| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate statistics from raw data |
| Difficulty | Moderate -0.8 This is a straightforward multi-part statistics question requiring basic calculations: computing a mean from raw data, drawing a histogram from grouped data, calculating mean and standard deviation from a frequency table, and applying a linear transformation. All techniques are standard S1 content with no problem-solving insight required—purely procedural application of formulas. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Grade | A* | A | B | C | D | E | F | G | U |
| Points | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| Mean GCSE score | Number of students |
| \(4.5 \leqslant X < 5.5\) | 8 |
| \(5.5 \leqslant X < 6.0\) | 14 |
| \(6.0 \leqslant X < 6.5\) | 19 |
| \(6.5 \leqslant X < 7.0\) | 13 |
| \(7.0 \leqslant X \leqslant 8.0\) | 6 |
| AS Grade | A | B | C | D | E | U |
| Score | 60 | 50 | 40 | 30 | 20 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mean score \(= \frac{2 \times 8 + 3 \times 7 + 4 \times 6 + 5 + 4}{11} = 6.36\) | M1 | for \(\sum fx / 11\) |
| \(= 6.36\) | A1 CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Linear sensible scales | G1 | |
| Frequency densities of 8, 28, 38, 26, 6 or \(4k, 14k, 19k, 13k, 3k\) for sensible values of \(k\) | G1 | either on script or on graph |
| Appropriate label for vertical scale e.g. 'Frequency density', 'frequency per \(\frac{1}{2}\) unit', 'students per mean GCSE score' | G1 | dep on reasonable attempt at fd; allow Key |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct mid points: 5, 5.75, 6.25, 6.75, 7.5 | B1 | |
| \(\sum fx = 372\), \(\sum fx^2 = 2334.875\) | B1FT | |
| Sample mean \(= 372/60 = 6.2\) | B1 CAO | |
| \(S_{xx} = 2334.875 - \frac{372^2}{60} = 28.475\) | M1 | for their \(S_{xx}\) |
| Sample s.d. \(= \sqrt{\frac{28.475}{59}} = 0.695\) | A1 CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Prediction of score \(= 13 \times 7.4 - 46 = 50.2\) | M1 | for \(13 \times 7.4 - 46\) |
| So predicted AS grade would be B | A1 | dep on 50.2 (or 50) seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Prediction of score \(= 13 \times 5.5 - 46 = 25.5\) | M1 | for \(13 \times 5.5 - 46\) |
| So predicted grade would be D/E (allow D or E) | A1 | dep on 25.5 (or 26 or 25) seen |
| Because score roughly halfway from 20 to 30, OR (for D) closer to D than E, OR (for E) past E but not up to D boundary | E1 | explanation of conversion — logical statement/argument supporting choice |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mean \(= 13 \times 6.2 - 46 = 34.6\) | B1 | FT their 6.2 |
| Standard deviation \(= 13 \times 0.695 = 9.035\) | M1 | for \(13 \times\) their 0.695 |
| A1 FT |
## Question 3:
**(i)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Mean score $= \frac{2 \times 8 + 3 \times 7 + 4 \times 6 + 5 + 4}{11} = 6.36$ | M1 | for $\sum fx / 11$ |
| $= 6.36$ | A1 CAO | |
**(ii)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Linear sensible scales | G1 | |
| Frequency densities of 8, 28, 38, 26, 6 or $4k, 14k, 19k, 13k, 3k$ for sensible values of $k$ | G1 | either on script or on graph |
| Appropriate label for vertical scale e.g. 'Frequency density', 'frequency per $\frac{1}{2}$ unit', 'students per mean GCSE score' | G1 | dep on reasonable attempt at fd; allow Key |
**(iii)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct mid points: 5, 5.75, 6.25, 6.75, 7.5 | B1 | |
| $\sum fx = 372$, $\sum fx^2 = 2334.875$ | B1FT | |
| Sample mean $= 372/60 = 6.2$ | B1 CAO | |
| $S_{xx} = 2334.875 - \frac{372^2}{60} = 28.475$ | M1 | for their $S_{xx}$ |
| Sample s.d. $= \sqrt{\frac{28.475}{59}} = 0.695$ | A1 CAO | |
**(iv)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Prediction of score $= 13 \times 7.4 - 46 = 50.2$ | M1 | for $13 \times 7.4 - 46$ |
| So predicted AS grade would be B | A1 | dep on 50.2 (or 50) seen |
**(v)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Prediction of score $= 13 \times 5.5 - 46 = 25.5$ | M1 | for $13 \times 5.5 - 46$ |
| So predicted grade would be D/E (allow D or E) | A1 | dep on 25.5 (or 26 or 25) seen |
| Because score roughly halfway from 20 to 30, OR (for D) closer to D than E, OR (for E) past E but not up to D boundary | E1 | explanation of conversion — logical statement/argument supporting choice |
**(vi)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Mean $= 13 \times 6.2 - 46 = 34.6$ | B1 | FT their 6.2 |
| Standard deviation $= 13 \times 0.695 = 9.035$ | M1 | for $13 \times$ their 0.695 |
| | A1 FT | |
---3 At East Cornwall College, the mean GCSE score of each student is calculated. This is done by allocating a number of points to each GCSE grade in the following way.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
Grade & A* & A & B & C & D & E & F & G & U \\
\hline
Points & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\
\hline
\end{tabular}
\end{center}
(i) Calculate the mean GCSE score, $X$, of a student who has the following GCSE grades:
$$\mathrm { A } ^ { * } , \mathrm {~A} ^ { * } , \mathrm {~A} , \mathrm {~A} , \mathrm {~A} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm { C } , \mathrm { D } .$$
60 students study AS Mathematics at the college. The mean GCSE scores of these students are summarised in the table below.
\begin{center}
\begin{tabular}{|l|l|}
\hline
Mean GCSE score & Number of students \\
\hline
$4.5 \leqslant X < 5.5$ & 8 \\
\hline
$5.5 \leqslant X < 6.0$ & 14 \\
\hline
$6.0 \leqslant X < 6.5$ & 19 \\
\hline
$6.5 \leqslant X < 7.0$ & 13 \\
\hline
$7.0 \leqslant X \leqslant 8.0$ & 6 \\
\hline
\end{tabular}
\end{center}
(ii) Draw a histogram to illustrate this information.\\
(iii) Calculate estimates of the sample mean and the sample standard deviation.
The scoring system for AS grades is shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
AS Grade & A & B & C & D & E & U \\
\hline
Score & 60 & 50 & 40 & 30 & 20 & 0 \\
\hline
\end{tabular}
\end{center}
The Mathematics department at the college predicts each student's AS score, $Y$, using the formula $Y = 13 X - 46$, where $X$ is the student's average GCSE score.\\
(iv) What AS grade would the department predict for a student with an average GCSE score of 7.4 ?\\
(v) What do you think the prediction should be for a student with an average GCSE score of 5.5? Give a reason for your answer.\\
(vi) Using your answers to part (iii), estimate the sample mean and sample standard deviation of the predicted AS scores of the 60 students in the department.
\hfill \mbox{\textit{OCR MEI S1 Q3 [18]}}