| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| 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 standard deviation. Part (i) requires simple algebraic manipulation of Σ(x-a) = Σx - na to find a, while part (ii) 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(-73.2/24 = -3.05\) | M1 | Accept \((-72.4 + \text{anything})/24\) |
| \(a = 8.95 + 3.05 = 12\) | A1 | Correct answer |
| OR \(8.95 \times 24 = 214.8\); \(\Sigma x - \Sigma a = -73.2\); \(\Sigma a = 288\), \(a = 12\) | M1, A1 [2] | For \(8.95 \times 24\) seen; Correct answer obtained using \(\Sigma x\) and \(\Sigma a\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\text{sd} = \sqrt{\frac{2115}{24} - (-3.05)^2}\) | M1 | For \(\frac{2115}{24} - (\pm \text{ their coded mean})^2\) |
| \(= 8.88\) | A1 | Correct answer |
| OR \(\text{sd} = \sqrt{\frac{3814.2}{24} - 8.95^2}\) | M1 | For \(\frac{\text{their } \Sigma x^2}{24} - 8.95^2\) where \(\Sigma x^2\) is obtained from expanding \(\Sigma(x-a)^2\) with \(2a\Sigma x\) seen |
| \(= 8.88\) | A1 [2] | Correct answer |
## Question 1:
**Part (i)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $-73.2/24 = -3.05$ | M1 | Accept $(-72.4 + \text{anything})/24$ |
| $a = 8.95 + 3.05 = 12$ | A1 | Correct answer |
| OR $8.95 \times 24 = 214.8$; $\Sigma x - \Sigma a = -73.2$; $\Sigma a = 288$, $a = 12$ | M1, A1 **[2]** | For $8.95 \times 24$ seen; Correct answer obtained using $\Sigma x$ and $\Sigma a$ |
**Part (ii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{sd} = \sqrt{\frac{2115}{24} - (-3.05)^2}$ | M1 | For $\frac{2115}{24} - (\pm \text{ their coded mean})^2$ |
| $= 8.88$ | A1 | Correct answer |
| OR $\text{sd} = \sqrt{\frac{3814.2}{24} - 8.95^2}$ | M1 | For $\frac{\text{their } \Sigma x^2}{24} - 8.95^2$ where $\Sigma x^2$ is obtained from expanding $\Sigma(x-a)^2$ with $2a\Sigma x$ seen |
| $= 8.88$ | A1 **[2]** | Correct answer |
---1 A summary of 24 observations of $x$ gave the following information:
$$\Sigma ( x - a ) = - 73.2 \quad \text { and } \quad \Sigma ( x - a ) ^ { 2 } = 2115 .$$
The mean of these values of $x$ is 8.95 .\\
(i) Find the value of the constant $a$.\\
(ii) Find the standard deviation of these values of $x$.
\hfill \mbox{\textit{CAIE S1 2007 Q1 [4]}}