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 need to order the data and determine:

• First Quartile (Q1): Located at the 25th percentile — for 27 data points, this is the 7th value, which is 35.
• Second Quartile (Q2 or Median): Located at the 50th percentile — this is the 14th value, which is 52.
• Third Quartile (Q3): Located at the 75th percentile — this is the 21st value, which is 60.

To calculate the mean (ar{x}):
ar{x} = \frac{Σx}{n} = \frac{1335}{27} \approx 49.4

For the standard deviation ($σ$), we use the formula:
$σ = \sqrt{\frac{Σx² - \frac{(Σx)²}{n}}{n}} = \sqrt{\frac{71801 - \frac{(1335)²}{27}}{27}} \approx 14.6.$

So, the mean is approximately 49.4 and the standard deviation is approximately 14.6.

Using the formula for skewness: $\text{Skewness} = \frac{\bar{x} - \text{mode}}{σ}$

Substituting the values we have: $\text{Skewness} = \frac{49.4 - 56}{14.6} \approx -0.448$

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

1. Mean < Median: The mean (49.4) is less than the median (52), which is a sign of negative skewness.
2. Presence of Outliers: If there are a few extreme values on the lower end, this can pull the mean down, further indicating negative skewness.