Consider $n$ numbers $x_1,\ldots,x_n$. Let ${\overline x}=n^{-1}\sum_{j=1}^n x_j$ be the sample mean. For notation, also let $A_k=\sum_{i=1}^n x_i^k$ for $k=1,2,3,\ldots$. Hence, ${\overline x}=A_1/n$ and $\sum_{i=1}^n (x_i-{\overline x})^2 = \sum_{i=1}^n x_i^2 -n {\overline x}^2 = A_2-A_1^2/n$. Check that you can derive these equations before trying the questions below.
Part a)
What is the summation of deviations: $\sum_{i=1}^n (x_i-{\overline x})\,\,$?

Part b)
Consider $\sum_{i=1}^n (x_i-{\overline x})^3$? Simplify this to an expression involving $A_1,A_2,A_3,n$.
Now, suppose $n=5$, $A_1={19}$, $A_2={85}$, $A_3={421}$. Applying your formula, what is the numerical value of $\sum_{i=1}^n (x_i-{\overline x})^3$?

Hint: