The sum of the deviations of the scores about the mean for any data set...

The sum of the deviations of the scores about the mean for any data set is always zero. This principle arises from the definition of the mean itself, which is calculated as the average of all the data points. When you find the deviation of each data point from the mean, some deviations will be positive (for values above the mean) and others will be negative (for values below the mean).

Mathematically, if you sum all these deviations, the positive and negative values cancel each other out, resulting in a total sum of zero. This property holds true regardless of the data set's distribution or sample size, ensuring that the mean accurately represents the center of the data.

Understanding this concept is fundamental in statistics as it reinforces the idea that the mean is a point of balance in the data set.