A student sits a national test and is told that the marks follow a normal distribution with mean 100. The student receives a mark of 124 and is told that he is at the \({68^{{\text{th}}}}\) percentile.

Calculate the variance of the distribution.

\(X:{\text{N}}(100,{\text{ }}{\sigma ^2})\)

\({\text{P}}(X < 124) = 0.68\)     (M1)(A1)

\(\frac{{24}}{\sigma } = 0.4676 \ldots \)     (M1)

\(\sigma  = 51.315 \ldots \)     (A1)

variance = 2630     A1

Notes:     Accept use of \({\text{P}}(X < 124.5) = 0.68\) leading to variance = 2744.

[5 marks]