| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate statistics from discrete frequency table |
| Difficulty | Easy -1.3 This is a straightforward S1 statistics question requiring standard calculations from a discrete frequency table: mode (read off), range (subtract), median/IQR (cumulative frequency), mean/SD (formula application), and a simple linear transformation. All techniques are routine recall with no problem-solving or insight required. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| Total number of eggs laid in a week ( \(\boldsymbol { x }\) ) | Number of weeks ( f) |
| 66 | 1 |
| 67 | 2 |
| 68 | 3 |
| 69 | 5 |
| 70 | 7 |
| 71 | 8 |
| 72 | 4 |
| 73 | 2 |
| 74 | 2 |
| 75 | 1 |
| Total | 35 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mode \(= 71\) | B1 | |
| Range \(= 75 - 66 = 9\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Median = 18th value \(= 70\) | B1 | |
| \(Q_1\) = 9th value \(= 69\), \(Q_3\) = 27th value \(= 71\) | M1 | For attempting to find \(Q_1\) and \(Q_3\) |
| \(IQR = 71 - 69 = 2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\bar{x} = \frac{\sum fx}{\sum f} = \frac{2450}{35} = 70\) | M1 A1 | M1 for \(\frac{\sum fx}{\sum f}\) |
| \(\sigma^2 = \frac{\sum fx^2}{\sum f} - \bar{x}^2 = \frac{171554}{35} - 70^2 = 4902.97... - 4900 = 2.971...\) | M1 | |
| \(\sigma = \sqrt{2.971...} \approx 1.72\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mean \(= 70 - 60 = 10\) | B1 | |
| Standard deviation \(= 1.72\) (unchanged) | B1 | SD unaffected by linear shift |
# Question 1:
## Part (a)(i) - Mode and Range
| Answer | Mark | Guidance |
|--------|------|----------|
| Mode $= 71$ | B1 | |
| Range $= 75 - 66 = 9$ | B1 | |
## Part (a)(ii) - Median and Interquartile Range
| Answer | Mark | Guidance |
|--------|------|----------|
| Median = 18th value $= 70$ | B1 | |
| $Q_1$ = 9th value $= 69$, $Q_3$ = 27th value $= 71$ | M1 | For attempting to find $Q_1$ and $Q_3$ |
| $IQR = 71 - 69 = 2$ | A1 | |
## Part (a)(iii) - Mean and Standard Deviation
| Answer | Mark | Guidance |
|--------|------|----------|
| $\bar{x} = \frac{\sum fx}{\sum f} = \frac{2450}{35} = 70$ | M1 A1 | M1 for $\frac{\sum fx}{\sum f}$ |
| $\sigma^2 = \frac{\sum fx^2}{\sum f} - \bar{x}^2 = \frac{171554}{35} - 70^2 = 4902.97... - 4900 = 2.971...$ | M1 | |
| $\sigma = \sqrt{2.971...} \approx 1.72$ | A1 | |
## Part (b) - Mean and SD of eggs kept
| Answer | Mark | Guidance |
|--------|------|----------|
| Mean $= 70 - 60 = 10$ | B1 | |
| Standard deviation $= 1.72$ (unchanged) | B1 | SD unaffected by linear shift |
---1 Henrietta lives on a small farm where she keeps some hens.\\
For a period of 35 weeks during the hens' first laying season, she records, each week, the total number of eggs laid by the hens.
Her records are shown in the table.
\begin{center}
\begin{tabular}{|l|l|}
\hline
Total number of eggs laid in a week ( $\boldsymbol { x }$ ) & Number of weeks ( f) \\
\hline
66 & 1 \\
\hline
67 & 2 \\
\hline
68 & 3 \\
\hline
69 & 5 \\
\hline
70 & 7 \\
\hline
71 & 8 \\
\hline
72 & 4 \\
\hline
73 & 2 \\
\hline
74 & 2 \\
\hline
75 & 1 \\
\hline
Total & 35 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item For these data:
\begin{enumerate}[label=(\roman*)]
\item state values for the mode and the range;
\item find values for the median and the interquartile range;
\item calculate values for the mean and the standard deviation.
\end{enumerate}\item Each week, for the 35 weeks, Henrietta sells 60 eggs to a local shop, keeping the remainder for her own use.
State values for the mean and the standard deviation of the number of eggs that she keeps.\\[0pt]
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2014 Q1 [11]}}