ANOVA: Explain why computing a variance of several numbers is like analyzing their differences
To arrive at the variance, one has to factor in the square from the difference of every point. This is gotten from the mean value of the entire set of data and then also includes the mean. If each of the points is found to be scattered more, then both the square values and mean will be high. If the points in the set of data are grouped together around the average value, the result will be low square values plus a lower value for the mean (Bakeman & Robinson, 2005).
This could be better understood if we analyze the formula for calculating the variance; ∑ (xi-µ) 2/n. In this formula, µ represents the mean and is also the measure of all the central point of the entire X’s. The numerator is, therefore, a measure of the position that is about their common center. This essentially is their difference. One has to square the difference because if one merely had a summation of their summation, the result would be a zero, i.e. ∑ (xi-µ)/n = 0 (Norris, Qureshi, & Howitt, 2014).
What’s more, having the differences squared would similarly give added weight to the bigger differences (>1) over, the smaller differences (<1). Reason being, whenever one squares a number, the result is certainly always a bigger number. The only exception, in this case, is if one uses a fraction whereby the result is always a smaller fraction. For instance, e.g. 3^2=9, but (1/3) ^2=1/9. Moreover, the division is done by n (or n-1 in the case of a sample). This is because one is looking for the average difference and not only the summation of the difference (Norris, Qureshi, & Howitt, 2014).
Bakeman, R. & Robinson, B. (2005). Understanding statistics in the behavioral sciences. Mahwah, NJ: Lawrence Erlbaum Associates.
Norris, G., Qureshi, F., & Howitt, D. (2014). Introduction to Statistics with SPSS for Social Science. Florence: Taylor and Francis.