| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Histogram from continuous grouped data |
| Difficulty | Moderate -0.8 This is a straightforward S1 statistics question requiring standard procedures: drawing a histogram from grouped data, calculating mean/SD from a frequency table, applying the outlier rule (mean ± 2SD), and comparing distributions. All techniques are routine textbook exercises with no problem-solving or novel insight required, making it easier than average but not trivial due to the computational work involved. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| Weight | \(30 \leqslant w < 50\) | \(50 \leqslant w < 60\) | \(60 \leqslant w < 70\) | \(70 \leqslant w < 80\) | \(80 \leqslant w < 90\) |
| Frequency | 11 | 10 | 18 | 14 | 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Frequency density table: | M1 | For fd's - at least 3 correct. Accept any suitable unit for fd such as eg freq per 10g. |
| A1 | ||
| Weight | Frequency | Group Width |
| \(30 \leq w < 50\) | 11 | 20 |
| \(50 \leq w < 60\) | 10 | 10 |
| \(60 \leq w < 70\) | 18 | 10 |
| \(70 \leq w < 80\) | 14 | 10 |
| \(80 \leq w < 90\) | 7 | 10 |
| Histogram: | G1 | Linear scales on both axes and labels. Vertical scale starting from zero (not broken - but can get final mark for heights if broken). Accept f/w or f/cw (freq/width or freq/class width). Ignore horizontal label. |
| G1 | width of bars. Must be drawn at 30, 50 etc NOT 29.5 or 30.5 etc NO GAPS ALLOWED. Must have linear scale. No inequality labels on their own such as \(30 < W < 50\) etc but allow if 30, 50, 60 etc occur at the correct boundary position. See additional notes. Allow this mark even if not using fd's. | |
| G1 | height of bars. Height of bars – must be linear vertical scale. FT of heights dep on at least 3 heights correct and all must agree with their fds. If fds not given and at least 3 heights correct then max M1A0G1G1G0. Allow restart with correct heights if given fd wrong (for last three marks only). | |
| Total | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Mean} = \frac{(40 \times 11) + (55 \times 10) + (65 \times 18) + (75 \times 14) + (85 \times 7)}{60} = \frac{3805}{60}\) | M1 | For midpoints. (at least 3 correct) No marks for mean or sd unless using midpoints. Answer must NOT be left as improper fraction as this is an estimate. Accept correct answers for mean and sd from calculator even if eg wrong \(S_{xx}\) given |
| \(= 63.4\) (or 63.42) | A1 | CAO (exact answer 63.41666...). Accept correct answers for mean and sd from calculator even if eg wrong \(S_{xx}\) given |
| \(\Sigma x^2 f = (40^2 \times 11) + (55^2 \times 10) + (65^2 \times 18) + (75^2 \times 14) + (85^2 \times 7)\) | M1 | For attempt at \(S_{xx}\). Should include sum of at least 3 correct multiples \(fx^2 - \Sigma x^2/n\). At least 1dp required. Use of mean 63.4 leading to answer of 14.29199... with \(S_{xx} = 12051.4\) gets full credit. |
| \(= 253225\) | ||
| \(S_{xx} = 253225 - \frac{3805^2}{60} = 11924.6\) | M1 | Allow M1 for anything which rounds to 11900. |
| \(s = \sqrt{\frac{11924.6}{59}} = \sqrt{202.11} = 14.2\) | A1 | Allow SC1 for RMSD 14.1 (14.0976…) from calculator. Only penalise once in part (ii) for over specification, even if mean and standard deviation both over specified. If using \((x - \bar{x})^2\) method, B2 if 14.2 or better (14.3 if use of 63.4), otherwise B0 |
| Total | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\bar{x} - 2s = 63.4 - (2 \times 14.2) = 35\) | M1 | For either. No marks in (iii) unless using \(\bar{x} + 2s\) or \(\bar{x} - 2s\). Only follow through numerical values, not variables such as \(s\), so if a candidate does not find \(s\) but then writes here 'limit is \(63.4 + 2 \times\) standard deviation', do NOT award M1. Do not penalise for over-specification. |
| \(\bar{x} + 2s = 63.4 + (2 \times 14.2) = 91.8\) | A1 | For both (FT). Must include an element of doubt and must mention both ends. |
| So there are probably some outliers at the lower end, but none at the upper end | E1 | Must have correct limits to get this mark. |
| Total | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Mean \(= \frac{3624.5}{50} = 72.5g\) (or exact answer 72.49g) | B1 | CAO Ignore units. |
| \(S_{xx} = 265416 - \frac{3624.5^2}{50} = 2676\) | M1 | For \(S_{xx}\). M1 for 265416 - 50 × their mean². BUT NOTE M0 if their \(S_{xx} < 0\) |
| \(s = \sqrt{\frac{2676}{49}} = \sqrt{54.61} = 7.39g\) | A1 | CAO ignore units. Allow 7.4 but NOT 7.3 (unless RMSD with working). For s² of 54.6 (or better) allow M1A0 with or without working. For RMSD of 7.3 (or better) allow M1A0 provided working seen. For RMSD² of 53.5 (or better) allow M1A0 provided working seen. |
| Total | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Variety A have lower average than Variety B oe | E1 | FT their means. Do not condone lower central tendency or lower mean. Allow 'on the whole' or similar in place of 'average'. |
| Variety A have higher variation than Variety B oe | E1 | FT their sd. Allow 'more spread' or similar but not 'higher range' or 'higher variance'. Condone less consistent. |
| Total | [2] |
## (i)
| **Answer** | **Marks** | **Guidance** |
|---|---|---|
| **Frequency density table:** | M1 | For fd's - at least 3 correct. Accept any suitable unit for fd such as eg freq per 10g. |
| | A1 | |
| Weight | Frequency | Group Width | Frequency density |
| $30 \leq w < 50$ | 11 | 20 | — |
| $50 \leq w < 60$ | 10 | 10 | 1 |
| $60 \leq w < 70$ | 18 | 10 | 1.8 |
| $70 \leq w < 80$ | 14 | 10 | 1.4 |
| $80 \leq w < 90$ | 7 | 10 | 0.7 |
| | | | |
| **Histogram:** | G1 | Linear scales on both axes and labels. Vertical scale starting from zero (not broken - but can get final mark for heights if broken). Accept f/w or f/cw (freq/width or freq/class width). Ignore horizontal label. |
| | G1 | width of bars. Must be drawn at 30, 50 etc NOT 29.5 or 30.5 etc NO GAPS ALLOWED. Must have linear scale. No inequality labels on their own such as $30 < W < 50$ etc but allow if 30, 50, 60 etc occur at the correct boundary position. See additional notes. Allow this mark even if not using fd's. |
| | G1 | height of bars. Height of bars – must be linear vertical scale. FT of heights dep on at least 3 heights correct and all must agree with their fds. If fds not given and at least 3 heights correct then max M1A0G1G1G0. Allow restart with correct heights if given fd wrong (for last three marks only). |
| **Total** | **[5]** | |
## (ii)
| **Answer** | **Marks** | **Guidance** |
|---|---|---|
| $\text{Mean} = \frac{(40 \times 11) + (55 \times 10) + (65 \times 18) + (75 \times 14) + (85 \times 7)}{60} = \frac{3805}{60}$ | M1 | For midpoints. (at least 3 correct) No marks for mean or sd unless using midpoints. Answer must NOT be left as improper fraction as this is an estimate. Accept correct answers for mean and sd from calculator even if eg wrong $S_{xx}$ given |
| $= 63.4$ (or 63.42) | A1 | CAO (exact answer 63.41666...). Accept correct answers for mean and sd from calculator even if eg wrong $S_{xx}$ given |
| $\Sigma x^2 f = (40^2 \times 11) + (55^2 \times 10) + (65^2 \times 18) + (75^2 \times 14) + (85^2 \times 7)$ | M1 | For attempt at $S_{xx}$. Should include sum of at least 3 correct multiples $fx^2 - \Sigma x^2/n$. At least 1dp required. Use of mean 63.4 leading to answer of 14.29199... with $S_{xx} = 12051.4$ gets full credit. |
| $= 253225$ | | |
| $S_{xx} = 253225 - \frac{3805^2}{60} = 11924.6$ | M1 | Allow M1 for anything which rounds to 11900. |
| $s = \sqrt{\frac{11924.6}{59}} = \sqrt{202.11} = 14.2$ | A1 | Allow SC1 for RMSD 14.1 (14.0976…) from calculator. Only penalise once in part (ii) for over specification, even if mean and standard deviation both over specified. If using $(x - \bar{x})^2$ method, B2 if 14.2 or better (14.3 if use of 63.4), otherwise B0 |
| **Total** | **[4]** | |
## (iii)
| **Answer** | **Marks** | **Guidance** |
|---|---|---|
| $\bar{x} - 2s = 63.4 - (2 \times 14.2) = 35$ | M1 | For either. No marks in (iii) unless using $\bar{x} + 2s$ or $\bar{x} - 2s$. Only follow through numerical values, not variables such as $s$, so if a candidate does not find $s$ but then writes here 'limit is $63.4 + 2 \times$ standard deviation', do NOT award M1. Do not penalise for over-specification. |
| $\bar{x} + 2s = 63.4 + (2 \times 14.2) = 91.8$ | A1 | For both (FT). Must include an element of doubt and must mention both ends. |
| So there are probably some outliers at the lower end, but none at the upper end | E1 | Must have correct limits to get this mark. |
| **Total** | **[3]** | |
## (iv)
| **Answer** | **Marks** | **Guidance** |
|---|---|---|
| Mean $= \frac{3624.5}{50} = 72.5g$ (or exact answer 72.49g) | B1 | CAO Ignore units. |
| $S_{xx} = 265416 - \frac{3624.5^2}{50} = 2676$ | M1 | For $S_{xx}$. M1 for 265416 - 50 × their mean². BUT NOTE M0 if their $S_{xx} < 0$ |
| $s = \sqrt{\frac{2676}{49}} = \sqrt{54.61} = 7.39g$ | A1 | CAO ignore units. Allow 7.4 but NOT 7.3 (unless RMSD with working). For s² of 54.6 (or better) allow M1A0 with or without working. For RMSD of 7.3 (or better) allow M1A0 provided working seen. For RMSD² of 53.5 (or better) allow M1A0 provided working seen. |
| **Total** | **[3]** | |
## (v)
| **Answer** | **Marks** | **Guidance** |
|---|---|---|
| Variety A have lower average than Variety B oe | E1 | FT their means. Do not condone lower central tendency or lower mean. Allow 'on the whole' or similar in place of 'average'. |
| Variety A have higher variation than Variety B oe | E1 | FT their sd. Allow 'more spread' or similar but not 'higher range' or 'higher variance'. Condone less consistent. |
| **Total** | **[2]** | |
---The weights, $w$ grams, of a random sample of 60 carrots of variety A are summarised in the table below.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Weight & $30 \leqslant w < 50$ & $50 \leqslant w < 60$ & $60 \leqslant w < 70$ & $70 \leqslant w < 80$ & $80 \leqslant w < 90$ \\
\hline
Frequency & 11 & 10 & 18 & 14 & 7 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\item Draw a histogram to illustrate these data. [5]
\item Calculate estimates of the mean and standard deviation of $w$. [4]
\item Use your answers to part (ii) to investigate whether there are any outliers. [3]
\end{enumerate}
The weights, $x$ grams, of a random sample of 50 carrots of variety B are summarised as follows.
$$n = 50 \quad \sum x = 3624.5 \quad \sum x^2 = 265416$$
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Calculate the mean and standard deviation of $x$. [3]
\item Compare the central tendency and variation of the weights of varieties A and B. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S1 2014 Q6 [17]}}