| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | October |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw box plot from raw data |
| Difficulty | Moderate -0.8 This is a routine S1 statistics question requiring standard procedures: reading a stem-and-leaf diagram, finding median and quartiles, applying the outlier rule, and drawing a box plot. All steps are algorithmic with no problem-solving insight needed. The final part (f) requires basic logical reasoning about ages but is straightforward given the constraint that the box plot doesn't change. |
| Spec | 2.02f Measures of average and spread2.02h Recognize outliers |
| Age | |||||||||||
| 1 | 3 | \(( 1 )\) | |||||||||
| 2 | 7 | 9 | \(( 2 )\) | ||||||||
| 3 | 1 | 2 | 8 | 8 | \(( 4 )\) | ||||||
| 4 | 5 | 5 | 6 | 7 | 8 | 8 | 9 | \(( 7 )\) | |||
| 5 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | 6 | 8 | \(( 10 )\) |
| 6 | 0 | 1 | 1 | 4 | 4 | 4 | 7 | \(( 7 )\) | |||
| 7 | 3 | 6 | \(( 2 )\) | ||||||||
| 8 | 7 | 8 | \(( 2 )\) | ||||||||
| Answer | Marks | Guidance |
|---|---|---|
| Median \(= \mathbf{53}\) | B1 | For 53 |
| Answer | Marks | Guidance |
|---|---|---|
| \(Q_1 = 45\), \(Q_3 = 61\) | M1 | Attempt at both and at least one correct; no need to be labelled |
| \(\text{IQR} = 61 - 45 = \mathbf{16}\) | A1cso | Both correct quartiles seen and \(61 - 45\) leading to 16 |
| Answer | Marks | Guidance |
|---|---|---|
| \(Q_1 - 1.5 \times \text{IQR} = 45 - 1.5 \times 16 [= 21]\) or \(Q_3 + 1.5 \times \text{IQR} = 61 + 1.5 \times 16 [= 85]\) | M1 | Attempting at least one of the limits; may be implied by 85 or 21 |
| Outliers are \(< 21\) or \(> 85\) | A1ft | Both outlier limits correct or correct ft using their quartiles |
| Three outliers at 13, 87 and 88 | A1 | Identifying the three outliers at 13, 87, 88 (dep on seeing both correct limits) |
| Answer | Marks | Guidance |
|---|---|---|
| Box plot with box only two whiskers one at each end; \(Q_1, Q_2, Q_3\) correctly drawn; upper whisker ending at 76 (or 85) and lower whisker ending at 27 (or 21); three outliers correctly shown | M1, A1ft, A1, A1 | Must be correctly paired; accuracy half a small square throughout |
| Answer | Marks | Guidance |
|---|---|---|
| e.g. Females are generally older than the men as median is higher (\(67 > 53\)) | B1 | Correct comment referring to ages with reference to a correctly named statistic; must include figures compared |
| Answer | Marks | Guidance |
|---|---|---|
| No change to box plot means one in each section so granddaughter [34–56] | B1 | For deducing granddaughter is at or below lower quartile but not below 34 |
| Answer | Marks | Guidance |
|---|---|---|
| Eldest daughter in range \([67 \sim 72]\) or Anja's age \([72 \sim 93]\) | M1 | Suitable range for eldest daughter or Anja above upper quartile |
| Since Anja 23 years older than eldest daughter, Anja in range [90–93] | A1 | For a range of [90~93] for Anja's age |
# Question 3:
## Part (a)
| Median $= \mathbf{53}$ | B1 | For 53 |
## Part (b)
| $Q_1 = 45$, $Q_3 = 61$ | M1 | Attempt at both and at least one correct; no need to be labelled |
| $\text{IQR} = 61 - 45 = \mathbf{16}$ | A1cso | Both correct quartiles seen **and** $61 - 45$ leading to 16 |
## Part (c)
| $Q_1 - 1.5 \times \text{IQR} = 45 - 1.5 \times 16 [= 21]$ or $Q_3 + 1.5 \times \text{IQR} = 61 + 1.5 \times 16 [= 85]$ | M1 | Attempting at least one of the limits; may be implied by 85 or 21 |
| Outliers are $< 21$ or $> 85$ | A1ft | Both outlier limits correct or correct ft using their quartiles |
| Three outliers at **13, 87 and 88** | A1 | Identifying the three outliers at 13, 87, 88 (dep on seeing both correct limits) |
## Part (d)
| Box plot with box only two whiskers one at each end; $Q_1, Q_2, Q_3$ correctly drawn; upper whisker ending at 76 (or 85) **and** lower whisker ending at 27 (or 21); three outliers correctly shown | M1, A1ft, A1, A1 | Must be correctly paired; accuracy half a small square throughout |
## Part (e)
| e.g. Females are generally older than the men as median is higher ($67 > 53$) | B1 | Correct comment referring to ages with reference to a correctly named statistic; must include figures compared |
## Part (f)(i)
| No change to box plot means one in each section so granddaughter **[34–56]** | B1 | For deducing granddaughter is at or below lower quartile but not below 34 |
## Part (f)(ii)
| Eldest daughter in range $[67 \sim 72]$ **or** Anja's age $[72 \sim 93]$ | M1 | Suitable range for eldest daughter or Anja above upper quartile |
| Since Anja 23 years older than eldest daughter, Anja in range **[90–93]** | A1 | For a range of [90~93] for Anja's age |
---\begin{enumerate}
\item The stem and leaf diagram shows the ages of the 35 male passengers on a cruise.
\end{enumerate}
\begin{center}
\begin{tabular}{ l | c l l l l l l l l l l l }
\multicolumn{12}{l}{Age} \\
1 & 3 & & & & & & & & & & $( 1 )$ \\
2 & 7 & 9 & & & & & & & & & $( 2 )$ \\
3 & 1 & 2 & 8 & 8 & & & & & & & $( 4 )$ \\
4 & 5 & 5 & 6 & 7 & 8 & 8 & 9 & & & & $( 7 )$ \\
5 & 2 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 8 & $( 10 )$ \\
6 & 0 & 1 & 1 & 4 & 4 & 4 & 7 & & & & $( 7 )$ \\
7 & 3 & 6 & & & & & & & & & $( 2 )$ \\
8 & 7 & 8 & & & & & & & & & $( 2 )$ \\
\end{tabular}
\end{center}
Key: 1 | 3 represents an age of 13 years\\
(a) Find the median age of the male passengers.\\
(b) Show that the interquartile range (IQR) of these ages is 16
An outlier is defined as a value that is more than\\
$1.5 \times$ IQR above the upper quartile\\
or\\
$1.5 \times$ IQR below the lower quartile\\
(c) Show that there are 3 outliers amongst these ages.\\
(d) On the grid in Figure 1 on page 9, draw a box plot for the ages of the male passengers on the cruise.
Figure 1 on page 9 also shows a box plot for the ages of the female passengers on the cruise.\\
(e) Comment on any difference in the distributions of ages of male and female passengers on the cruise.\\
State the values of any statistics you have used to support your comment.\\
(1)
Anja, along with her 2 daughters and a granddaughter, now join the cruise.\\
Anja's granddaughter is younger than both of Anja's daughters.\\
Anja had her 23rd birthday on the day her eldest daughter was born.\\
When their 4 ages are included with the other female passengers on the cruise, the box plot does not change.\\
(f) State, giving reasons, what you can say about\\
(i) the granddaughter's age\\
(ii) Anja's age.\\
(3)\\
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{29ac0c0b-f963-40a1-beba-7146bbb2d021-09_1025_1593_1541_182}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\hfill \mbox{\textit{Edexcel S1 2021 Q3 [14]}}