Question 4 - A-Level Maths

CAIE S1 2024 June — Question 4 7 marks

Exam Board	CAIE
Module	S1 (Statistics 1)
Year	2024
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 question requires reading a back-to-back stem-and-leaf diagram, finding median and IQR (standard procedure for n=19), drawing box plots, and making a basic comment about mean vs median. All techniques are routine for S1 with no problem-solving or novel insight required, making it easier than average A-level maths questions.
Spec	2.02f Measures of average and spread 2.02g Calculate mean and standard deviation

4 The back-to-back stem-and-leaf diagram shows the annual salaries of 19 employees at each of two companies, Petral and Ravon.

Petral						Ravon
\multirow{7}{*}{99}		3	0	0	30	2	6
	8	2	2	1	31	1	5
	5	5	4	0	32	0	0	2
		7	5	3	33	0	4	8	9
			1	0	34	1	1	3	4	6
					35	3
				8	36	7	9

Key: 2 | 31 | 5 means \$31 200 for a Petral employee and \$31500 for a Ravon employee.

Find the median and the interquartile range of the salaries of the Petral employees.
The median salary of the Ravon employees is $\$ 33800$, the lower quartile is $\$ 32000$ and the upper quartile is $\$ 34400$.
Represent the data shown in the back-to-back stem-and-leaf diagram by a pair of box-and-whisker plots in a single diagram. \includegraphics[max width=\textwidth, alt={}, center]{f979a442-da05-410b-84dc-3da3286514a0-07_707_1395_477_335}
Comment on whether the mean or the median would be a better representation of the data for the employees at Petral.

Show mark scheme Show mark scheme source

Question 4(a):

Answer	Marks	Guidance
Median $= 32000$	B1	Clearly identified, e.g. Q2, med. Accept 32k.
$[\text{UQ} = 33500,\ \text{LQ} = 31200]$; $[\text{IQR} =]\ 33500 - 31200$	M1	$33300 \leqslant \text{UQ} \leqslant 33700 - 31100 \leqslant \text{LQ} \leqslant 31200$. Implied if both quartile values are stated and an appropriate IQR is calculated accurately.
$= 2300$	A1	WWW. Ignore \$ signs. If M0 scored, SC B1 for 2300 WWW. If key ignored consistently: B0 Median $= 320$; SC M1 $325 \leqslant \text{UQ} \leqslant 335 - 311 \leqslant \text{LQ} \leqslant 312$; SC A1 23.

Total: 3 marks

Question 4(b):

Box-and-whisker plot on provided grid.

Answer	Marks	Guidance
R	P
Min	30 200	30 000
LQ	32 000	31 200
Med	33 800	32 000
UQ	34 400	33 500
Max	36 900	36 800
B1	All five key values for $R$ plotted accurately in standard format using a linear scale with at least three linear values. Labelled $R$. Condone whiskers through box or at corners of boxes or extending $\frac{1}{2}$ square beyond limit. Scale no less than 1 cm = \$1000. Daylight rule applied to vertical lines of box.
B1FT	All five key values for $P$, FT from (a), plotted accurately in standard format using a linear scale with at least three linear values. Labelled $P$. Condone whiskers through box or at corners of boxes or extending $\frac{1}{2}$ square beyond limit. Scale no less than 1 cm = \$1000. Daylight rule applied to vertical lines of box.
B1	Whiskers not through box (condone $\frac{1}{2}$ square in box) for either, not drawn at corners of boxes. Single linear scale for the diagram and labelled 'salaries' (OE) and \$. If only one plot attempted, SC B1 for meeting all the requirements above.

Total: 3 marks

Question 4(c):

Answer	Marks	Guidance
Median because there is an extreme value (\$36 800)	B1	Do not accept 'values'. Must identify median and reference either the extreme value (anomaly, outlier, 36 800) or the skew in context (e.g. concentrated in lower values, positive skew).

Total: 1 mark

## Question 4(a):

Median $= 32000$ | B1 | Clearly identified, e.g. Q2, med. Accept 32k.

$[\text{UQ} = 33500,\ \text{LQ} = 31200]$; $[\text{IQR} =]\ 33500 - 31200$ | M1 | $33300 \leqslant \text{UQ} \leqslant 33700 - 31100 \leqslant \text{LQ} \leqslant 31200$. Implied if both quartile values are stated and an appropriate IQR is calculated accurately.

$= 2300$ | A1 | WWW. Ignore \$ signs. If M0 scored, **SC B1** for 2300 WWW. If key ignored consistently: B0 Median $= 320$; **SC M1** $325 \leqslant \text{UQ} \leqslant 335 - 311 \leqslant \text{LQ} \leqslant 312$; **SC A1** 23.

**Total: 3 marks**

---

## Question 4(b):

Box-and-whisker plot on provided grid.

| | R | P |
|---|---|---|
| Min | 30 200 | 30 000 |
| LQ | 32 000 | 31 200 |
| Med | 33 800 | 32 000 |
| UQ | 34 400 | 33 500 |
| Max | 36 900 | 36 800 |

| B1 | All five key values for $R$ plotted accurately in standard format using a linear scale with at least three linear values. Labelled $R$. Condone whiskers through box or at corners of boxes or extending $\frac{1}{2}$ square beyond limit. Scale no less than 1 cm = \$1000. Daylight rule applied to vertical lines of box.

| B1FT | All five key values for $P$, FT from **(a)**, plotted accurately in standard format using a linear scale with at least three linear values. Labelled $P$. Condone whiskers through box or at corners of boxes or extending $\frac{1}{2}$ square beyond limit. Scale no less than 1 cm = \$1000. Daylight rule applied to vertical lines of box.

| B1 | Whiskers not through box (condone $\frac{1}{2}$ square in box) for either, not drawn at corners of boxes. Single linear scale for the diagram and labelled 'salaries' (OE) and \$. If only one plot attempted, **SC B1** for meeting all the requirements above.

**Total: 3 marks**

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## Question 4(c):

Median because there is an extreme value (\$36 800) | B1 | Do not accept 'values'. Must identify median and reference either the extreme value (anomaly, outlier, 36 800) or the skew in context (e.g. concentrated in lower values, positive skew).

**Total: 1 mark**

---

Show LaTeX source

4 The back-to-back stem-and-leaf diagram shows the annual salaries of 19 employees at each of two companies, Petral and Ravon.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{5}{|c|}{Petral} &  & \multicolumn{5}{|c|}{Ravon} \\
\hline
\multirow{7}{*}{99} &  & 3 & 0 & 0 & 30 & 2 & 6 &  &  &  \\
\hline
 & 8 & 2 & 2 & 1 & 31 & 1 & 5 &  &  &  \\
\hline
 & 5 & 5 & 4 & 0 & 32 & 0 & 0 & 2 &  &  \\
\hline
 &  & 7 & 5 & 3 & 33 & 0 & 4 & 8 & 9 &  \\
\hline
 &  &  & 1 & 0 & 34 & 1 & 1 & 3 & 4 & 6 \\
\hline
 &  &  &  &  & 35 & 3 &  &  &  &  \\
\hline
 &  &  &  & 8 & 36 & 7 & 9 &  &  &  \\
\hline
\end{tabular}
\end{center}

Key: 2 | 31 | 5 means \$31 200 for a Petral employee and \$31500 for a Ravon employee.
\begin{enumerate}[label=(\alph*)]
\item Find the median and the interquartile range of the salaries of the Petral employees.\\

The median salary of the Ravon employees is $\$ 33800$, the lower quartile is $\$ 32000$ and the upper quartile is $\$ 34400$.
\item Represent the data shown in the back-to-back stem-and-leaf diagram by a pair of box-and-whisker plots in a single diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{f979a442-da05-410b-84dc-3da3286514a0-07_707_1395_477_335}
\item Comment on whether the mean or the median would be a better representation of the data for the employees at Petral.
\end{enumerate}

\hfill \mbox{\textit{CAIE S1 2024 Q4 [7]}}

This paper (7 questions)

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Q1 5 Q2 5 Q3 6 Q4 7 Q5 7 Q6 10 Q7 10

Answer	Marks	Guidance
Median \(= 32000\)	B1	Clearly identified, e.g. Q2, med. Accept 32k.
\([\text{UQ} = 33500,\ \text{LQ} = 31200]\); \([\text{IQR} =]\ 33500 - 31200\)	M1	\(33300 \leqslant \text{UQ} \leqslant 33700 - 31100 \leqslant \text{LQ} \leqslant 31200\). Implied if both quartile values are stated and an appropriate IQR is calculated accurately.
\(= 2300\)	A1	WWW. Ignore \\( signs. If M0 scored, SC B1 for 2300 WWW. If key ignored consistently: B0 Median \)= 320\(; SC M1 \)325 \leqslant \text{UQ} \leqslant 335 - 311 \leqslant \text{LQ} \leqslant 312$; SC A1 23.