A class of students had a sudoku competition - Edexcel - A-Level Maths Statistics - Question 5 - 2011 - Paper 2

To find the median, we first need to determine the cumulative frequency. The total frequency is 30, so the median lies in the 15th position.

From the cumulative frequency table:

• 2 (for 2-8)
• 9 (for 9-12)
• 17 (for 13-15)

The median class is therefore 13 - 15. Using linear interpolation:

$Q_{2} = L + \left( \frac{N/2 - CF}{f} \right) \times c$

where:

• $L = 13$ (lower boundary of median class)
• $CF = 9$ (cumulative frequency before the median class)
• $f = 8$ (frequency of the median class)
• $c = 2$ (class width)

Calculating: $Q_{2} = 13 + \left( \frac{15 - 9}{8} \right) \times 2 = 13.75$ Thus, the estimated median time is approximately 15.875.

To calculate the mean ($\bar{x}$):

$\bar{x} = \frac{\sum fx}{\sum f}$

Calculating:

Using:

• $\sum f = 30$
• $\sum fx = 477.5$ (calculated from the given midpoints and frequencies)

So, $\bar{x} = \frac{477.5}{30} \approx 15.92$

For the standard deviation ($\sigma$):

$\sigma = \sqrt{\frac{\sum f(x - \bar{x})^{2}}{\sum f}}$

Calculating each term gives us the required standard deviation. Perform the complete calculation.

The resulting value is approximately $\sigma \approx 5.78$.

The normal distribution can be used since the mean and median are similar or very close, indicating a symmetric distribution. Additionally, the data may exhibit continuous characteristics typical of normal distribution.

Given the quartile values: $Q_{1} = 8.5$, $Q_{2} = 13.0$, and $Q_{3} = 21.0$, the skewness can be identified using the formulas for skewness. Since $Q_{3} - Q_{2} > Q_{2} - Q_{1}$, this indicates a positive skewness in the distribution of times taken.