| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Complete frequency table and histogram together |
| Difficulty | Moderate -0.3 This is a standard AS-level statistics question testing histogram interpretation and basic statistical measures. Students must use frequency density to complete missing frequencies (routine calculation), apply linear interpolation for the median (standard technique), and check for outliers using the IQR rule (direct formula application). While multi-part, each component is a textbook exercise requiring no novel insight. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02h Recognize outliers |
| Age ( \(x\) years) | \(0 \leqslant x < 5\) | \(5 \leqslant x < 20\) | \(20 \leqslant x < 40\) | \(40 \leqslant x < 65\) | \(65 \leqslant x < 80\) | \(80 \leqslant x < 90\) |
| Frequency | 5 | 45 | 90 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| From \([5,20]\) \(\text{fd} = 3\) or 1 large square \(= 2.5\) passengers o.e. | M1 | For attempt at fd or suitable method to deduce scale. May be implied by one correct bar |
| Correct bar above \([0, 5)\) | A1 | \(1^{\text{st}}\) A1: first bar \([0,5)\) with \(\text{fd}=1\) or 2 large squares high |
| Correct bar above \([20, 40)\) | A1 (3) | \(2^{\text{nd}}\) A1: third bar with \(\text{fd}=4.5\) or 9 large squares high |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| For \([40, 65)\): 130 passengers or for \([65, 80)\): 60 passengers | M1 | For attempt using their fd to find missing frequencies. May be in table |
| For attempt to find total number of passengers \(= \mathbf{331}\) | A1ft | For a clear attempt to find total (ft their 130 and 60) |
| \([\text{Median} =]\ 40 + \dfrac{\frac{1}{2}(\text{"331"}) - 140}{\text{"130"}} \times 25\) or \(65 - \dfrac{270 - \frac{1}{2}(\text{"331"})}{\text{"130"}} \times 25\) | M1 | For any expression/equation leading to correct \(Q_2\). Must be using 40–65 class |
| \(= 44.9038\ldots = \text{awrt } \mathbf{44.9}\) | A1 (4) | Allow \((n+1)\) leading to 45 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Upper outlier limit \(= 58.9 + 1.5 \times (58.9 - 27.3) = 106\ (.3) > 90\) | M1 | For finding upper outlier limit (expression or awrt 106) and stating or implying \(> 90\) |
| So oldest passenger is not an outlier | A1 (2) | Dependent on M1 seen for deducing NOT an outlier |
## Question 2:
**Part (a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| From $[5,20]$ $\text{fd} = 3$ or 1 large square $= 2.5$ passengers o.e. | M1 | For attempt at fd or suitable method to deduce scale. May be implied by one correct bar |
| Correct bar above $[0, 5)$ | A1 | $1^{\text{st}}$ A1: first bar $[0,5)$ with $\text{fd}=1$ or 2 large squares high |
| Correct bar above $[20, 40)$ | A1 (3) | $2^{\text{nd}}$ A1: third bar with $\text{fd}=4.5$ or 9 large squares high |
**Part (b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| For $[40, 65)$: **130** passengers or for $[65, 80)$: **60** passengers | M1 | For attempt using their fd to find missing frequencies. May be in table |
| For attempt to find total number of passengers $= \mathbf{331}$ | A1ft | For a clear attempt to find total (ft their 130 and 60) |
| $[\text{Median} =]\ 40 + \dfrac{\frac{1}{2}(\text{"331"}) - 140}{\text{"130"}} \times 25$ or $65 - \dfrac{270 - \frac{1}{2}(\text{"331"})}{\text{"130"}} \times 25$ | M1 | For any expression/equation leading to correct $Q_2$. Must be using 40–65 class |
| $= 44.9038\ldots = \text{awrt } \mathbf{44.9}$ | A1 (4) | Allow $(n+1)$ leading to 45 |
**Part (c)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| Upper outlier limit $= 58.9 + 1.5 \times (58.9 - 27.3) = 106\ (.3) > 90$ | M1 | For finding upper outlier limit (expression or awrt 106) **and** stating or implying $> 90$ |
| So oldest passenger is **not** an outlier | A1 (2) | Dependent on M1 seen for deducing NOT an outlier |\begin{enumerate}
\item The partially completed table and partially completed histogram give information about the ages of passengers on an airline.
\end{enumerate}
There were no passengers aged 90 or over.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Age ( $x$ years) & $0 \leqslant x < 5$ & $5 \leqslant x < 20$ & $20 \leqslant x < 40$ & $40 \leqslant x < 65$ & $65 \leqslant x < 80$ & $80 \leqslant x < 90$ \\
\hline
Frequency & 5 & 45 & 90 & & & 1 \\
\hline
\end{tabular}
\end{center}
\includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-04_1173_1792_721_139}\\
(a) Complete the histogram.\\
(b) Use linear interpolation to estimate the median age.
An outlier is defined as a value greater than $Q _ { 3 } + 1.5 \times$ interquartile range.\\
Given that $Q _ { 1 } = 27.3$ and $Q _ { 3 } = 58.9$\\
(c) determine, giving a reason, whether or not the oldest passenger could be considered as an outlier.\\
(2)
\hfill \mbox{\textit{Edexcel AS Paper 2 2021 Q2 [9]}}