Question 2 - A-Level Maths

Edexcel S1 2013 June — Question 2 11 marks

Exam Board	Edexcel
Module	S1 (Statistics 1)
Year	2013
Session	June
Marks	11
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Measures of Location and Spread
Type	Direct cumulative frequency graph reading
Difficulty	Easy -1.3 This is a routine statistics question requiring standard procedures: reading values from a cumulative frequency graph, calculating median/quartiles from a stem-and-leaf diagram, identifying outliers using the 1.5×IQR rule, drawing a box plot, and making basic comparative statements. All techniques are direct applications of S1 syllabus content with no problem-solving or novel insight required.
Spec	2.02a Interpret single variable data: tables and diagrams 2.02f Measures of average and spread 2.02g Calculate mean and standard deviation 2.02h Recognize outliers 2.02j Clean data: missing data, errors

The marks of a group of female students in a statistics test are summarised in Figure 1

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6faf2dd2-a114-40b7-88ae-4a75dbfb4706-04_629_1102_342_429} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

Write down the mark which is exceeded by \(75 \%\) of the female students. The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.
Mark (2|6 means 26) Totals
1 4 (1)
2 6 (1)
3 447 (3)
4 066778 (6)
5 001113677 (9)
6 223338 (6)
7 008 (3)
8 5 (1)
9 0 (1)
Find the median and interquartile range of the marks of the male students. An outlier is a mark that is
either more than \(1.5 \times\) interquartile range above the upper quartile or more than \(1.5 \times\) interquartile range below the lower quartile.
In the space provided on Figure 1 draw a box plot to represent the marks of the male students, indicating clearly any outliers.
Compare and contrast the marks of the male and the female students.

Show mark scheme Show mark scheme source

Question 2:

Part (a)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
25 (allow any \(x\) where \(24 < x < 26\))	B1

Part (b)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
\(Q_2\) (or median or \(m\)) \(= \mathbf{51}\)	B1	Mark (b) and (c) together BUT must see clear statement that median (or \(m\) or \(Q_2\)) = 51 and IQR = 17
\(\mathbf{IQR} = 63 - 46 = \mathbf{17}\) (or \(Q_3 - Q_1 = 17\))	M1, A1	M1 for 2 quartiles (at least one correct) and attempt to find the difference. Must see \(63 -\) their \(46\). A1 for 17 only. [Answer only of IQR = 17 scores M1A1]

Part (c)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
Outliers given by \(46 - 1.5 \times 17 = 20.5\) or \(63 + 1.5 \times 17 = 88.5\)	M1	1st M1 for correct attempt to calc at least one limit for outliers, ft their quartiles or IQR, or award for sight of 20.5 or 88.5
Outlier limits are \(\mathbf{20.5}\) and \(\mathbf{88.5}\)	A1	1st A1 for identifying both limits of 20.5 and 88.5
Box plot with whiskers to 20.5 and 88.5; outliers at 14 and 90	M1 A1ft B1	2nd M1 for box with upper and lower whisker(s) with at least 2 correct values (or correct ft). 2nd A1ft for 20.5 or 26, 46, 51, 63 and 85 or 88.5 in appropriate places. B1 for only 2 outliers appropriately marked at 14 and 90. Apply \(\pm 0.5\) square accuracy for diagram. A box plot not on graph paper can only score 1st M1A1

Part (d)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
Medians: Median for females lower than males	B1ft	1st B1ft for one correct comment comparing median, IQR, range or skewness
IQR: IQR for females smaller than males. Allow "lower/higher" but not "wider"	B1ft	2nd B1ft for second correct comment comparing median, IQR, range or skewness. Do not allow contradictory statements
Range: Range of females is less than males
Skewness: Male and female marks are both positively skew		In (d) ft from their diagrams (if no diagram then use their values)

# Question 2:

## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| 25 (allow any $x$ where $24 < x < 26$) | B1 | |

## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $Q_2$ (or median or $m$) $= \mathbf{51}$ | B1 | Mark (b) and (c) together BUT must see clear statement that median (or $m$ or $Q_2$) = 51 and IQR = 17 |
| $\mathbf{IQR} = 63 - 46 = \mathbf{17}$ (or $Q_3 - Q_1 = 17$) | M1, A1 | M1 for 2 quartiles (at least one correct) and attempt to find the difference. Must see $63 -$ their $46$. A1 for 17 only. [Answer only of IQR = 17 scores M1A1] |

## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Outliers given by $46 - 1.5 \times 17 = 20.5$ or $63 + 1.5 \times 17 = 88.5$ | M1 | 1st M1 for correct attempt to calc at least one limit for outliers, ft their quartiles or IQR, or award for sight of 20.5 or 88.5 |
| Outlier limits are $\mathbf{20.5}$ and $\mathbf{88.5}$ | A1 | 1st A1 for identifying both limits of 20.5 and 88.5 |
| Box plot with whiskers to 20.5 and 88.5; outliers at 14 and 90 | M1 A1ft B1 | 2nd M1 for box with upper and lower whisker(s) with at least 2 correct values (or correct ft). 2nd A1ft for 20.5 or 26, 46, 51, 63 and 85 or 88.5 in appropriate places. B1 for only 2 outliers appropriately marked at 14 and 90. **Apply $\pm 0.5$ square accuracy for diagram**. A box plot not on graph paper can only score 1st M1A1 |

## Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| **Medians:** Median for females lower than males | B1ft | 1st B1ft for one correct comment comparing median, IQR, range or skewness |
| **IQR:** IQR for females smaller than males. Allow "lower/higher" but not "wider" | B1ft | 2nd B1ft for second correct comment comparing median, IQR, range or skewness. Do not allow contradictory statements |
| **Range:** Range of females is less than males | | |
| **Skewness:** Male and female marks are both positively skew | | In (d) ft from their diagrams (if no diagram then use their values) |

Show LaTeX source

\begin{enumerate}
  \item The marks of a group of female students in a statistics test are summarised in Figure 1
\end{enumerate}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{6faf2dd2-a114-40b7-88ae-4a75dbfb4706-04_629_1102_342_429}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

(a) Write down the mark which is exceeded by $75 \%$ of the female students.

The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.

\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Mark & (2|6 means 26) & Totals \\
\hline
1 & 4 & (1) \\
\hline
2 & 6 & (1) \\
\hline
3 & 447 & (3) \\
\hline
4 & 066778 & (6) \\
\hline
5 & 001113677 & (9) \\
\hline
6 & 223338 & (6) \\
\hline
7 & 008 & (3) \\
\hline
8 & 5 & (1) \\
\hline
9 & 0 & (1) \\
\hline
\end{tabular}
\end{center}

(b) Find the median and interquartile range of the marks of the male students.

An outlier is a mark that is\\
either more than $1.5 \times$ interquartile range above the upper quartile or more than $1.5 \times$ interquartile range below the lower quartile.\\
(c) In the space provided on Figure 1 draw a box plot to represent the marks of the male students, indicating clearly any outliers.\\
(d) Compare and contrast the marks of the male and the female students.\\

\hfill \mbox{\textit{Edexcel S1 2013 Q2 [11]}}

This paper (6 questions)

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Q1 13 Q2 11 Q3 12 Q4 14 Q5 15 Q6 10

Mark	(2\|6 means 26)	Totals
1	4	(1)
2	6	(1)
3	447	(3)
4	066778	(6)
5	001113677	(9)
6	223338	(6)
7	008	(3)
8	5	(1)
9	0	(1)