| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Complete frequency table from histogram only |
| Difficulty | Moderate -0.8 This is a standard S1 histogram question requiring students to use the key principle that frequency = frequency density × class width, then construct a cumulative frequency graph and read off standard statistics. While it involves multiple parts, each step follows routine procedures taught explicitly in the syllabus with no problem-solving or novel insight required. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{freq} = fd \times cw\ \ 10, 40, 120, 30\) | M1, A1 | Attempt to multiply at least 3 fds by their 'class widths' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| cf heights correct: 10, 50, 170, 200 | B1 | 3 or more correct cfs heights on graph 10, 50, 170, 200 |
| Labels correct: cf and length(cm), linear scales from zero | B1 | Allow 0.5 on horizontal axis |
| Plotting at upper end points | M1 | Attempt at least three plotted at upper end points (either 5 or 5.5, 10 or 10.5 etc.) |
| Starting at \((0,0)\), polygon or smooth curve increasing through plotted points at lengths 5, 10, 20, 25 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| median \(= 14.2\) | B1 | Accept \(13.2 - 15.2\) |
| \('18.5' - '10'\) | M1 | Subtract their LQ from their UQ if reasonable from their graph |
| IQ range \(= 8.5\) | A1FT | Correct FT using \(LQ = 10\) and \(UQ\) between 17.5 and 19.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| mean \(= (2.5\times10 + 7.5\times40 + 15\times120 + 22.5\times30) / 200\) | M1 | Using mid points \((\pm 0.5)\) and their frequencies from 7(i) in correct formula |
| \(= 14\) | A1 |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{freq} = fd \times cw\ \ 10, 40, 120, 30$ | M1, A1 | Attempt to multiply at least 3 fds by their 'class widths' |
---
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| cf heights correct: 10, 50, 170, 200 | B1 | 3 or more correct cfs **heights** on graph 10, 50, 170, 200 |
| Labels correct: cf and length(cm), linear scales from zero | B1 | Allow 0.5 on horizontal axis |
| Plotting at upper end points | M1 | Attempt at least three plotted at upper end points (either 5 or 5.5, 10 or 10.5 etc.) |
| Starting at $(0,0)$, polygon or smooth curve increasing through plotted points at lengths 5, 10, 20, 25 | A1 | |
---
## Question 7(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| median $= 14.2$ | B1 | Accept $13.2 - 15.2$ |
| $'18.5' - '10'$ | M1 | Subtract their LQ from their UQ if reasonable from their graph |
| IQ range $= 8.5$ | A1FT | Correct FT using $LQ = 10$ and $UQ$ between 17.5 and 19.5 |
---
## Question 7(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| mean $= (2.5\times10 + 7.5\times40 + 15\times120 + 22.5\times30) / 200$ | M1 | Using mid points $(\pm 0.5)$ and their frequencies from 7(i) in correct formula |
| $= 14$ | A1 | |7 The following histogram represents the lengths of worms in a garden.\\
\includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-10_789_1195_301_466}\\
(i) Calculate the frequencies represented by each of the four histogram columns.\\
(ii) On the grid on the next page, draw a cumulative frequency graph to represent the lengths of worms in the garden.\\
\includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-11_1111_1409_251_408}\\
(iii) Use your graph to estimate the median and interquartile range of the lengths of worms in the garden.\\
(iv) Calculate an estimate of the mean length of worms in the garden.\\
{www.cie.org.uk} after the live examination series.
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\hfill \mbox{\textit{CAIE S1 2017 Q7 [11]}}