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:

1. First Quartile (Q1): The first quartile is the median of the first half of the data. Counting up to the 25th percentile gives: Q1 = 35

2. Second Quartile (Q2) or Median: The median divides the data into two halves. Here, it is: Q2 = 52

3. Third Quartile (Q3): The median of the second half of the data is: Q3 = 60

Thus, the values of the three quartiles are:

• Q1 = 35
• Q2 = 52
• Q3 = 60

To calculate the mean (ar{x}):

$\bar{x} = \frac{\Sigma x}{n} = \frac{1335}{27} \approx 49.4$

For standard deviation (σ):

$\sigma = \sqrt{\frac{\Sigma x^2}{n} - \left(\frac{\Sigma x}{n}\right)^2} = \sqrt{\frac{71801}{27} - (49.4)^2}$

Calculating gives:

$\sigma = \sqrt{214.54} \approx 14.6$

To evaluate skewness, we use the skewness formula:

$\text{Skewness} = \frac{\bar{x} - \text{Mode}}{\sigma}$

Using our values:

$\text{Skewness} = \frac{49.4 - 56}{14.6} \approx -0.448$

Since the skewness value is negative, this indicates that the data are negatively skewed.

1. Distribution Shape: The histogram or frequency distribution may show a longer tail on the left side, indicating more values concentrated on the higher end of the scale.

2. Comparison of Mean and Median: The mean is lower than the median (Mean < Median), which is another indicator of negative skewness.