| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw histogram then estimate mean/standard deviation |
| Difficulty | Moderate -0.8 This is a straightforward statistics question requiring standard histogram construction with unequal class widths (calculating frequency densities) and mean estimation from grouped data using midpoints. Both are routine A-level statistics procedures with no conceptual challenges or problem-solving required. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Time, \(t\) minutes | \(0 \leqslant t < 5\) | \(5 \leqslant t < 10\) | \(10 \leqslant t < 20\) | \(20 \leqslant t < 30\) | \(30 \leqslant t < 50\) |
| Frequency | 23 | 102 | 135 | 76 | 24 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Class widths: \(5, 5, 10, 10, 20\) | M1 | At least 4 frequency densities calculated (f/cw), accept unsimplified and class widths \(\pm 1\) of true values. May be implied by graph. |
| Frequency densities: \(4.6, 20.4, 13.5, 7.6, 1.2\) | A1 | All heights correct on graph NOT FT |
| Bar ends at \(0, 5, 10, 20, 30, 50\) clear intention not to draw at \(4.5\) or \(5.5\) etc. | B1 | Bar ends at \(0, 5, 10, 20, 30, 50\) clear intention not to draw at \(4.5\) or \(5.5\) etc. |
| Axes labelled: Frequency density (fd), time (t) and mins | B1 | Linear scales between \(0\) and \(20.4\) or above on vertical axis, and \(0\) and \(50\) or above on horizontal axis. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\dfrac{2.5\times23 + 7.5\times102 + 15\times135 + 25\times76 + 40\times24}{360}\) | M1 | Uses at least 4 midpoint attempts (e.g. \(2.5 \pm 0.5\)) in correct formula, accept unsimplified expression, denominator either correct or *their* \(\Sigma\)frequencies. |
| \(\left[\dfrac{5707.5}{360} =\right] 15.9,\ 15\dfrac{41}{48}\) | A1 | Evaluated. |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Class widths: $5, 5, 10, 10, 20$ | M1 | At least 4 frequency densities calculated (f/cw), accept unsimplified and class widths $\pm 1$ of true values. May be implied by graph. |
| Frequency densities: $4.6, 20.4, 13.5, 7.6, 1.2$ | A1 | All heights correct on graph **NOT FT** |
| Bar ends at $0, 5, 10, 20, 30, 50$ clear intention not to draw at $4.5$ or $5.5$ etc. | B1 | Bar ends at $0, 5, 10, 20, 30, 50$ clear intention not to draw at $4.5$ or $5.5$ etc. |
| Axes labelled: Frequency density (fd), time (t) and mins | B1 | Linear scales between $0$ and $20.4$ or above on vertical axis, and $0$ and $50$ or above on horizontal axis. |
---
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\dfrac{2.5\times23 + 7.5\times102 + 15\times135 + 25\times76 + 40\times24}{360}$ | M1 | Uses at least 4 midpoint attempts (e.g. $2.5 \pm 0.5$) in correct formula, accept unsimplified expression, denominator either correct or *their* $\Sigma$frequencies. |
| $\left[\dfrac{5707.5}{360} =\right] 15.9,\ 15\dfrac{41}{48}$ | A1 | Evaluated. |
---3 The times taken, in minutes, by 360 employees at a large company to travel from home to work are summarised in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Time, $t$ minutes & $0 \leqslant t < 5$ & $5 \leqslant t < 10$ & $10 \leqslant t < 20$ & $20 \leqslant t < 30$ & $30 \leqslant t < 50$ \\
\hline
Frequency & 23 & 102 & 135 & 76 & 24 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Draw a histogram to represent this information.\\
\includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-04_1198_1200_836_516}
\item Calculate an estimate of the mean time taken by an employee to travel to work.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q3 [6]}}