Question

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

Answer 1

Given that $\sum x=35$ and $n=5$.
$\therefore \overline{x}=\frac{\sum x}{n}=\frac{35}{5}=7$.
Let us find $\sum x^2$
Now, $\sum (x-9{)}^2=82$
$\Rightarrow \sum (x^2-18x+81)=82$
$\Rightarrow \sum x^2-(18\sum x)+(81\sum 1)=82$
$\Rightarrow \sum x^2-630+405=82$ $\because \sum x=35$ and $\sum 1=5$
$\Rightarrow \sum x^2=307$.
To find $\sum (x-\overline{x}{)}^2$, let us consider
$\sum (x-9{)}^2=82$
$\Rightarrow \sum (x-7-2{)}^2=82$
$\Rightarrow \sum {\left[\right(x-7)-2]}^2=82$
$\Rightarrow \sum (x-7{)}^2-2\sum \left[\right(x-7)\times 2]+\sum 4=82$
$\Rightarrow \sum (x-\overline{x}{)}^2-4\sum (x-\overline{x})+4\sum 1=82$
$\Rightarrow \sum (x-\overline{x}{)}^2-4(0)+(4\times 5)=82$ $\because \sum 1=5$ and $\sum (x-\overline{x})=0$
$\Rightarrow \sum (x-\overline{x}{)}^2=62$
$\therefore \sum x^2=307$ and $\sum (x-\overline{x}{)}^2=62$.