Combining the mean and standard deviation of two classes

Class \(A\) has \(20\) students with mean mark \(62\) and standard deviation \(8\) marks. Class \(B\) has \(30\) students with mean mark \(70\) and standard deviation \(6\) marks. For both classes, \(\sigma=\sqrt{\dfrac{\sum x^2}{n}-\overline{x}^2}\) is used to calculate the standard deviation.

1. Find the mean mark of all \(50\) students.[2]
2. By making \(\sum x^2\) the subject of the formula for \(\sigma\), show that for any set of \(n\) values, \(\sum x^2=n(\sigma^2+\overline{x}^2)\).[2]
3. Hence find the standard deviation of all \(50\) marks, and explain why it is larger than the weighted average of the two standard deviations.[3]