| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate variance/SD from coded sums |
| Difficulty | Moderate -0.8 This is a straightforward application of standard coding formulas for variance/SD. Part (a) uses the direct formula σ² = [Σ(x-q)²]/n - [Σ(x-q)/n]², requiring only substitution. Part (b) involves the simple relationship Σx = Σ(x-q) + nq. Both parts are routine algebraic manipulation with no conceptual challenge beyond recalling standard results. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var} = \left[\frac{\Sigma(x-q)^2}{50} - \left(\frac{\Sigma(x-q)}{50}\right)^2\right] = \frac{14235}{50} - \left(\frac{700}{50}\right)^2\) | M1 | \(\frac{14235}{a} - \left(\frac{700}{a}\right)^2\); where \(a = 49, 50, 51\) |
| \([= 284.7 - 196 = 88.7]\) | ||
| \([\text{sd} = \sqrt{88.7} =]\ 9.42\) | A1 | 9.4180677 rounded to at least 3SF |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\Sigma x - 50q = 700\) \([2865 - 50q = 700]\) | M1 | Forming equation with \(\Sigma x\), \(50q\) and \(700\) |
| \(q = 43.3,\ 43\frac{3}{10}\) | A1 | If M0 scored, SC B1 for \(43.3\) WWW |
| 2 |
**Question 1:**
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var} = \left[\frac{\Sigma(x-q)^2}{50} - \left(\frac{\Sigma(x-q)}{50}\right)^2\right] = \frac{14235}{50} - \left(\frac{700}{50}\right)^2$ | M1 | $\frac{14235}{a} - \left(\frac{700}{a}\right)^2$; where $a = 49, 50, 51$ |
| $[= 284.7 - 196 = 88.7]$ | | |
| $[\text{sd} = \sqrt{88.7} =]\ 9.42$ | A1 | 9.4180677 rounded to at least 3SF |
| | **2** | |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\Sigma x - 50q = 700$ $[2865 - 50q = 700]$ | M1 | Forming equation with $\Sigma x$, $50q$ and $700$ |
| $q = 43.3,\ 43\frac{3}{10}$ | A1 | If M0 scored, **SC B1** for $43.3$ WWW |
| | **2** | |1 A summary of 50 values of $x$ gives
$$\Sigma ( x - q ) = 700 , \quad \Sigma ( x - q ) ^ { 2 } = 14235$$
where $q$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find the standard deviation of these values of $x$.
\item Given that $\Sigma x = 2865$, find the value of $q$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2023 Q1 [4]}}