| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Outliers from box plot or summary statistics |
| Difficulty | Moderate -0.3 This is a standard S1 statistics question testing routine application of outlier definitions (1.5×IQR rule), box plot construction, and skewness interpretation. While multi-part with 5 sections, each part follows textbook procedures: calculating outlier boundaries, drawing box plots from given statistics, and using quartile changes to deduce data ranges. Part (e) requires slightly more thought about how adding data affects quartiles, but this is still a familiar S1 exercise requiring no novel insight—slightly easier than average overall. |
| Spec | 2.02h Recognize outliers |
| VIXV SIHIANI III IM IONOO | VIAV SIHI NI JYHAM ION OO | VI4V SIHI NI JLIYM ION OO |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| Upper quartile \(= 34\) | B1 | For \(Q_3 = 34\) either stated or used/implied |
| Lower limit \(= 24 - 15 = 9\) or upper limit is \(34 + 15 = 49\) | M1 | For one correct calculation (ft their 34 for upper limit) |
| Outliers are: 8, 52.5 and 56 | A1ft, A1ft | 2nd A1ft for lower outlier at 8; 3rd A1ft for upper outliers at 52.5 and 56 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| Box with \(Q_1=24\), \(Q_2=30\), \(Q_3=34\) and two whiskers | B1 | Box correct |
| Lower whisker ending at 10 (or 9) and outlier at 8 only | B1 | |
| Upper whisker ending at 45 (or 49) and outliers at 52.5 and 56 | B1 | Usual accuracy: to within 0.5 of a square |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(Q_2 - Q_1\ (=6) > (``4"=)\ Q_3 - Q_2\) or e.g. "\(Q_3\) closer to \(Q_2\) than \(Q_1\)" | M1 | For correct comparison of \(Q_2-Q_1\) and \(Q_3-Q_2\) |
| So negative (skew) | A1ft | Correct deduction based on their \(Q_3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| IQR now \(34 - 26 = 8\); new limits \(26 - 1.5\times 8 = \mathbf{14}\) and \(34 + 1.5\times 8 = \mathbf{46}\) | M1 | Recognising new IQR and at least one correct new limit |
| Box plot with correct lower whisker at 15.5 (or 14) and outliers at 8 and 10 | A1ft | 1st A1ft correct lower whisker ending at 15.5 (or 14) and 2 correct outliers at 8 and 10 |
| Fully correct box plot with upper whisker to 45 (or 46) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(Q_1\) has increased so both above 24; Median same; so between 26 and 30 inc | B1 | For range [26, 30] |
| \(Q_3\) unchanged so must be either side of \(Q_3\); so between 34 and 45 inc | B1 | For range [34, 45); condone [...] or (...) |
# Question 4:
## Part (a):
| Working | Marks | Notes |
|---------|-------|-------|
| Upper quartile $= 34$ | B1 | For $Q_3 = 34$ either stated or used/implied |
| Lower limit $= 24 - 15 = 9$ or upper limit is $34 + 15 = 49$ | M1 | For one correct calculation (ft their 34 for upper limit) |
| Outliers are: 8, 52.5 and 56 | A1ft, A1ft | 2nd A1ft for lower outlier at 8; 3rd A1ft for upper outliers at 52.5 and 56 |
## Part (b):
| Working | Marks | Notes |
|---------|-------|-------|
| Box with $Q_1=24$, $Q_2=30$, $Q_3=34$ and two whiskers | B1 | Box correct |
| Lower whisker ending at 10 (or 9) and outlier at 8 only | B1 | |
| Upper whisker ending at 45 (or 49) and outliers at 52.5 and 56 | B1 | Usual accuracy: to within 0.5 of a square |
## Part (c):
| Working | Marks | Notes |
|---------|-------|-------|
| $Q_2 - Q_1\ (=6) > (``4"=)\ Q_3 - Q_2$ or e.g. "$Q_3$ closer to $Q_2$ than $Q_1$" | M1 | For correct comparison of $Q_2-Q_1$ and $Q_3-Q_2$ |
| So **negative** (skew) | A1ft | Correct deduction based on their $Q_3$ |
## Part (d):
| Working | Marks | Notes |
|---------|-------|-------|
| IQR now $34 - 26 = 8$; new limits $26 - 1.5\times 8 = \mathbf{14}$ and $34 + 1.5\times 8 = \mathbf{46}$ | M1 | Recognising new IQR and at least one correct new limit |
| Box plot with correct lower whisker at 15.5 (or 14) and outliers at 8 and 10 | A1ft | 1st A1ft correct lower whisker ending at 15.5 (or 14) and 2 correct outliers at 8 and 10 |
| Fully correct box plot with upper whisker to 45 (or 46) | A1 | |
## Part (e):
| Working | Marks | Notes |
|---------|-------|-------|
| $Q_1$ has increased so both above 24; Median same; so **between 26 and 30** inc | B1 | For range [26, 30] |
| $Q_3$ unchanged so must be either side of $Q_3$; so **between 34 and 45** inc | B1 | For range [34, 45); condone [...] or (...) |
---4. A group of students took some tests. A teacher is analysing the average mark for each student. Each student obtained a different average mark.
For these average marks, the lower quartile is 24 , the median is 30 and the interquartile range (IQR) is 10\\
The three lowest average marks are 8, 10 and 15.5 and the three highest average marks are 45, 52.5 and 56
The teacher defines an outlier to be a value that is either\\
more than $1.5 \times$ IQR below the lower quartile or\\
more than $1.5 \times$ IQR above the upper quartile
\begin{enumerate}[label=(\alph*)]
\item Determine any outliers in these data.
\item On the grid below draw a box plot for these data, indicating clearly any outliers.\\
\includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-12_350_1223_1128_370}
\item Use the quartiles to describe the skewness of these data.
Give a reason for your answer.
Two more students also took the tests. Their average marks, which were both less than 45, are added to the data and the box plot redrawn.
The median and the upper quartile are the same but the lower quartile is now 26
\item Redraw the box plot on the grid below.\\
(3)\\
\includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-12_350_1221_2106_367}
\item Give ranges of values within which each of these students' average marks must lie.
Turn over for spare grids if you need to redraw your answer for part (b) or part (d).
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
VIXV SIHIANI III IM IONOO & VIAV SIHI NI JYHAM ION OO & VI4V SIHI NI JLIYM ION OO \\
\hline
\end{tabular}
\end{center}
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Copy of grid for part (b)}
\includegraphics[alt={},max width=\textwidth]{81d5e460-9559-4d25-aa08-6440559aec83-15_356_1226_1726_367}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Copy of grid for part (d)}
\includegraphics[alt={},max width=\textwidth]{81d5e460-9559-4d25-aa08-6440559aec83-15_353_1226_2240_367}
\end{center}
\end{figure}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2020 Q4 [14]}}