| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | June |
| 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-k)/n = x̄ - k, 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 but not trivial since students must remember the coding relationships. |
| Spec | 2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{\sum x}{30} - k = \frac{315}{30} = 10.5\) | M1 | Dividing 315 by \(\pm 30\) and \(+\) or \(-\) from 50.5; need both and no more |
| \(k = 5.5 - 10.5 = 40\) | A1 | Correct answer from correct working |
| OR: \(\sum x = 50.5 \times 30 = 1515\), \(1515 - 30k = 315\) | M1 | Mult by 50.5 by 30 and \(+\) or \(-\) 315 and dividing by \(\pm 30\); need all these |
| \(k = 40\) | A1 | Correct answer from correct working. 1200 gets M0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{var} = 4022/30 - 10.5^2 (= 23.817)\) | M1 | Subst in correct coded variance formula |
| \(\text{sd} = 4.88\) | A1 | |
| OR: \(\sum x^2 - 2(40)\sum x + 30(40)^2 = 4022\), \(\sum x^2 = 77222\), \(\text{Var} = 77222/30 - 50.5^2 (= 23.817)\) | M1 | Expanding with \(\pm 40\Sigma x\) and \(\pm 30(40)^2\) seen |
| \(\text{sd} = 4.88\) | A1 |
## Question 1:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{\sum x}{30} - k = \frac{315}{30} = 10.5$ | M1 | Dividing 315 by $\pm 30$ and $+$ or $-$ from 50.5; need both and no more |
| $k = 5.5 - 10.5 = 40$ | A1 | Correct answer from correct working |
| OR: $\sum x = 50.5 \times 30 = 1515$, $1515 - 30k = 315$ | M1 | Mult by 50.5 by 30 and $+$ or $-$ 315 and dividing by $\pm 30$; need all these |
| $k = 40$ | A1 | Correct answer from correct working. 1200 gets **M0** |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{var} = 4022/30 - 10.5^2 (= 23.817)$ | M1 | Subst in correct coded variance formula |
| $\text{sd} = 4.88$ | A1 | |
| OR: $\sum x^2 - 2(40)\sum x + 30(40)^2 = 4022$, $\sum x^2 = 77222$, $\text{Var} = 77222/30 - 50.5^2 (= 23.817)$ | M1 | Expanding with $\pm 40\Sigma x$ and $\pm 30(40)^2$ seen |
| $\text{sd} = 4.88$ | A1 | |
---1 Kadijat noted the weights, $x$ grams, of 30 chocolate buns. Her results are summarised by
$$\Sigma ( x - k ) = 315 , \quad \Sigma ( x - k ) ^ { 2 } = 4022$$
where $k$ is a constant. The mean weight of the buns is 50.5 grams.\\
(i) Find the value of $k$.\\
(ii) Find the standard deviation of $x$.\\
\hfill \mbox{\textit{CAIE S1 2017 Q1 [4]}}