| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2019 |
| Session | March |
| 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 standard formulas relating coded sums to variance and mean. Part (i) uses the variance formula σ² = Σ(x-c)²/n - [Σ(x-c)/n]² to find Σ(x-c), and part (ii) applies the basic coding relationship mean = c + Σ(x-c)/n. Both parts require only direct substitution into well-known formulas with minimal algebraic manipulation, making this easier than average for A-level. |
| Spec | 2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sigma^2 = \frac{\sum(x-c)^2}{n} - \left(\frac{\sum(x-c)}{n}\right)^2\) and \(3.2^2 = \frac{3099.2}{40} - \left(\frac{\sum(x-c)}{40}\right)^2\) | M1 | Use correct formula with values substituted |
| \(\left(\frac{\sum(x-c)}{40}\right)^2 = 67.24\); \(\sum(x-c) = 40 \times \sqrt{67.24}\) | M1 | Rearrange to make \(\left(\frac{\sum(x-c)}{40}\right)^2\) the subject, unsimplified |
| \(= 328\) | A1 | Exact value, cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sum x - 40c = \text{their (i)}\); Mean \(= \frac{\text{their(i)}}{40} + 50 = 58.2\) | B1FT | FT their (i) |
## Question 2:
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sigma^2 = \frac{\sum(x-c)^2}{n} - \left(\frac{\sum(x-c)}{n}\right)^2$ and $3.2^2 = \frac{3099.2}{40} - \left(\frac{\sum(x-c)}{40}\right)^2$ | M1 | Use correct formula with values substituted |
| $\left(\frac{\sum(x-c)}{40}\right)^2 = 67.24$; $\sum(x-c) = 40 \times \sqrt{67.24}$ | M1 | Rearrange to make $\left(\frac{\sum(x-c)}{40}\right)^2$ the subject, unsimplified |
| $= 328$ | A1 | Exact value, cao |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sum x - 40c = \text{their (i)}$; Mean $= \frac{\text{their(i)}}{40} + 50 = 58.2$ | B1FT | FT their (i) |
---2 For 40 values of the variable $x$, it is given that $\Sigma ( x - c ) ^ { 2 } = 3099.2$, where $c$ is a constant. The standard deviation of these values of $x$ is 3.2 .\\
(i) Find the value of $\Sigma ( x - c )$.\\
(ii) Given that $c = 50$, find the mean of these values of $x$.\\
\hfill \mbox{\textit{CAIE S1 2019 Q2 [4]}}