Question

For a collection of data of $\Sigma x=35,n=5,\Sigma (x-9{)}^2=82$ then find $\Sigma x^2$ and $\Sigma (x-\stackrel{-}{x}{)}^2$.

Answer 1

$\sum {(x-9)}^2=82$

$\Rightarrow \sum x^2+81-2\sum 9x=82$ by expanding the above summation

$\Rightarrow \sum x^2+81-18\sum x=82$

$\Rightarrow \sum x^2+81-18\times 35=82$

$\Rightarrow \sum x^2=82-81+18\times 35$

$\therefore \sum x^2=631$

$\overline{x}=\frac{\sum x}{n}=\frac{35}{5}=7$

$\sum {(x-\overline{x})}^2$

$=\sum {(x-7)}^2$

$=\sum x^2+49-2\sum 7x$ by expanding the above summation

$=631+49-14\times 35$ from above

$=680-490=190$