| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2020 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate variance from summary statistics |
| Difficulty | Moderate -0.3 This is a straightforward multi-part statistics question testing standard A-level content: reading box plots, calculating mean/SD from summary statistics using formulas, and interpreting these measures. Parts (a)-(d) are routine calculations, (e) requires simple comparison, while (f)-(g) involve basic conceptual understanding of how additional data affects measures. All techniques are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \([68 - 7 =] \mathbf{61}\) (only) | B1 |
| (b) | \([25 - 14 =] \mathbf{11}\) | B1 |
| (c) | \(\left[\mu \text{ or } \bar{x} = \frac{607.5}{27} =\right] \mathbf{22.5}\) | B1 |
| (d) | \(\sigma = \sqrt{\frac{17623.25}{27} - "22.5"^2}\) or \(\sqrt{146.4629...}\) = \(12.10218...\) awrt \(\mathbf{12.1}\) | M1; A1 |
| (e) | \(\mu + 3\sigma = "22.5" + 3 \times "12.1..." = \) awrt 59 so only one outlier | B1ft |
| (f) | Median increases implies that both values must be \(> 20\). Mean is the same means that \(a + b = 45\). So possible values are: e.g. \(b = 21\) and \(a = 24\) (o.e.) | M1; M1; A1 |
| (g) | Both values will be less than 1 standard deviation from the mean and so the standard deviation of all 29 values will be smaller. | B1 |
| (a) | $[68 - 7 =] \mathbf{61}$ (only) | B1 | B1 for correctly interpreting the box plot to find the range (more than 1 answer is B0). |
|---|---|---|---|
| (b) | $[25 - 14 =] \mathbf{11}$ | B1 | B1 for correct understanding of IQR and answer of 11. |
| (c) | $\left[\mu \text{ or } \bar{x} = \frac{607.5}{27} =\right] \mathbf{22.5}$ | B1 | B1 for 22.5 only (or exact equivalent such as $\frac{45}{2}$). Allow 22 mins and 30 secs. |
| (d) | $\sigma = \sqrt{\frac{17623.25}{27} - "22.5"^2}$ or $\sqrt{146.4629...}$ = $12.10218...$ awrt $\mathbf{12.1}$ | M1; A1 | M1 for a correct expression including square root. Allow $\sqrt{146}$ or better. Ft their mean. NB Allow use of $s = 12.3327...$ or awrt 12.3. A1 for awrt 12.1. |
| (e) | $\mu + 3\sigma = "22.5" + 3 \times "12.1..." = $ awrt 59 so only **one outlier** | B1ft | B1ft for a correct calculation or value based on their $\mu$ and $\sigma$ and compatible conclusion. |
| (f) | Median increases implies that both values must be $> 20$. Mean is the same means that $a + b = 45$. So possible values are: e.g. $b = 21$ and $a = 24$ (o.e.) | M1; M1; A1 | 1st M1 Correct start to the problem and a correct statement about the values based on median. Allow if their final two values are both $> 20$. 2nd M1 for a correct explanation leading to equation $a + b = 45$ (o.e. e.g. equidistant from mean). A1 for a correct pair of values (both $> 20$ with a sum of 45) **and at least some attempt to explain** how their values satisfy at least one of the conditions (both $> 20$ or $a + b = 45$). Ignore $a =$ or $b =$ labels. |
| (g) | Both values will be less than 1 standard deviation from the mean and so the standard deviation of all 29 values will be smaller. | B1 | B1 for a correct explanation. Must mention that both values are less than 1 sd (ft their answer to (d)) from the mean. |\begin{enumerate}
\item Each member of a group of 27 people was timed when completing a puzzle.
\end{enumerate}
The time taken, $x$ minutes, for each member of the group was recorded.\\
These times are summarised in the following box and whisker plot.\\
\includegraphics[max width=\textwidth, alt={}, center]{2b63aa7f-bc50-4422-8dc0-e661b521c221-08_353_1436_458_319}\\
(a) Find the range of the times.\\
(b) Find the interquartile range of the times.
For these 27 people $\sum x = 607.5$ and $\sum x ^ { 2 } = 17623.25$\\
(c) calculate the mean time taken to complete the puzzle,\\
(d) calculate the standard deviation of the times taken to complete the puzzle.
Taruni defines an outlier as a value more than 3 standard deviations above the mean.\\
(e) State how many outliers Taruni would say there are in these data, giving a reason for your answer.
Adam and Beth also completed the puzzle in $a$ minutes and $b$ minutes respectively, where $a > b$.\\
When their times are included with the data of the other 27 people
\begin{itemize}
\item the median time increases
\item the mean time does not change\\
(f) Suggest a possible value for $a$ and a possible value for $b$, explaining how your values satisfy the above conditions.\\
(g) Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).
\end{itemize}
\hfill \mbox{\textit{Edexcel Paper 3 2020 Q3 [10]}}