HardMathematicsHigh School (11–12)Statistics and ProbabilityNumerical

An exam board checks the marks of 25 candidates…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

An exam board checks the marks of $25$ candidates. The first report gave a mean of $40$ and a standard deviation of $5$ . A clerk then finds that two marks were copied wrongly. A mark of $30$ should have been $35$ , and a mark of $50$ should have been $45$ . The board wants the figures corrected before the report goes to the schools. It also wants to know whether the candidates are now more or less consistent, so it asks for the coefficient of variation of the corrected marks. You do not have the individual marks, only the totals that follow from the mean and the standard deviation. Find the corrected mean, the corrected variance, the corrected standard deviation and the corrected coefficient of variation.

Need a reminder? The key ideaShow

$\bar{x} = \dfrac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$ and $\sigma^2 = \dfrac{n_1\left(\sigma_1^2 + d_1^2\right) + n_2\left(\sigma_2^2 + d_2^2\right)}{n_1 + n_2}$ , where $d_i = \bar{x}_i - \bar{x}$ . To correct a reading, change $\sum x$ and $\sum x^2$ .

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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