| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Interpret or analyse given back-to-back stem-and-leaf |
| Difficulty | Moderate -0.8 This is a straightforward stem-and-leaf interpretation question requiring basic statistical skills: reading the diagram correctly, finding median/quartiles by counting positions (n=27 makes this simple), calculating IQR, drawing box plots, and making a standard comment about mean vs median. All techniques are routine recall with no problem-solving or novel insight required. |
| Spec | 2.02f Measures of average and spread |
| Company \(A\) | Company \(B\) | ||||||||||||
| \multirow{6}{*}{9} | 4 | 1 | 1 | 0 | 25 | 4 | 4 | 5 | 6 | 6 | 7 | ||
| 7 | 2 | 1 | 0 | 26 | 0 | 1 | 3 | 5 | 5 | 7 | 9 | 9 | |
| 4 | 2 | 1 | 0 | 27 | 1 | 3 | 4 | 6 | 6 | 8 | 8 | ||
| 5 | 4 | 2 | 0 | 28 | 0 | 1 | 2 | 2 | 2 | ||||
| 9 | 8 | 5 | 29 | ||||||||||
| 1 | 30 | 9 | |||||||||||
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Median \(= 2710\) | B1 | Must be identified, condone Q2. Ignore units throughout |
| \(2840 - 2610\) | M1 | \(2820 \leq UQ \leq 2850 - 2600 \leq LQ \leq 2620\) |
| \(230\) | A1 | www. If M0 scored SC B1 for 230 www. If key ignored consistently: B0 Median \(= 271\), SC M1 \(282 \leq UQ \leq 285 - 260 \leq LQ \leq 262\), SC A1 23 |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Box-and-whisker plot on provided grid | B1 | All 5 key values for \(B\) plotted accurately in standard format using a linear scale with 3 identified values. Labelled \(B\). Scale at least 1 cm = \$100 |
| B: 2540, 2600, 2690, 2780, 3090 | B1FT | All 5 key values for \(A\), FT from (a), plotted accurately in standard format using a linear scale with 3 identified values. Labelled \(A\). Scale at least 1cm = \$100 |
| A: 2500, 2610, 2710, 2840, 3010 | ||
| B1 | Whiskers not through box for both, not drawn at corners of boxes, single linear scale for the diagram and labelled 'salaries' (oe) and \$ | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Mean less appropriate than median because of extreme value for company \(B\) [at \\(3090]. No extreme value in company B. No, \\)3090 is an anomaly. | B1 | Must refer to company B, may be implied by appropriate use of \\(3090. Must include an indication that the mean is not appropriate. No contradictory statements, e.g. acceptable comment with 'but mean could be used for company A'. Condone reference to \\)309 |
| Total | 1 |
## Question 3:
**Part (a)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Median $= 2710$ | B1 | Must be identified, condone Q2. Ignore units throughout |
| $2840 - 2610$ | M1 | $2820 \leq UQ \leq 2850 - 2600 \leq LQ \leq 2620$ |
| $230$ | A1 | www. If M0 scored **SC B1** for 230 www. If key ignored consistently: B0 Median $= 271$, **SC M1** $282 \leq UQ \leq 285 - 260 \leq LQ \leq 262$, **SC A1** 23 |
| **Total** | **3** | |
**Part (b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Box-and-whisker plot on provided grid | B1 | All 5 key values for $B$ plotted accurately in standard format using a linear scale with 3 identified values. Labelled $B$. Scale at least 1 cm = \$100 |
| B: 2540, 2600, 2690, 2780, 3090 | B1FT | All 5 key values for $A$, FT from (a), plotted accurately in standard format using a linear scale with 3 identified values. Labelled $A$. Scale at least 1cm = \$100 |
| A: 2500, 2610, 2710, 2840, 3010 | | |
| | B1 | Whiskers not through box for both, not drawn at corners of boxes, single linear scale for the diagram and labelled 'salaries' (oe) and \$ |
| **Total** | **3** | |
**Part (c)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Mean less appropriate than median because of extreme value for company $B$ [at \$3090]. No extreme value in company B. No, \$3090 is an anomaly. | B1 | Must refer to company B, may be implied by appropriate use of \$3090. Must include an indication that the mean is not appropriate. No contradictory statements, e.g. acceptable comment with 'but mean could be used for company A'. Condone reference to \$309 |
| **Total** | **1** | |
---3 The following back-to-back stem-and-leaf diagram represents the monthly salaries, in dollars, of 27 employees at each of two companies, $A$ and $B$.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{5}{|c|}{Company $A$} & & \multicolumn{8}{|c|}{Company $B$} \\
\hline
\multirow{6}{*}{9} & 4 & 1 & 1 & 0 & 25 & 4 & 4 & 5 & 6 & 6 & 7 & \multicolumn{2}{|c|}{} \\
\hline
& 7 & 2 & 1 & 0 & 26 & 0 & 1 & 3 & 5 & 5 & 7 & 9 & 9 \\
\hline
& 4 & 2 & 1 & 0 & 27 & 1 & 3 & 4 & 6 & 6 & 8 & 8 & \\
\hline
& 5 & 4 & 2 & 0 & 28 & 0 & 1 & 2 & 2 & 2 & & & \\
\hline
& & 9 & 8 & 5 & 29 & & & & & & & & \\
\hline
& & & & 1 & 30 & 9 & & & & & & & \\
\hline
\end{tabular}
\end{center}
Key: 1 |27| 6 means $\$ 2710$ for company $A$ and $\$ 2760$ for company $B$
\begin{enumerate}[label=(\alph*)]
\item Find the median and the interquartile range of the monthly salaries of employees in company $A$.\\
The lower quartile, median and upper quartile for company $B$ are $\$ 2600 , \$ 2690$ and $\$ 2780$ respectively.
\item Draw two box-and-whisker plots in a single diagram to represent the information for the salaries of employees at companies $A$ and $B$.\\
\includegraphics[max width=\textwidth, alt={}, center]{f2666d82-4711-499a-98c0-3421e4c228fb-07_810_1406_573_411}
\item Comment on whether the mean would be a more appropriate measure than the median for comparing the given information for the two companies.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2023 Q3 [7]}}