| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Interpret or analyse given back-to-back stem-and-leaf |
| Difficulty | Moderate -0.3 This is a straightforward stem-and-leaf interpretation question requiring standard statistical calculations (median, quartiles, IQR) and basic box plots. Part (ii) involves simple algebra to find unknown values given the IQR. While it has multiple parts and requires careful counting, all techniques are routine for S1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread |
| Country \(X\) | Country \(Y\) | |||
| (10) | 9766644432 | 80 | ||
| (18) | 888776655544333220 | 81 | 1122333556789 | (13) |
| (16) | 9998877655322100 | 82 | 001233394566788 | (15) |
| (16) | 8765553322211100 | 83 | 01224444556677789 | (17) |
| (11) | 87655443311 | 84 | 001244556677789 | (15) |
| 85 | \(12 r 335566788\) | (12) | ||
| 86 | 01223555899 | (11) | ||
| Answer | Marks | Guidance |
|---|---|---|
| (i) for X: Median = 0.825 cm IQ range = 0.019 cm (0.833 – 0.814) | B1 | Correct median |
| B1 | Correct IQ range | |
| (ii) \(q = 4\) \(r = 2\) | B1 | |
| B1 | Must be 4 and 2 not 3 and 1 | |
| SR \(q = 0.824\) and \(r = 0.852\) | B1 | |
| (iii) [Box plot diagram with Y and X variables shown, with labels X, Y and length/cm, linear scale from 0.80 to 0.87 and both on one diagram] | B1 | Labels X, Y and length/cm, linear scale from 0.80 to 0.87 and both on one diagram |
| B1ft | Correct median and quartiles for X fl theirs must be a box | |
| B1ft | Correct median and quartiles for Y fl theirs must be a box | |
| B1 | Whiskers correct no line through middle | |
| (iv) Y has longer insets on average Y has larger range | B1 | Correct statement about lengths |
| B1 | Correct statement about spreads |
**(i)** for X: Median = 0.825 cm IQ range = 0.019 cm (0.833 – 0.814) | B1 | Correct median |
| B1 | Correct IQ range |
**(ii)** $q = 4$ $r = 2$ | B1 | |
| B1 | Must be 4 and 2 not 3 and 1 |
SR $q = 0.824$ and $r = 0.852$ | B1 | |
**(iii)** [Box plot diagram with Y and X variables shown, with labels X, Y and length/cm, linear scale from 0.80 to 0.87 and both on one diagram] | B1 | Labels X, Y and length/cm, linear scale from 0.80 to 0.87 and both on one diagram |
| B1ft | Correct median and quartiles for X fl theirs must be a box |
| B1ft | Correct median and quartiles for Y fl theirs must be a box |
| B1 | Whiskers correct no line through middle |
**(iv)** Y has longer insets on average Y has larger range | B1 | Correct statement about lengths |
| B1 | Correct statement about spreads |
---6 The lengths of some insects of the same type from two countries, $X$ and $Y$, were measured. The stem-and-leaf diagram shows the results.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
& Country $X$ & \multicolumn{3}{|c|}{Country $Y$} \\
\hline
(10) & 9766644432 & 80 & & \\
\hline
(18) & 888776655544333220 & 81 & 1122333556789 & (13) \\
\hline
(16) & 9998877655322100 & 82 & 001233394566788 & (15) \\
\hline
(16) & 8765553322211100 & 83 & 01224444556677789 & (17) \\
\hline
(11) & 87655443311 & 84 & 001244556677789 & (15) \\
\hline
& & 85 & $12 r 335566788$ & (12) \\
\hline
& & 86 & 01223555899 & (11) \\
\hline
\end{tabular}
\end{center}
Key: 5 | 81 | 3 means an insect from country $X$ has length 0.815 cm and an insect from country $Y$ has length 0.813 cm .\\
(i) Find the median and interquartile range of the lengths of the insects from country $X$.\\
(ii) The interquartile range of the lengths of the insects from country $Y$ is 0.028 cm . Find the values of $q$ and $r$.\\
(iii) Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.\\
(iv) Compare the lengths of the insects from the two countries.
\hfill \mbox{\textit{CAIE S1 2010 Q6 [10]}}