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 need to order the ages of residents in the Balmoral Hotel:

1. 6
2. 15
3. 44
4. 47
5. 50
6. 63

• The lower quartile (Q1) is the median of the first half of the data: Q1 = 45.
• The median (Q2) is the middle value of the ordered data: Q2 = 50.5.
• The upper quartile (Q3) is the median of the second half of the data: Q3 = 63.

To calculate the mean:

ar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

Where:

• $\sum x = 1469$ (the total age of residents),
• $n = 28$ (the number of residents).

Thus,

$\bar{x} = \frac{1469}{28} \approx 52.46$.

(ii) For the standard deviation, we use:

S_d = \sqrt{ \frac{\sum x^2 - (\frac{\sum x)^2}}{n} }

Given that $\sum x^2 = 81 213$:

$S_d = \sqrt{\frac{81213 - \frac{1469^2}{28}}{28}} \approx 12.21$

Using the formula for skewness:

$skewness = \frac{mean - mode}{standard deviation}$

For the Balmoral Hotel, substituting the known values:

• mean = 52.46,
• mode = 50,
• standard deviation = 12.21,

we find:

$skewness = \frac{52.46 - 50}{12.21} \approx 0.20$

The distribution of ages between the residents of the Balmoral Hotel and the Abbey Hotel shows that:

• The Balmoral Hotel has a lower mode (50 years) compared to the Abbey Hotel's mode (39 years), indicating that residents in Balmoral Hotel tend to be older.
• The standard deviation for the Balmoral Hotel is higher ( 12.21) compared to the Abbey Hotel (12.7), suggesting a wider spread of ages among Balmoral residents.
• Overall, the Balmoral Hotel appears to have an older resident cohort with a higher age variability.