| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate statistics from raw data |
| Difficulty | Easy -1.2 This is a straightforward S1 statistics question requiring only standard calculations (mean, standard deviation, median, IQR) from given data, plus brief explanations about appropriateness of measures. All techniques are routine recall with no problem-solving or novel insight required, making it easier than average A-level maths questions. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| 17 | 19 | 22 | 26 | 28 | 31 | 34 | 36 | 38 | 39 |
| 41 | 42 | 43 | 47 | 50 | 51 | 53 | 55 | 57 | 58 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Mean \(\bar{x} = 39.3\) to \(39.4\) | B1 | AWFW (39.35) |
| Standard Deviation \((s_n, s_{n-1}) = 12.3\) to \(12.7\) | B2 | AWFW (12.358 or 12.679) |
| If neither correct but working shown: Mean \(\bar{x} = \frac{\sum x}{20}\) | (M1) | \(\sum x = 787\), \(\sum x^2 = 34023\); Used |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Median \(= 42\) | B2 | CAO |
| Median \(= 41.5\) or \(39\) or \(40\) | (B1) | CAO |
| Interquartile Range \(= 55 - 31 = 24\) | B2 | CAO; allow B1 for identification of 31 and 55; B0 if method shown is incorrect |
| Interquartile Range \(= 21\) to \(27\) | (B1) | AWFW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Mode: e.g. Does not exist; If exists, must be \(> 60\) or \(58\); All/too many different values; Sparse data | B1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Range: e.g. Maximum value is unknown \(/ > 60\) or \(58\) | B1 | OE; accept 'slowest' but not 'smallest' |
# Question 1:
## Part 1(a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Mean $\bar{x} = 39.3$ to $39.4$ | B1 | AWFW (39.35) |
| Standard Deviation $(s_n, s_{n-1}) = 12.3$ to $12.7$ | B2 | AWFW (12.358 or 12.679) |
| If **neither** correct **but** working shown: Mean $\bar{x} = \frac{\sum x}{20}$ | (M1) | $\sum x = 787$, $\sum x^2 = 34023$; Used |
## Part 1(b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Median $= 42$ | B2 | CAO |
| Median $= 41.5$ or $39$ or $40$ | (B1) | CAO |
| Interquartile Range $= 55 - 31 = 24$ | B2 | CAO; allow B1 for identification of 31 and 55; B0 if method shown is incorrect |
| Interquartile Range $= 21$ to $27$ | (B1) | AWFW |
## Part 1(c)(i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| **Mode:** e.g. Does not exist; If exists, must be $> 60$ or $58$; All/too many different values; Sparse data | B1 | OE |
## Part 1(c)(ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| **Range:** e.g. **Maximum value** is unknown $/ > 60$ or $58$ | B1 | OE; accept 'slowest' but not 'smallest' |
---1 The times, in seconds, taken by 20 people to solve a simple numerical puzzle were
\begin{center}
\begin{tabular}{ l l l l l l l l l l }
17 & 19 & 22 & 26 & 28 & 31 & 34 & 36 & 38 & 39 \\
41 & 42 & 43 & 47 & 50 & 51 & 53 & 55 & 57 & 58 \\
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the mean and the standard deviation of these times.
\item In fact, 23 people solved the puzzle. However, 3 of them failed to solve it within the allotted time of 60 seconds.
Calculate the median and the interquartile range of the times taken by all 23 people.\\
(4 marks)
\item For the times taken by all 23 people, explain why:
\begin{enumerate}[label=(\roman*)]
\item the mode is not an appropriate numerical measure;
\item the range is not an appropriate numerical measure.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2007 Q1 [9]}}