The marks, x, of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below - Edexcel - A-Level Maths Statistics - Question 4 - 2012 - Paper 1

To find the quartiles:

• Lower Quartile (Q1): This is the value at the 25th percentile. For 45 data points, Q1 corresponds to the value at the 11.25th position, which rounds to 11, hence:

  Q1 = 46.

• Median (Q2): The median is the value at the 50th percentile, which corresponds to the 23rd position:

  Q2 = 46.

• Upper Quartile (Q3): This is the value at the 75th percentile, at the 33.75th position, rounding to 34:

  Q3 = 56.

To calculate the mean:

$\text{Mean} = \frac{\sum x}{n} = \frac{2497}{45} \approx 55.49$

For the standard deviation:

First, find variance:

$\text{Variance} = \frac{\sum x^2 - \left(\frac{\sum x}{n}\right)^2 n}{n} = \frac{143369 - \left(\frac{2497}{45}\right)^2 \times 45}{45}$

This simplifies to find the standard deviation roughly as:

$\text{Standard Deviation} \approx 10.45$.

The mean is less than the median, and the median is less than the mode, which indicates a negative skew. This is because most of the marks are concentrated on the higher end due to the lower values pulling the mean down.

To find the new mean, subtract 5 and scale by 90%:

• New Mean = (55.49 - 5) * 0.9 = 45.44.

For standard deviation:

• New Standard Deviation = 10.45 * 0.9 = 9.405.