| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Cumulative frequency graph construction then interpretation |
| Difficulty | Easy -1.2 This is a straightforward grouped data question requiring standard S1 techniques: finding median/quartile positions (cumulative frequency), drawing a histogram with unequal class widths (frequency density), and calculating mean from grouped data using midpoints. All are routine textbook exercises with no problem-solving or novel insight required. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02b Histogram: area represents frequency2.02f Measures of average and spread |
| \(0 < t \leqslant 10\) | \(10 < t \leqslant 15\) | \(15 < t \leqslant 20\) | \(20 < t \leqslant 25\) | \(25 < t \leqslant 40\) | \(40 < t \leqslant 60\) | ||
| Frequency | 19 | 12 | 28 | 22 | 18 | 13 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Median in 15–20 mins | B1 | |
| UQ in 25–40 mins | B1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| fd: \(1.9, 2.4, 5.6, 4.4, 1.2, 0.65\) or Scaled freq \(9.5, 12, 28, 22, 6, 3.25\) | M1 | Attempt at fd or scaled freq \([f/(\text{attempt at cw})]\) |
| Correct heights seen on diagram | A1 | |
| Correct bar widths, visually no gaps | B1 | |
| Labels (time/mins and fd or freq per 5 min) and correct bar ends | B1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{5\times19 + 12.5\times12 + 17.5\times28 + 22.5\times22 + 32.5\times18 + 50\times13}{112} = \frac{2465}{112}\) | M1 | Attempt at \(\Sigma xf / 112\) using midpoints, NOT classwidths, NOT upper class bounds |
| \(= 22.0\) minutes | A1 [2] | Correct answer, accept 22 |
## Question 3:
**(i)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Median in 15–20 mins | B1 | |
| UQ in 25–40 mins | B1 [2] | |
**(ii)**
| Answer | Mark | Guidance |
|--------|------|----------|
| fd: $1.9, 2.4, 5.6, 4.4, 1.2, 0.65$ or Scaled freq $9.5, 12, 28, 22, 6, 3.25$ | M1 | Attempt at fd or scaled freq $[f/(\text{attempt at cw})]$ |
| Correct heights seen on diagram | A1 | |
| Correct bar widths, visually no gaps | B1 | |
| Labels (time/mins and fd or freq per 5 min) and correct bar ends | B1 [4] | |
**(iii)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{5\times19 + 12.5\times12 + 17.5\times28 + 22.5\times22 + 32.5\times18 + 50\times13}{112} = \frac{2465}{112}$ | M1 | Attempt at $\Sigma xf / 112$ using midpoints, NOT classwidths, NOT upper class bounds |
| $= 22.0$ minutes | A1 [2] | Correct answer, accept 22 |
---3 The table summarises the times that 112 people took to travel to work on a particular day.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
\begin{tabular}{ l }
Time to travel to \\
work $( t$ minutes $)$ \\
\end{tabular} & $0 < t \leqslant 10$ & $10 < t \leqslant 15$ & $15 < t \leqslant 20$ & $20 < t \leqslant 25$ & $25 < t \leqslant 40$ & $40 < t \leqslant 60$ \\
\hline
Frequency & 19 & 12 & 28 & 22 & 18 & 13 \\
\hline
\end{tabular}
\end{center}
(i) State which time interval in the table contains the median and which time interval contains the upper quartile.\\
(ii) On graph paper, draw a histogram to represent the data.\\
(iii) Calculate an estimate of the mean time to travel to work.
\hfill \mbox{\textit{CAIE S1 2012 Q3 [8]}}