| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw histogram then estimate mean/standard deviation |
| Difficulty | Moderate -0.8 This is a routine S1 statistics question requiring standard procedures: calculating frequency densities for unequal class widths to draw a histogram, then using midpoints to estimate mean and standard deviation from grouped data. These are textbook techniques with no problem-solving or conceptual challenge beyond careful arithmetic. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Height (m) | \(21 - 40\) | \(41 - 45\) | \(46 - 50\) | \(51 - 60\) | \(61 - 80\) |
| Frequency | 18 | 15 | 21 | 52 | 28 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| fd: 0.9, 3.4, 4.2, 5.2, 1.4 | M1 | Attempt at scaled freq \([f/(\text{attempt at cw})]\) |
| Correct heights on diagram, scale no less than 1cm to 1 unit | A1 | |
| Correct bar widths visually no gaps | B1 | |
| Labels (ht/metres and fd or freq per 20m etc.) and end points at 20.5 etc.; condone 2 end point errors; scale no less than 1cm to 5m for 20, 30… unless clearly accurate, linear scale between 20.5 and 80 | B1 (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((30.5\times 18+43\times 15+48\times 21+55.5\times 52+70.5\times 28)/134\) | M1 | Attempt at unsimplified, mid points (at least 4 within 0.5) |
| \(= \frac{7062}{134} = 52.701\) | M1, A1 | Attempt at \(\Sigma fx\) their mid points \(\div 134\); correct mean rounding to 53 |
| \(\text{Var} = (30.5^2\times 18+43^2\times 15+48^2\times 21+55.5^2\times 52+70.5^2\times 28)/134 - 52.701^2 = 149.496\) | M1 | Attempts at \(\Sigma fx^2\) their mid points \(\div\) their \(\Sigma f - \text{mean}^2\) |
| \(\text{sd} = 12.2\) | A1 (5) | Correct answer, nfww |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| fd: 0.9, 3.4, 4.2, 5.2, 1.4 | M1 | Attempt at scaled freq $[f/(\text{attempt at cw})]$ |
| Correct heights on diagram, scale no less than 1cm to 1 unit | A1 | |
| Correct bar widths visually no gaps | B1 | |
| Labels (ht/metres and fd or freq per 20m etc.) and end points at 20.5 etc.; condone 2 end point errors; scale no less than 1cm to 5m for 20, 30… unless clearly accurate, linear scale between 20.5 and 80 | B1 (4) | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(30.5\times 18+43\times 15+48\times 21+55.5\times 52+70.5\times 28)/134$ | M1 | Attempt at unsimplified, mid points (at least 4 within 0.5) |
| $= \frac{7062}{134} = 52.701$ | M1, A1 | Attempt at $\Sigma fx$ their mid points $\div 134$; correct mean rounding to 53 |
| $\text{Var} = (30.5^2\times 18+43^2\times 15+48^2\times 21+55.5^2\times 52+70.5^2\times 28)/134 - 52.701^2 = 149.496$ | M1 | Attempts at $\Sigma fx^2$ their mid points $\div$ their $\Sigma f - \text{mean}^2$ |
| $\text{sd} = 12.2$ | A1 (5) | Correct answer, nfww |
---6 The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
Height (m) & $21 - 40$ & $41 - 45$ & $46 - 50$ & $51 - 60$ & $61 - 80$ \\
\hline
Frequency & 18 & 15 & 21 & 52 & 28 \\
\hline
\end{tabular}
\end{center}
(i) Draw a histogram on graph paper to illustrate the data.\\
(ii) Calculate estimates of the mean and standard deviation of these heights.
\hfill \mbox{\textit{CAIE S1 2015 Q6 [9]}}