Well , the numerical difference between our results is due to the fact that on the first hand you gave an approx. mean , it should be (51/49)+57 which is not exactly 58.04 . Base change is a term we commonly use in statistics , your variables were x= 51,54,57,60,63 I said to change it to y= x - 57 , so the new variables are -6(=51-57) , -3(=54-57) ...etc. Also y= 57 - x → fy= fx - 57f ( for all f ,x,y) summing over all the values we get ∑ fy
= ∑fx - 57 ∑f , dividing by ∑f we get mean(y) = (∑fy)/∑f = ( ∑fx)/∑f - 57
= mean(x) - 57 . And now for the most important part we note that x - mean(x) = y - mean(y) ; so that f ( x - mean(x) )^2 = f (y - mean(y) )^2
= fy^2 - 2f y mean(y) + f(mean(y))^2 , one can not but notice that with respect to summation over all values of f and y , the mean is constant. So summing we get ∑ f(x - mean(x) )^2
= ∑ f (y)^2 - 2 mean(y)∑ fy + (mean(y))^2 ∑ f =∑f (y)^2 - 2 (mean(y))^2 ∑f + (mean(y) )^2 ∑f = ∑f (y)^2 - (mean(y))^2 ∑f ; since (mean(y))^2 ∑f = ( ( ∑fy) / (∑f ) )^2 ∑ f = (∑fy) (∑fy)/(∑f)
=mean(y) ∑fy , we can also write the formula as ∑f (y)^2 - mean(y) ∑fy ; the rest follows . I hope it is clear to you now.