| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Removing data values |
| Difficulty | Moderate -0.8 This is a straightforward application of formulas for mean and standard deviation when removing a data value. Part (i) requires simple algebraic manipulation of the mean formula, while part (ii) involves using the standard deviation formula to find Σx² then recalculating. Both parts are routine calculations with no conceptual difficulty beyond knowing the formulas. |
| Spec | 2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| New mean \(= \dfrac{172.6 \times 28 - 161.8}{27} = 173\) | M1 | Mult by 28, subtract 161.8 and dividing by 27 or 28 |
| A1 | Correct answer — Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Original \(\Sigma x^2 = (4.58^2 + 172.6^2) \times 28\) | M1 | Subst in formula to find \(\Sigma x^2\) and attempt to make \(\Sigma x^2\) subject, with 2 terms both squared |
| \(= 834728.6\ (835000)\) | A1 | Correct answer |
| Remaining \(\Sigma x^2 = 834728.6 - 161.8^2 = 808549.36\) | M1 | Subtract \(161.8^2\) from their original \(\Sigma x^2\) |
| \(\text{sd of remaining} = \sqrt{\dfrac{808549.36}{27} - 173^2} = 4.16\) | A1 | Correct answer, accept 4.15 or 3.93 — Total: 4 |
## Question 4:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| New mean $= \dfrac{172.6 \times 28 - 161.8}{27} = 173$ | M1 | Mult by 28, subtract 161.8 and dividing by 27 or 28 |
| | A1 | Correct answer — **Total: 2** |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Original $\Sigma x^2 = (4.58^2 + 172.6^2) \times 28$ | M1 | Subst in formula to find $\Sigma x^2$ and attempt to make $\Sigma x^2$ subject, with 2 terms both squared |
| $= 834728.6\ (835000)$ | A1 | Correct answer |
| Remaining $\Sigma x^2 = 834728.6 - 161.8^2 = 808549.36$ | M1 | Subtract $161.8^2$ from their original $\Sigma x^2$ |
| $\text{sd of remaining} = \sqrt{\dfrac{808549.36}{27} - 173^2} = 4.16$ | A1 | Correct answer, accept 4.15 or 3.93 — **Total: 4** |
---4 The heights, $x \mathrm {~cm}$, of a group of 28 people were measured. The mean height was found to be 172.6 cm and the standard deviation was found to be 4.58 cm . A person whose height was 161.8 cm left the group.\\
(i) Find the mean height of the remaining group of 27 people.\\
(ii) Find $\Sigma x ^ { 2 }$ for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people.
\hfill \mbox{\textit{CAIE S1 2014 Q4 [6]}}