scores is $\dfrac{1}{n} \sum |y_i - \bar{y}|$.

Substituting the values, we get $\dfrac{1}{n} \sum |(x_i + 2) - (\bar{x} + 2)| = \dfrac{1}{n} \sum |x_i - \bar{x}|$.

This is exactly equal to the original mean deviation about the mean. Thus, the mean deviation about the mean does not change.

Similarly, all measures of dispersion like variance, standard deviation, and mean deviation about the median remain unchanged when a constant is added to each observation.

Answer: The mean deviation about the mean does not change.