| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Coding to simplify calculation |
| Difficulty | Moderate -0.8 This is a routine S1 statistics question testing standard procedures: finding midpoints, using given summations to calculate mean and standard deviation with coding, linear interpolation for median, and reverse-coding. All techniques are straightforward textbook applications with no problem-solving insight required. The multi-part structure (6 parts) adds length but not conceptual difficulty. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02j Clean data: missing data, errors |
| Coded Time (v) | Frequency ( \(\boldsymbol { f }\) ) | Coded Time Midpoint (m) |
| \(0 \leqslant v < 5\) | 20 | 2.5 |
| \(5 \leqslant v < 10\) | 24 | \(a\) |
| \(10 \leqslant v < 15\) | 16 | 12.5 |
| \(15 \leqslant v < 20\) | 14 | 17.5 |
| \(20 \leqslant v < 30\) | 6 | \(b\) |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | 7.5 and 25 | B1 |
| (b) | Mean \(= 10.3125\) | B1 |
| (c) | \(\sigma = \sqrt{\frac{120125}{80} - '10.3125'^2}\) | M1 |
| \(= 6.6188...\) \((s = 6.6605...)\) | A1 | A1 for awrt 6.62 (Allow \(s =\) awrt 6.66) |
| (d) | \(\text{Median} = \{5\} + \frac{20}{24} \times 5\) or \( | \{10\} |
| \(= 9.16666\) | A1 | A1 for awrt 9.17 or (if using \(n+1\)) for awrt 9.27 |
| (e) | Mean > median \(\therefore\) positive skew | M1A1 |
| (f) | \(t = 10v + 5\) | M1 |
| Mean \(= 10 \times 10.3125 + 5 = 108.125\) | A1 | |
| \(\sigma = 10 \times 6.6188\) | M1 | 2nd M1 for \(10 \times\) 'their sd' |
| \(= 66.188...\) (66.605 from \(s\)) | A1 | A1 awrt 66.2 |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | 7.5 and 25 | B1 | B1 both values correct (may be seen in table) |
| (b) | Mean $= 10.3125$ | B1 | B1 for awrt 10.3 (Do not allow improper fractions) |
| (c) | $\sigma = \sqrt{\frac{120125}{80} - '10.3125'^2}$ | M1 | M1 for correct expression including square root (allow ft from their mean) |
| | $= 6.6188...$ $(s = 6.6605...)$ | A1 | A1 for awrt 6.62 (Allow $s =$ awrt 6.66) |
| (d) | $\text{Median} = \{5\} + \frac{20}{24} \times 5$ or $|\{10\}| - \frac{4}{24} \times 5$ | M1 | M1 for correct fraction: $\frac{20}{24} \times 5$ or if using $n+1$ for $\frac{20.5}{24} \times 5$ |
| | $= 9.16666$ | A1 | A1 for awrt 9.17 or (if using $n+1$) for awrt 9.27 |
| (e) | Mean > median $\therefore$ positive skew | M1A1 | M1 for correct comparison of 'their b' and 'their d' (must have answer to both b and d). Comparison may be part of bigger expression e.g. $3(\text{mean} - \text{median})/\text{s.d.}$ Allow use of $Q_3 - Q_2 > Q_2 - Q_1$ only if $Q_1 = 5$ and $Q_3 = 15$ are both seen. A1 for positive skew (which must follow from their values) |
| (f) | $t = 10v + 5$ | M1 | 1st M1 for $10 \times$ 'their mean'$+5$ |
| | Mean $= 10 \times 10.3125 + 5 = 108.125$ | A1 | |
| | $\sigma = 10 \times 6.6188$ | M1 | 2nd M1 for $10 \times$ 'their sd' |
| | $= 66.188...$ (66.605 from $s$) | A1 | A1 awrt 66.2 |
**Total: 12 marks**
Notes: For part (d), if using $n+1$, A1 for awrt 9.27
---4. A researcher recorded the time, $t$ minutes, spent using a mobile phone during a particular afternoon, for each child in a club.
The researcher coded the data using $v = \frac { t - 5 } { 10 }$ and the results are summarised in the table below.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Coded Time (v) & Frequency ( $\boldsymbol { f }$ ) & Coded Time Midpoint (m) \\
\hline
$0 \leqslant v < 5$ & 20 & 2.5 \\
\hline
$5 \leqslant v < 10$ & 24 & $a$ \\
\hline
$10 \leqslant v < 15$ & 16 & 12.5 \\
\hline
$15 \leqslant v < 20$ & 14 & 17.5 \\
\hline
$20 \leqslant v < 30$ & 6 & $b$ \\
\hline
\end{tabular}
\end{center}
$$\text { (You may use } \sum f m = 825 \text { and } \sum f m ^ { 2 } = 12012.5 \text { ) }$$
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $a$ and the value of $b$.
\item Calculate an estimate of the mean of $v$.
\item Calculate an estimate of the standard deviation of $v$.
\item Use linear interpolation to estimate the median of $v$.
\item Hence describe the skewness of the distribution. Give a reason for your answer.
\item Calculate estimates of the mean and the standard deviation of the time spent using a mobile phone during the afternoon by the children in this club.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2016 Q4 [12]}}