| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | November |
| Marks | 6 |
| 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 formulas for variance/SD from coded data. Students need to recall that Var(x) = E[(x-a)²] - [E(x-a)]² and apply algebraic manipulation to find Σx². The question requires routine substitution into well-practiced formulas with no conceptual challenges or problem-solving insight. |
| Spec | 2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(133/n + 25 = 28.325\) | M1 | Equation involving 133, 25 and 28.325 |
| \(n = 40\) | A1 | Correct answer for \(n\) |
| \(3762/40 - 3.325^2 = 82.99\) | M1 | Using coded mean in variance formula |
| standard deviation \(= 9.11\) | A1 [4] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(82.99 = \Sigma x^2/40 - 28.325^2\) | M1 | Using uncoded material in variance formula |
| \(\Sigma x^2 = (82.99 + 28.325^2) \times 40\) | ||
| \(= 35412\ (35400)\) | A1 | Correct answer |
| OR | ||
| \(\Sigma(x-25)^2 = \Sigma x^2 - 50\Sigma x + 40 \times 25^2\) | M1 | Expanding and substituting for \(\Sigma x\) |
| \(\Sigma x^2 = 3762 + 50 \times 1133 + 25000\) | ||
| \(= 35412\) | A1 [2] | Correct answer |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $133/n + 25 = 28.325$ | M1 | Equation involving 133, 25 and 28.325 |
| $n = 40$ | A1 | Correct answer for $n$ |
| $3762/40 - 3.325^2 = 82.99$ | M1 | Using coded mean in variance formula |
| standard deviation $= 9.11$ | A1 [4] | Correct answer |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $82.99 = \Sigma x^2/40 - 28.325^2$ | M1 | Using uncoded material in variance formula |
| $\Sigma x^2 = (82.99 + 28.325^2) \times 40$ | | |
| $= 35412\ (35400)$ | A1 | Correct answer |
| **OR** | | |
| $\Sigma(x-25)^2 = \Sigma x^2 - 50\Sigma x + 40 \times 25^2$ | M1 | Expanding and substituting for $\Sigma x$ |
| $\Sigma x^2 = 3762 + 50 \times 1133 + 25000$ | | |
| $= 35412$ | A1 [2] | Correct answer |
---2 The values, $x$, in a particular set of data are summarised by
$$\Sigma ( x - 25 ) = 133 , \quad \Sigma ( x - 25 ) ^ { 2 } = 3762 .$$
The mean, $\bar { x }$, is 28.325 .\\
(i) Find the standard deviation of $x$.\\
(ii) Find $\Sigma x ^ { 2 }$.
\hfill \mbox{\textit{CAIE S1 2011 Q2 [6]}}