| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate mean from coded sums |
| Difficulty | Moderate -0.8 This is a straightforward application of coding formulas for mean and variance. Part (a) requires simple algebraic manipulation of Σ(x-k)/n = x̄ - k, and part (b) uses the standard variance formula with coded data. Both are direct recall of standard results with minimal problem-solving required, making it easier than average. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{40 \times 34}{40} - k = \frac{520}{40}\) | M1 | Forms an equation involving \(\Sigma x\), \(\Sigma(x-k)\) and \(k\). Accept at a numeric stage with \(k\). |
| \(k\ [= 34 - 13] = 21\) | A1 | Evaluated. |
Total: 2 marks
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var} = \left[\frac{\sum(x-k)^2}{40} - \left(\frac{\sum(x-k)}{40}\right)^2\right] = \frac{9640}{40} - \left(\frac{520}{40}\right)^2 = [241 - 13^2 =]\) | M1 | Values substituted into an appropriate variance formula, accept unsimplified |
| \(72\) | A1 |
**Question 2:**
**Part 2(a):**
$\left[\frac{\sum x}{40} - k = \frac{\sum(x-k)}{40}\right]$
$\frac{40 \times 34}{40} - k = \frac{520}{40}$ | M1 | Forms an equation involving $\Sigma x$, $\Sigma(x-k)$ and $k$. Accept at a numeric stage with $k$.
$k\ [= 34 - 13] = 21$ | A1 | Evaluated.
Total: **2 marks**
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var} = \left[\frac{\sum(x-k)^2}{40} - \left(\frac{\sum(x-k)}{40}\right)^2\right] = \frac{9640}{40} - \left(\frac{520}{40}\right)^2 = [241 - 13^2 =]$ | M1 | Values substituted into an appropriate variance formula, accept unsimplified |
| $72$ | A1 | |
---2 A summary of 40 values of $x$ gives the following information:
$$\Sigma ( x - k ) = 520 , \quad \Sigma ( x - k ) ^ { 2 } = 9640$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Given that the mean of these 40 values of $x$ is 34 , find the value of $k$.
\item Find the variance of these 40 values of $x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q2 [4]}}