| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate statistics from discrete frequency table |
| Difficulty | Moderate -0.8 This is a routine S1 question testing standard calculations from a frequency table with grouped data. It requires straightforward application of formulas for mode, median, quartiles, mean, and standard deviation, plus brief interpretation. The presence of grouped classes adds minor complexity but this is a standard textbook exercise with no problem-solving or novel insight required. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| Number of books on loan | 0 | 1 | 2 | 3 | 4 | \(5 - 9\) | \(10 - 14\) | 15 |
| Number of members | 4 | 13 | 24 | 17 | 15 | 11 | 5 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mode \(= 2\) | B1 | CAO |
| Range \(= 15\) | B1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| CF: 4 17 41 58 73 84 89 95; \(x\): 0 1 2 3 4 9 14 15 | — | — |
| Median (\(48^{th}\)) \(= 3\) | B2 | CAO; B0 if shown method is incorrect |
| IQR: \((72^{nd} - 24^{th}) = 4 - 2 = 2\) | B2 | CAO; Allow B1 for identification of 4 and 2; B0 if shown method is incorrect |
| If neither correct but CF attempted and matched correctly with \(\geq 5\) \(x\)-values | (M1)(A1) | Allow for median \(= 2 + \frac{x}{17}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mean \((\bar{x}) = 4.2\) | B2 | CAO; \(\sum fx = 399\) |
| Standard Deviation \((s_n, s_{n-1}) = 3.88\) to \(3.91\) | B2 | AWFW; \(\sum fx^2 = 3111\); (3.887 or 3.907) |
| If neither correct but mid-points of 7 and 12 seen | (B1) | — |
| and use of mean \((\bar{x}) = \frac{\sum fx}{95}\) | (M1) | Allow for \(4.1 \leq \bar{x} \leq 4.3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Unknown values (16) have no effect on median and IQR or median and IQR are exact values but \(\bar{x}\) and \(s\) are estimates | B1 | — |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use all available data or Enable further analyses | B1 | — |
# Question 4:
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Mode $= 2$ | B1 | CAO |
| Range $= 15$ | B1 | CAO |
## Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| CF: 4 17 41 58 73 84 89 95; $x$: 0 1 2 3 4 9 14 15 | — | — |
| Median ($48^{th}$) $= 3$ | B2 | CAO; B0 if shown method is incorrect |
| IQR: $(72^{nd} - 24^{th}) = 4 - 2 = 2$ | B2 | CAO; Allow B1 for identification of 4 and 2; B0 if shown method is incorrect |
| If neither correct but CF attempted and matched correctly with $\geq 5$ $x$-values | (M1)(A1) | Allow for median $= 2 + \frac{x}{17}$ |
## Part (a)(iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Mean $(\bar{x}) = 4.2$ | B2 | CAO; $\sum fx = 399$ |
| Standard Deviation $(s_n, s_{n-1}) = 3.88$ to $3.91$ | B2 | AWFW; $\sum fx^2 = 3111$; (3.887 or 3.907) |
| If neither correct but mid-points of 7 and 12 seen | (B1) | — |
| and use of mean $(\bar{x}) = \frac{\sum fx}{95}$ | (M1) | Allow for $4.1 \leq \bar{x} \leq 4.3$ |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Unknown values (16) have no effect on median and IQR **or** median and IQR are exact values but $\bar{x}$ and $s$ are estimates | B1 | — |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Use all available data **or** Enable further analyses | B1 | — |
---4 A library allows each member to have up to 15 books on loan at any one time.
The table shows the numbers of books currently on loan to a random sample of 95 members of the library.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
Number of books on loan & 0 & 1 & 2 & 3 & 4 & $5 - 9$ & $10 - 14$ & 15 \\
\hline
Number of members & 4 & 13 & 24 & 17 & 15 & 11 & 5 & 6 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item For these data:
\begin{enumerate}[label=(\roman*)]
\item state values for the mode and range;
\item determine values for the median and interquartile range;
\item calculate estimates of the mean and standard deviation.
\end{enumerate}\item Making reference to your answers to part (a), give a reason for preferring:
\begin{enumerate}[label=(\roman*)]
\item the median and interquartile range to the mean and standard deviation for summarising the given data;
\item the mean and standard deviation to the mode and range for summarising the given data.\\
(1 mark)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2007 Q4 [12]}}