If the deviations are taken from the actual mean the sum of the deviation will be equal to the :

Question

Accepted Answer

The correct option is 2:

This concept is a fundamental property of the arithmetic mean in statistics. The mean (or average) is the single point in a data set where the total sum of the positive differences (deviations) between the data points and the mean perfectly balances the total sum of the negative differences. If you subtract the mean from every value in a distribution, you will have some positive deviations (for values greater than the mean) and some negative deviations (for values less than the mean). When all these deviations are added together, the positive and negative values will always cancel each other out, resulting in a total sum of zero. Mathematically, for a data set $X$ with mean $\bar{X}$, $\sum (X_i - \bar{X}) = 0$. This zero-sum property is one of the defining algebraic characteristics of the mean and confirms its role as the center of gravity for the data set.

The incorrect option is 1 (One): The sum of deviations is only equal to one by chance for a very specific, and highly unlikely, data set. This is not a universal property of the mean.

The incorrect option is 3 (Not known): The result is not "Not known"; it is a mathematically proven and universally accepted fact in statistics. If you know the actual mean, the sum of deviations is guaranteed to be zero.

The incorrect option is 4 (Infinite): The sum of deviations is finite and, specifically, equal to zero, provided the data set itself is finite. "Infinite" would only occur in theoretical, undefined contexts or non-finite distributions.