| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| 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 (i) requires simple algebraic manipulation of Σ(x-100) = n(x̄-100) to find n. Part (ii) uses the standard variance identity Σ(x-a)² = Σ(x-b)² - n(a-b)². Both are routine textbook exercises requiring recall of formulas rather than problem-solving, making this easier than average. |
| Spec | 2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(72/n + 100 = 104.8\) or \(72 + 100n = 104.8n\); \(n = 15\) | M1 A1 [2] | \(72/n\) or \(100n\) and \(104.8n\) seen or implied correct answer |
| (ii) \(sd^2 = 499.2/15 - (72/15)^2\) (= 10.24) or \(sd^2 = \sum(x - 104.8)^2/15 - (\sum(x - 104.8)/15)^2\) or \(\sum(x - 104.8)^2 = 153.6\) (154) | M1 A1 [3] | numerical use of a correct sd/variance formula, their \(n\); numerical use of different correct sd/var formula, their \(n\); correct final answer |
| [OR1] \(\sum(x - 100)^2 - 2 \times 4.8 \times \sum(x - 100) + 15 \times 4.8^2 = 153.6\) (154) | numerical 1st and 2nd terms; numerical 3rd term; correct final answer | |
| [OR2] \(\sum x^2 = \sum(x - 100)^2 + 200 \times \sum x - 150000\); \(\sum(x - 104.8)^2 = \sum x^2 - 209.6\sum x + 15 \times 104.8^2 = 153.6\) (154) | numerical use of a correct expansion to find \(\sum x^2\); numerical use of a correct expansion for \(\sum(x - 104.8)^2\); correct final answer |
(i) $72/n + 100 = 104.8$ or $72 + 100n = 104.8n$; $n = 15$ | M1 A1 [2] | $72/n$ or $100n$ and $104.8n$ seen or implied correct answer
(ii) $sd^2 = 499.2/15 - (72/15)^2$ (= 10.24) or $sd^2 = \sum(x - 104.8)^2/15 - (\sum(x - 104.8)/15)^2$ or $\sum(x - 104.8)^2 = 153.6$ (154) | M1 A1 [3] | numerical use of a correct sd/variance formula, their $n$; numerical use of different correct sd/var formula, their $n$; correct final answer
[OR1] $\sum(x - 100)^2 - 2 \times 4.8 \times \sum(x - 100) + 15 \times 4.8^2 = 153.6$ (154) | | numerical 1st and 2nd terms; numerical 3rd term; correct final answer
[OR2] $\sum x^2 = \sum(x - 100)^2 + 200 \times \sum x - 150000$; $\sum(x - 104.8)^2 = \sum x^2 - 209.6\sum x + 15 \times 104.8^2 = 153.6$ (154) | | numerical use of a correct expansion to find $\sum x^2$; numerical use of a correct expansion for $\sum(x - 104.8)^2$; correct final answer2 The heights, $x \mathrm {~cm}$, of a group of young children are summarised by
$$\Sigma ( x - 100 ) = 72 , \quad \Sigma ( x - 100 ) ^ { 2 } = 499.2 .$$
The mean height is 104.8 cm .\\
(i) Find the number of children in the group.\\
(ii) Find $\Sigma ( x - 104.8 ) ^ { 2 }$.
\hfill \mbox{\textit{CAIE S1 2012 Q2 [5]}}