The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below - Edexcel - A-Level Maths Statistics - Question 2 - 2008 - Paper 2

To find the quartiles, we first list the ages in order:

• 6, 7, 15, 21, 44, 50, 63

• Lower Quartile ($Q_1$): This is the median of the first half of the data. For 3 lower values (6, 7, 15), the lower quartile is 7.

• Median ($Q_2$): The median is the middle value of the ordered list. Thus, the median of the seven values is 50.

• Upper Quartile ($Q_3$): This is the median of the upper half of the data. For the upper values (50, 63), it is also 63.

The mean can be calculated using the formula:

ar{x} = \frac{\sum x}{n}

Where $= 7$ (the number of residents), and $\\sum x = 6 + 7 + 15 + 21 + 44 + 50 + 63 = 206$.

Thus,

ar{x} = \frac{206}{7} \approx 29.43

The standard deviation ($s$) is given by the formula:

$s = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}$

Calculating this:

$s = \sqrt{\frac{81 213}{7} - (29.43)^2} \approx \sqrt{11602.43 - 867.53} \approx \sqrt{10734.9} \approx 103.57$

To evaluate the measure of skewness:

$\text{Skewness} = \frac{\text{mean} - \text{mode}}{\text{standard deviation}}$

Substituting the values we calculated:

• Mean = ar{x} \approx 50
• Mode = 50
• Standard deviation = 103.57

Thus,

$\text{Skewness} = \frac{50 - 50}{103.57} = 0$

In comparing the age distributions:

• The Abbey Hotel has a mode of 39 whereas the Balmoral Hotel has a mode of 50, indicating that the most common age in Abbey is younger than in Balmoral.
• The Abbey Hotel shows a standard deviation of 12.7 compared to 103.57 for the Balmoral Hotel, suggesting that the ages in the Abbey Hotel are more concentrated around the mean than in the Balmoral Hotel.
• Therefore, the distribution in Balmoral may be considered wider and less uniform than that of Abbey.