| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2023 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw cumulative frequency graph from cumulative frequency table |
| Difficulty | Easy -1.8 This is a routine statistics question requiring only standard procedures: plotting given cumulative frequency points on provided axes, reading a value from the graph, and calculating mean/standard deviation from grouped data using standard formulas. All techniques are direct applications with no problem-solving or conceptual challenges. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Weight \(( x \mathrm {~kg} )\) | \(x \leqslant 40\) | \(x \leqslant 60\) | \(x \leqslant 65\) | \(x \leqslant 70\) | \(x \leqslant 85\) | \(x \leqslant 100\) |
| Cumulative frequency | 0 | 14 | 38 | 60 | 106 | 120 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Cumulative frequency curve plotted] | M1 | At least 3 points plotted accurately at class upper end points \((40,0)\), \((60,14)\), \((65,38)\), \((70,60)\), \((85,106)\), \((100,120)\). Linear cumulative frequency scale \(0 \leq cf \leq 120\) and linear weight scale \(40 \leq \text{weight(kg)} \leq 100\) with at least 3 values identified on each axis. Condone scale reversed |
| A1 | All points plotted correctly, curve drawn (within tolerance) and joined to \((40,0)\). Axes labelled cumulative frequency (cf), weight (w) and kg (kilograms) – or a suitable title | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([120 \times 0.65 =]\ 78\) seen | M1 | May be implied by use on graph |
| \(76\) [kg] | A1 | \(75 <\) hours \(< 79\). Indication of use of graph required |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Frequencies: \([0]\ 14\ 24\ 22\ 46\ 14\) | B1 | At least 5 correct frequencies seen (condone omission of 0) |
| Midpoints: \(20\ 50\ 62.5\ 67.5\ 77.5\ 92.5\) | B1 | At least 5 correct midpoints seen (condone omission of 20) |
| \(\text{Mean} = \frac{0\times20+14\times50+24\times62.5+22\times67.5+46\times77.5+14\times92.5}{120}\) \(= \frac{[0]+700+1500+1485+3565+12950}{120}\left[=\frac{8545}{120}\right]\) | M1 | Correct formula for mean using *their* midpoints and *their* frequencies, implied by \(\frac{8545}{120}\) if correct midpoints & frequencies seen. May be gained in variance calculation. If midpoints not clearly identified, condone midpoints \(\pm 0.5\) |
| \(= 71.2\) | A1 | Accept \(\frac{1709}{24}\), \(71\frac{5}{24}\) or \(71.208333\) to at least 3SF. If M0 scored, SC B1 for \(\frac{1709}{24}\), \(71\frac{5}{24}\) or \(71.208333\) to at least 3SF www |
| \(\text{Variance} = \frac{0\times20^2+14\times50^2+24\times62.5^2+22\times67.5^2+46\times77.5^2+14\times92.5^2}{120} - 71.2^2\) \(= \frac{[0]+35000+93750+100237.5+276287.5+119787.5}{120} - \left(\frac{8545}{120}\right)^2\) \([= 138.23]\) | M1 | Correct formula for variance using *their* midpoints, *their* frequencies and *their* mean. Implied by \(\frac{625062.5}{120} - \left(\frac{8545}{120}\right)^2\) if correct midpoints & frequencies seen |
| Standard deviation \(= 11.8\) | A1 | \(11.757016\) to at least 3SF |
| 6 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Cumulative frequency curve plotted] | M1 | At least 3 points plotted accurately at class upper end points $(40,0)$, $(60,14)$, $(65,38)$, $(70,60)$, $(85,106)$, $(100,120)$. Linear cumulative frequency scale $0 \leq cf \leq 120$ and linear weight scale $40 \leq \text{weight(kg)} \leq 100$ with at least 3 values identified on each axis. Condone scale reversed |
| | A1 | All points plotted correctly, curve drawn (within tolerance) and joined to $(40,0)$. Axes labelled cumulative frequency (cf), weight (w) and kg (kilograms) – or a suitable title |
| | **2** | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[120 \times 0.65 =]\ 78$ seen | M1 | May be implied by use on graph |
| $76$ [kg] | A1 | $75 <$ hours $< 79$. Indication of use of graph required |
| | **2** | |
## Question 4(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Frequencies: $[0]\ 14\ 24\ 22\ 46\ 14$ | B1 | At least 5 correct frequencies seen (condone omission of 0) |
| Midpoints: $20\ 50\ 62.5\ 67.5\ 77.5\ 92.5$ | B1 | At least 5 correct midpoints seen (condone omission of 20) |
| $\text{Mean} = \frac{0\times20+14\times50+24\times62.5+22\times67.5+46\times77.5+14\times92.5}{120}$ $= \frac{[0]+700+1500+1485+3565+12950}{120}\left[=\frac{8545}{120}\right]$ | M1 | Correct formula for mean using *their* midpoints and *their* frequencies, implied by $\frac{8545}{120}$ if correct midpoints & frequencies seen. May be gained in variance calculation. If midpoints not clearly identified, condone midpoints $\pm 0.5$ |
| $= 71.2$ | A1 | Accept $\frac{1709}{24}$, $71\frac{5}{24}$ or $71.208333$ to at least 3SF. If M0 scored, **SC B1** for $\frac{1709}{24}$, $71\frac{5}{24}$ or $71.208333$ to at least 3SF www |
| $\text{Variance} = \frac{0\times20^2+14\times50^2+24\times62.5^2+22\times67.5^2+46\times77.5^2+14\times92.5^2}{120} - 71.2^2$ $= \frac{[0]+35000+93750+100237.5+276287.5+119787.5}{120} - \left(\frac{8545}{120}\right)^2$ $[= 138.23]$ | M1 | Correct formula for variance using *their* midpoints, *their* frequencies and *their* mean. Implied by $\frac{625062.5}{120} - \left(\frac{8545}{120}\right)^2$ if correct midpoints & frequencies seen |
| Standard deviation $= 11.8$ | A1 | $11.757016$ to at least 3SF |
| | **6** | |4 The weights, $x \mathrm {~kg}$, of 120 students in a sports college are recorded. The results are summarised in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Weight $( x \mathrm {~kg} )$ & $x \leqslant 40$ & $x \leqslant 60$ & $x \leqslant 65$ & $x \leqslant 70$ & $x \leqslant 85$ & $x \leqslant 100$ \\
\hline
Cumulative frequency & 0 & 14 & 38 & 60 & 106 & 120 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Draw a cumulative frequency graph to represent this information.\\
\includegraphics[max width=\textwidth, alt={}, center]{82c36c11-878c-47d1-a07f-fbf8b2a22d97-06_1390_1389_660_418}
\item It is found that $35 \%$ of the students weigh more than $W \mathrm {~kg}$.
Use your graph to estimate the value of $W$.
\item Calculate estimates for the mean and standard deviation of the weights of the 120 students. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2023 Q4 [10]}}