| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Adding data values |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas for transforming sums and updating statistics when adding a data value. Part (i) uses basic algebraic manipulation of summations (Σ(x-50) = Σx - 50n), and part (ii) applies the standard deviation formula with one additional value. Both parts are routine calculations requiring no problem-solving insight, making this easier than average but not trivial due to the multi-step nature. |
| Spec | 2.02g Calculate mean and standard deviation5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Sigma(x - 50) = 824 - 16 \times 50 = 24\) | B1 | Correct answer |
| \(\frac{\Sigma(x-50)^2}{16} - \left(\frac{\Sigma(x-50)}{16}\right)^2 = 6.5^2\) | M1 | Consistent substituting in the correct coded variance formula OR valid method for \(\Sigma x^2\) then expanding \(\Sigma(x-50)^2\), 3 terms at least 2 correct |
| \(\Sigma(x - 50)^2 = 712\) | A1 [3] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| new mean \(= 896/17 (= 52.7)\) | B1 | Correct answer |
| \(\text{new var} = \frac{712 + 22^2}{17} - \left(\frac{24 + (72-50)}{17}\right)^2\) | M1 | Using the correct coded variance formula with \(n = 17\) and new coded mean² OR their \((\Sigma x^2 + 72^2)/17\) – their new mean² |
| new sd \(= 7.94\) | A1 [3] | Rounding to correct answer, accept 7.95 or 7.98 or 7.91 |
**(i)**
$\Sigma(x - 50) = 824 - 16 \times 50 = 24$ | B1 | Correct answer
$\frac{\Sigma(x-50)^2}{16} - \left(\frac{\Sigma(x-50)}{16}\right)^2 = 6.5^2$ | M1 | Consistent substituting in the correct coded variance formula OR valid method for $\Sigma x^2$ then expanding $\Sigma(x-50)^2$, 3 terms at least 2 correct
$\Sigma(x - 50)^2 = 712$ | A1 [3] | Correct answer
**(ii)**
new mean $= 896/17 (= 52.7)$ | B1 | Correct answer
$\text{new var} = \frac{712 + 22^2}{17} - \left(\frac{24 + (72-50)}{17}\right)^2$ | M1 | Using the correct coded variance formula with $n = 17$ and new coded mean² OR their $(\Sigma x^2 + 72^2)/17$ – their new mean²
new sd $= 7.94$ | A1 [3] | Rounding to correct answer, accept 7.95 or 7.98 or 7.912 Esme noted the test marks, $x$, of 16 people in a class. She found that $\Sigma x = 824$ and that the standard deviation of $x$ was 6.5.\\
(i) Calculate $\Sigma ( x - 50 )$ and $\Sigma ( x - 50 ) ^ { 2 }$.\\
(ii) One person did the test later and her mark was 72. Calculate the new mean and standard deviation of the marks of all 17 people.
\hfill \mbox{\textit{CAIE S1 2010 Q2 [6]}}