| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2023 |
| Session | October |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Find median and quartiles from stem-and-leaf diagram |
| Difficulty | Easy -1.2 This is a straightforward S1 question requiring standard procedures: reading values from a stem-and-leaf diagram to find median and quartiles (using n/2 and n/4 positions), applying the given outlier formula, drawing a box plot, and making contextual comparisons. All techniques are routine recall with no problem-solving or novel insight required, making it easier than average. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| Weight (kg) | Totals | |
| 1 | 6 | (1) |
| 2 | 36 | (2) |
| 3 | 246 | (3) |
| 4 | 2556678 | (7) |
| 5 | 34777899 | (8) |
| 6 | 022338 | (7) |
| 7 | 28 | (2) |
| 8 | 26 | (2) |
| 9 | 4 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(Q_2 = 57\) | B1 | Cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(Q_1 = 45\), \(Q_3 = 63\) | B1 B1 | Cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(63 + 1.5(63-45) = 90\) or \(45 - 1.5(63-45) = 18\) | M1 | For use of either \(Q_3 + 1.5(Q_3 - Q_1)\) or \(Q_1 - 1.5(Q_3 - Q_1)\) ft part (a) |
| \(= 90\) or \(= 18\) | A1ft | For either 90 or 18 ft part (a) |
| 16 and 94 are outliers | A1* | For identifying both outliers with no incorrect/missing working (can ft part (a)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Boxplot drawn with 2 whiskers | M1 | A boxplot drawn with 2 whiskers |
| \(Q_1\), \(Q_2\) and \(Q_3\) plotted correctly | A1ft | For \(Q_1\), \(Q_2\), \(Q_3\) plotted correctly ft part (a) |
| Whiskers drawn correctly | A1ft | Whiskers drawn at 18 and 90 ft part (b) or 23 and 86 |
| Outliers marked at 16 and 94 | A1 | Outliers marked at 16 and 94 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Median/\(Q_2\) for February is less/lower than median/\(Q_2\) for December | B1ft | Correct comparison of medians ft boxplot; must mention word median/\(Q_2\); no figures required but if quoted must be correct |
| IQR/range for February is less/lower than December | B1ft | Correct comparison of range/IQR ft boxplot; must mention IQR or range; no figures required but if quoted must be correct |
| Correct interpretation of either average or spread e.g. on average February weigh less than December; weights of February are less varied than December; they weighed more later in the year; most of the distribution has shifted right | B1ft | Correct interpretation of either average or spread ft boxplot. NB Ignore any reference to skew or outliers |
# Question 2:
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $Q_2 = 57$ | B1 | Cao |
## Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $Q_1 = 45$, $Q_3 = 63$ | B1 B1 | Cao |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $63 + 1.5(63-45) = 90$ or $45 - 1.5(63-45) = 18$ | M1 | For use of either $Q_3 + 1.5(Q_3 - Q_1)$ or $Q_1 - 1.5(Q_3 - Q_1)$ ft part (a) |
| $= 90$ or $= 18$ | A1ft | For either 90 or 18 ft part (a) |
| 16 and 94 are outliers | A1* | For identifying both outliers with no incorrect/missing working (can ft part (a)) |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Boxplot drawn with 2 whiskers | M1 | A boxplot drawn with 2 whiskers |
| $Q_1$, $Q_2$ and $Q_3$ plotted correctly | A1ft | For $Q_1$, $Q_2$, $Q_3$ plotted correctly ft part (a) |
| Whiskers drawn correctly | A1ft | Whiskers drawn at 18 and 90 ft part (b) **or** 23 and 86 |
| Outliers marked at 16 and 94 | A1 | Outliers marked at 16 and 94 |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| Median/$Q_2$ for February is less/lower than median/$Q_2$ for December | B1ft | Correct comparison of medians ft boxplot; must mention word median/$Q_2$; no figures required but if quoted must be correct |
| IQR/range for February is less/lower than December | B1ft | Correct comparison of range/IQR ft boxplot; must mention IQR or range; no figures required but if quoted must be correct |
| Correct interpretation of either average or spread e.g. on average February weigh less than December; weights of February are less varied than December; they weighed more later in the year; most of the distribution has shifted right | B1ft | Correct interpretation of either average or spread ft boxplot. **NB** Ignore any reference to skew or outliers |
---\begin{enumerate}
\item The weights, to the nearest kilogram, of a sample of 33 red kangaroos taken in December are summarised in the stem and leaf diagram below.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
\multicolumn{2}{|c|}{Weight (kg)} & Totals \\
\hline
1 & 6 & (1) \\
\hline
2 & 36 & (2) \\
\hline
3 & 246 & (3) \\
\hline
4 & 2556678 & (7) \\
\hline
5 & 34777899 & (8) \\
\hline
6 & 022338 & (7) \\
\hline
7 & 28 & (2) \\
\hline
8 & 26 & (2) \\
\hline
9 & 4 & (1) \\
\hline
\end{tabular}
\end{center}
Key: 3 | 2 represents 32 kg\\
(a) Find\\
(i) the value of the median\\
(ii) the value of $Q _ { 1 }$ and the value of $Q _ { 3 }$\\
for the weights of these red kangaroos.
For these data an outlier is defined as a value that is\\
greater than $Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$\\
or smaller than $Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$\\
(b) Show that there are 2 outliers for these data.
Figure 1 on page 7 shows a box plot for the weights of the same 33 red kangaroos taken in February, earlier in the year.\\
(c) In the space on Figure 1, draw a box plot to represent the weights of these red kangaroos in December.\\
(d) Compare the distribution of the weights of red kangaroos taken in February with the distribution of the weights of red kangaroos taken in December of the same year. You should interpret your comparisons in the context of the question.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{f94b29e0-081f-45e8-99a7-ac835eec91e5-07_2267_51_307_36}
\end{center}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f94b29e0-081f-45e8-99a7-ac835eec91e5-07_766_1803_1777_132}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Turn over for a spare grid if you need to redraw your box plot.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f94b29e0-081f-45e8-99a7-ac835eec91e5-09_901_1833_1653_114}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\begin{verbatim}
(Total for Question 2 is 13 marks)
\end{verbatim}
\hfill \mbox{\textit{Edexcel S1 2023 Q2 [13]}}