(a) To draw the histogram, calculate the class widths: \( 10,\ 5,\ 15,\ 20,\ 10 \). Then calculate the frequency density for each class:

\( \text{Frequency density} = \dfrac{\text{frequency}}{\text{class width}} \)

\( 1.8,\ 4.8,\ 2,\ 1,\ 0.8 \)

Ensure all heights are correct on the diagram using a linear scale. The correct bar ends are: \( 10.5,\ 15.5,\ 30.5,\ 50.5,\ 60.5 \).

(b) The greatest possible value of the interquartile range (IQR) is calculated using the class intervals \( 11\text{–}15 \) and \( 31\text{–}50 \). Thus, the greatest IQR \( = 50 - 11 = 39 \).

(c) To calculate the mean:

\( \text{Mean} = \frac{ 18 \times 5.5 + 24 \times 13 + 30 \times 23 + {}\\ 20 \times 40.5 + 8 \times 55.5 }{100}\) \(= \frac{2355}{100} = 23.6 \)

To calculate the variance:

\( \text{Var} = \frac{ 18 \times 5.5^2 + 24 \times 13^2 + 30 \times 23^2 + {}\\ 20 \times 40.5^2 + 8 \times 55.5^2 }{100}\) \(- \text{mean}^2 \)

\( = \frac{77917.5}{100} - 23.6^2\) \(= 224.57 \)

Standard deviation \( = \sqrt{224.57} = 15.0 \).