Explain why computing a variance of several numbers is like analyzing their differences.

Question

Explain why computing a variance of several numbers is like analyzing their differences.

Answer 1

Lisa, look at the formula for the variance:

 

∑ (x_i-µ)²/n

 

µ is the mean and is a measure of the center of all of the x's (i.e. a measure of the average position of all the x_i's), so the numerator is a measure of the relative position of each individual to their collective center, ie. their difference.  We square this difference because if we merely added their differences they would sum to zero, i.e. ∑ (x_i-µ)/n = 0, and squaring the differences also gives extra weight to larger differences (>1) than smaller differences (<1).  This is because when you square a number it always gets much larger, unless the number is a fraction, in which case it gets smaller, e.g. 2^2=4, but (1/2)^2=1/4.  Also, we divide by n (or n-1 in the case of a sample) because we want the average difference, not just what the differences total to.

 

I hope this helps you identify some ideas for why variance is about the differences among the numbers in a data set.