The birth weights, in kg, of 1500 babies are summarised in the table below - Edexcel - A-Level Maths Statistics - Question 3 - 2010 - Paper 1

To find the mean birth weight, we use the formula:

$\text{Mean} = \frac{\sum f x}{\sum f}$

Substituting the values:

$\text{Mean} = \frac{4841}{1500} = 3.2273... \approx 3.23 \text{ kg}$

The formula for standard deviation (SD) is:

$\text{SD} = \sqrt{\frac{\sum f x^2}{\sum f} - \left( \frac{\sum f x}{\sum f} \right)^2}$

Substituting the values:

$\text{SD} = \sqrt{\frac{15889.5}{1500} - (3.2273...)^2}$

Calculating:

First, calculate the mean squared:

$(3.2273...)^2 \approx 10.4149$

Now substituting back:

$\text{SD} = \sqrt{10.5923 - 10.4149} \approx \sqrt{0.1774} \approx 0.4210... \approx 0.42$

To find the median: Total frequency (n) = 1500, so we need to find ( n/2 = 750 ). Reviewing the cumulative frequency, we find:

• 1 (0.0 - 1.0)
• 7 (1.0 - 2.0)
• 67 (2.0 - 2.5)
• 347 (2.5 - 3.0)
• 1167 (3.0 - 3.5)

Since 750 falls between 347 and 1167, the median is within the range of 3.0 - 3.5 kg. Using linear interpolation:

$ext{Median} = 3.00 + \left( \frac{750 - 347}{820 - 347} \right) \times (3.5 - 3.0)$

Calculating:

$= 3.00 + \left( \frac{403}{473} \right) \times 0.5 \approx 3.00 + 0.4260 \approx 3.426 \text{ kg}$

The skewness can be assessed by comparing the mean and median. Since the mean (3.23 kg) is slightly lower than the median (approximately 3.426 kg), the distribution is negatively skewed. This implies that there are a tail of lighter weights influencing the mean downward more than the median.