Mathematics 9709/61 — May/June 2013

Walkthrough

We have $n = 30$ values of $x$, but only information about the coded values $x - c$. The constant $c$ shifts all data by the same amount, so it changes the mean but not the spread. Therefore the standard deviation of $x$ equals the standard deviation of $x - c$.

For any values $y_1, \ldots, y_n$, the variance can be written as

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

Here $y = x - c$, so substitute the given sums:

$$\text{sd}^2 = \frac{1957.5}{30} - \left(\frac{234}{30}\right)^2.$$

Now compute each piece:

$$\frac{1957.5}{30} = 65.25, \qquad \frac{234}{30} = 7.8.$$

Thus

$$\text{sd}^2 = 65.25 - 7.8^2 = 65.25 - 60.84 = 4.41.$$

Finally, standard deviation is the positive square root:

$$\text{sd} = \sqrt{4.41} = 2.1.$$

Key Takeaways

• The variance formula can be used with $\Sigma y$ and $\Sigma y^2$ without needing the raw data.
• Adding or subtracting a constant from every value does not change variance or standard deviation.
• For coded data, the variance is still the mean of the squares minus the square of the mean of the coded values.

Common Mistakes

• Forgetting to divide both sums by 30 before combining them.
• Evaluating the formula correctly but forgetting to take the square root at the end.
• Squaring $\frac{234}{30}$ incorrectly; note $7.8^2 = 60.84$.

Things to Be Careful About

• The mark scheme awards M1 for correct substitution into the variance formula and A1 for the accurate value $2.1$ (or $2.10$).
• Standard deviation is always non-negative, so choose the positive square root.
• Keep the working exact: $4.41$ is exactly $2.1^2$, so $2.1$ is the exact answer.

Walkthrough

The coded values are $x - c$. Their mean is

$$\frac{\Sigma (x - c)}{n} = \frac{234}{30} = 7.8.$$

If we add $c$ to every coded value, we recover the original values $x$. Adding a constant to every value adds the same constant to the mean, so

$$\bar{x} = \frac{\Sigma (x - c)}{n} + c = 7.8 + c.$$

We are told $\bar{x} = 86$. Substitute this into the equation:

$$86 = c + 7.8.$$ $$c = 86 - 7.8 = 78.2.$$

Key Takeaways

• The mean of $x$ can be recovered from coded data by adding the coding constant to the mean of the coded data.
• The relation $\bar{x} = c + \frac{\Sigma (x - c)}{n}$ is the direct link between the original mean and the coded sums.

Common Mistakes

• Writing $86 = \frac{234}{30} - c$ or using the wrong sign between the two terms.
• Quoting $c = 78.2$ without any method; the mark scheme requires the $\frac{234}{30}$ step to be seen.
• Confusing the sum $\Sigma(x-c) = 234$ with its mean $7.8$.

Things to Be Careful About

• The mark scheme gives M1 as soon as $\frac{234}{30}$ is seen in the equation, then A1 for $c = 78.2$.
• Remember that subtracting the coding constant would give the opposite answer; adding it back is correct because $x = (x-c) + c$.
• Since $c$ is a constant, $86 - 7.8 = 78.2$ exactly; no rounding is needed.