Mathematics 9709/51 — May/June 2023

Walkthrough

The question gives a coded summary of 50 values: $\sum(x-q)=700$ and $\sum(x-q)^2=14\,235$. Because subtracting $q$ from every value only shifts the data by a constant, it does not affect the standard deviation. Therefore the standard deviation of $x$ is the same as the standard deviation of $x-q$.

Use the variance formula for a set of values $y=x-q$:

$$\text{Var}(y) = \frac{\sum y^2}{n} - \left(\frac{\sum y}{n}\right)^2$$

Here $\sum y^2 = 14235$ and $\sum y = 700$, with $n=50$. Substituting gives

$$\text{Var} = \frac{14235}{50} - \left(\frac{700}{50}\right)^2 = 284.7 - 196 = 88.7$$

So the standard deviation is

$$\sqrt{88.7} = 9.42 \quad (3\text{ s.f.})$$

Key Takeaways

• The variance and standard deviation are unchanged when a constant is added to every data value.
• For coded data, use the formula $\text{Var} = \frac{\sum y^2}{n} - \left(\frac{\sum y}{n}\right)^2$.
• The final answer is a measure of spread, not the mean, so square roots must be taken carefully.

Common Mistakes

• Forgetting to subtract the square of the mean of the coded values.
• Giving the variance $88.7$ as the standard deviation instead of taking the square root.
• Using the raw sum $\sum x$ rather than the coded sums $\sum(x-q)$ and $\sum(x-q)^2$.

Things to Be Careful About

• Use the correct denominator $n=50$; the mark scheme also accepts a denominator of $49$ or $51$ in this context.
• The final standard deviation should be rounded to at least 3 significant figures; $9.42$ is acceptable.
• Keep the units and signs consistent throughout the substitution.

Walkthrough

This part gives the raw total $\sum x = 2865$ and asks for the coding constant $q$. The coded sum is made term by term:

$$\sum(x-q) = (x_1-q)+(x_2-q)+\cdots+(x_{50}-q)$$

Since there are 50 terms, this equals $\sum x - 50q$. Setting this equal to 700 gives the equation

$$\sum x - 50q = 700$$

Substitute $\sum x = 2865$:

$$2865 - 50q = 700$$

Then solve:

$$50q = 2165 \quad \Rightarrow \quad q = 43.3$$

Key Takeaways

• $\sum(x-q) = \sum x - nq$.
• Adding a constant to every value changes the mean by that constant, but leaves the standard deviation unchanged.
• Given the coded sum and the raw total, the coding constant can be recovered by solving a linear equation.

Common Mistakes

• Forgetting the factor $50$ in front of $q$.
• Subtracting $q$ only once instead of from each value.
• Making a sign error when rearranging $2865 - 50q = 700$.

If no method is shown, the mark scheme awards SC B1 for the final value $43.3$; always show the equation to earn full method marks.

Things to Be Careful About

• $q$ is not the mean of $x$; the mean of $x$ is $2865/50 = 57.3$.
• Ensure $\sum(x-q)$ is expanded correctly before substituting.
• The answer $43.3$ can also be written as $43\frac{3}{10}$.