Exactly $20$ years ago, the mean age of a group of boys was $15.7$ years and the sum of the squares of their ages was $16000$. If the sum of the squares of their ages has increased by $8224$ in this $20$-year period, find the number of boys in the group.

A transport company runs $21$ coaches between two cities every week. In the past, the mean and variance of the journey times were $4$ hours $35$ minutes and $53.29$ minutes?

What would be the mean and standard deviation of the times if all the coaches departed $10$ minutes later and arrived$5$ minutes earlier than in the past?
Are there any situations in which achieving this might actually be possible?

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 $\Sigma u^2=58.$