Each member of a group of 27 people was timed when completing a puzzle - Edexcel - A-Level Maths Statistics - Question 3 - 2020 - Paper 1

The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). From the box plot, Q3 is 25 minutes and Q1 is 14 minutes. Therefore, the IQR is:

$\text{IQR} = Q3 - Q1 = 25 - 14 = 11 \text{ minutes}$

The mean is obtained by dividing the total sum of the times by the number of individuals. Here,

$\mu = \frac{\sum x}{n} = \frac{607.5}{27} = 22.5 \text{ minutes}$

The standard deviation is calculated using the formula:

$\sigma = \sqrt{\frac{\sum x^2}{n} - \left( \frac{\sum x}{n} \right)^2}$

Substituting the given values:

$\sigma = \sqrt{\frac{17623.25}{27} - \left( \frac{607.5}{27} \right)^2} = \sqrt{12.102181 - 22.5^2} = 12.1 \text{ minutes}$

First, we find the mean plus three standard deviations:

$\mu + 3\sigma = 22.5 + 3 \times 12.1 = 58.8 \text{ minutes}$

Since the maximum time recorded is 68 minutes, which is greater than 58.8 minutes, there is 1 outlier.

Since we want median to increase and mean to remain constant, we can choose:

• Let a = 45 minutes
• Let b = 4 minutes

In this case, both values satisfy the condition where both a and b are within the limits of the existing data and allow for the median to increase.

Adding Adam and Beth's times (4 and 45 minutes, respectively) to the 27 times reduces the variability among the data, especially if both are closer to the mean than the existing data values. This could lead to a smaller standard deviation for the combined dataset than for the original 27.