| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw box plot from cumulative frequency |
| Difficulty | Moderate -0.8 This is a straightforward application of standard statistical procedures: reading quartiles from a cumulative frequency graph, drawing a box plot, and applying the 1.5×IQR outlier rule. All steps are routine recall and direct calculation with no problem-solving insight required, making it easier than average. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(LQ = 15\), Median \(= 18\), \(UQ = 26\) | B1 | \(LQ = 15\), Median \(= 18\), and \(UQ = 26\) |
| (box plot drawn) | B1 | Linear scale and labels |
| B1\(\sqrt{}\) | Quartiles and median box, ft on their values, but \(M - LQ < UQ - M\) | |
| Whiskers from 5 to LQ and UQ to 80 | B1\(\sqrt{}\) [4] | Whiskers from 5 to LQ and UQ to 80, ft on their values |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Most \((3/4)\) are earning less than \(26K\), not many earning high salaries, etc | B1 [1] | Any sensible answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| IQ range \(= 11\) | B1 | \(IQR = 11\) |
| high outlier is above \(26 + 1.5 \times 11\) | M1 | Their \(UQ + 1.5 \times\) their IQ range |
| \(= 42500\) euros | A1 [3] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Low outlier is below \(15 - 1.5 \times 11 = -1.5\) | B1\(\sqrt{}\) [1] | Correct reason, must involve subtraction, ft on their LQ and \(1.5 \times\) their IQR |
## Question 5:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $LQ = 15$, Median $= 18$, $UQ = 26$ | B1 | $LQ = 15$, Median $= 18$, and $UQ = 26$ |
| (box plot drawn) | B1 | Linear scale and labels |
| | B1$\sqrt{}$ | Quartiles and median box, ft on their values, but $M - LQ < UQ - M$ |
| Whiskers from 5 to LQ and UQ to 80 | B1$\sqrt{}$ [4] | Whiskers from 5 to LQ and UQ to 80, ft on their values |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Most $(3/4)$ are earning less than $26K$, not many earning high salaries, etc | B1 [1] | Any sensible answer |
### Part (iii)(a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| IQ range $= 11$ | B1 | $IQR = 11$ |
| high outlier is above $26 + 1.5 \times 11$ | M1 | Their $UQ + 1.5 \times$ their IQ range |
| $= 42500$ euros | A1 [3] | Correct answer |
### Part (iii)(b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Low outlier is below $15 - 1.5 \times 11 = -1.5$ | B1$\sqrt{}$ [1] | Correct reason, must involve subtraction, ft on their LQ and $1.5 \times$ their IQR |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{b72ace6b-d3d4-401d-bffe-403c9127f2a8-3_1157_1001_258_573}
The cumulative frequency graph shows the annual salaries, in thousands of euros, of a random sample of 500 adults with jobs, in France. It has been plotted using grouped data. You may assume that the lowest salary is 5000 euros and the highest salary is 80000 euros.\\
(i) On graph paper, draw a box-and-whisker plot to illustrate these salaries.\\
(ii) Comment on the salaries of the people in this sample.\\
(iii) An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
\begin{enumerate}[label=(\alph*)]
\item How high must a salary be in order to be classified as an outlier?
\item Show that none of the salaries is low enough to be classified as an outlier.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2011 Q5 [9]}}