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 arrange the data in ascending order:

27, 29, 32, 32, 33, 36, 39, 46, 56, 68, 72, 73, 74, 75, 81.

• The first quartile (Q1), is the median of the first half of the data: Q1 = 35.
• The second quartile (Q2), or median, is 49.4.
• The third quartile (Q3) is the median of the second half of the data: Q3 = 60.

To calculate the mean: $\bar{x} = \frac{\Sigma x}{n} = \frac{1335}{27} = 49.4$.

For the standard deviation: $\sigma^2 = \frac{\Sigma x^2}{n} - (\bar{x})^2 = \frac{71801}{27} - (49.4)^2 = 214.543...$ Thus, standard deviation, $\sigma = \sqrt{214.543...} \approx 14.6$.

Using the skewness formula: $\text{Skewness} = \frac{mean - mode}{standard deviation} = \frac{49.4 - 56}{14.6} = -0.448.$

Since the skewness is less than zero, it indicates that the data are negatively skewed.

1. The tail of the data distribution is longer on the left side, indicating that a lower number of people leaving is more frequent than higher numbers.
2. The presence of outliers on the lower end of the data can further illustrate the negative skewness, moving the mean lower than the mode.