Over a period of time, the number of people x leaving a hotel each morning was recorded - Edexcel - A-Level Maths Statistics - Question 1 - 2006 - Paper 1

To find the quartiles, we first order the data:

• Ordered data: 27, 29, 32, 32, 35, 36, 42, 49, 52, 56, 58, 61, 63, 64, 68, 71, 78, 81.

• Q1 (first quartile) is the median of the first half: 35.

• Q2 (median) is the median of the dataset: 52.

• Q3 (third quartile) is the median of the second half: 60.

The mean is calculated as:
$\bar{x} = \frac{\Sigma x}{n} = \frac{1335}{27} = 49.4$

The standard deviation is calculated as:
$\sigma = \sqrt{\frac{\Sigma x^2}{n} - \bar{x}^2} = \sqrt{\frac{71801}{27} - (49.4)^2} = 14.6 \text{ or } 14.9$

To evaluate the skewness, we use the formula:
$\text{Skewness} = \frac{mean - mode}{standard deviation} = \frac{49.4 - 56}{14.6} = -0.448$
This result indicates that the data are negatively skewed, as the skewness is less than zero.

1. The distribution of data has a longer tail on the left side, indicating that there are fewer low values.

2. The quartiles show Q1 is less than Q2 and Q2 is less than Q3, confirming that the spread of the data is skewed towards the higher values.