Each year Upchester United plays against Upchester City in a local derby match. The number of goals scored in a match by United is denoted by $u$ and the number of goals scored in a match by City is denoted by $c.$ The number of goals scored in the past $15$ matches are summarised by $\Sigma {\left(u-1\right)}^2=25,\Sigma \left(u-1\right)=9,$ $\Sigma c^2=39$ and $\Sigma c=19.$

How many goals have been scored altogether in these $15$ matches?

Each year Upchester United plays against Upchester City in a local derby match. The number of goals scored in a match by United is denoted by $u$ and the number of goals scored in a match by City is denoted by $c.$ The number of goals scored in the past $15$ matches are summarised by $\Sigma {\left(u-1\right)}^2=25,\Sigma \left(u-1\right)=9,$ $\Sigma c^2=39$ and $\Sigma c=19.$

Show that $\Sigma u^2=58.$

Each year Upchester United plays against Upchester City in a local derby match. The number of goals scored in a match by United is denoted by $u$ and the number of goals scored in a match by City is denoted by $c.$ The number of goals scored in the past $15$ matches are summarised by $\Sigma {\left(u-1\right)}^2=25,\Sigma \left(u-1\right)=9,$ $\Sigma c^2=39$ and $\Sigma c=19.$

Find, correct to $3$ decimal places, the variance of the number of goals scored by the two teams together in these $15$ matches.

Twenty values of $x$ are summarised by $\Sigma {\left(x-1\right)}^2=132$ and $\Sigma \left(x-1\right)=44.$

Eighty values of $y$ are summarised by $\Sigma {\left(y+1\right)}^2=17704$and $\Sigma \left(y+1\right)=1184.$

Show that $\Sigma x=64$ and that $\Sigma x^2=240$

Twenty values of $x$ are summarised by $\Sigma {\left(x-1\right)}^2=132$ and $\Sigma \left(x-1\right)=44.$

Eighty values of $y$ are summarised by $\Sigma {\left(y+1\right)}^2=17704$and $\Sigma \left(y+1\right)=1184.$

Calculate the value of $\Sigma y$  and of $\Sigma y^2$

Twenty values of $x$ are summarised by $\Sigma {\left(x-1\right)}^2=132$ and $\Sigma \left(x-1\right)=44.$

Eighty values of $y$ are summarised by $\Sigma {\left(y+1\right)}^2=17704$and $\Sigma \left(y+1\right)=1184.$

Find the exact variance of the $100$ values of $x$ and $y$ combined.

The heights, $x \mathrm{cm},$ of $200$boys and the heights, $y \mathrm{cm},$ of $300$ girls are summarised by the following totals:

$\Sigma {\left(x-160\right)}^2=18240, \Sigma \left(x-160\right)=1820, \Sigma {\left(y-150\right)}^2=20100, \Sigma \left(y-150\right)=2250.$

Find the mean height of these$500$ children.

The heights, $x \mathrm{cm},$ of $200$boys and the heights, $y \mathrm{cm},$ of $300$ girls are summarised by the following totals:

$\Sigma {\left(x-160\right)}^2=18240, \Sigma \left(x-160\right)=1820, \Sigma {\left(y-150\right)}^2=20100, \Sigma \left(y-150\right)=2250.$

By first evaluating $\Sigma x^2$ and $\Sigma y^2,$ find the variance of the heights of the $500$ children, including appropriate units with your answer.