| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Cumulative frequency graph construction then interpretation |
| Difficulty | Moderate -0.8 This is a standard S1 question testing routine procedures: finding median by interpolation from grouped data, calculating mean and standard deviation from given summations using formulas, and commenting on skewness using the mean-median relationship. All techniques are direct textbook applications with no problem-solving or novel insight required, making it easier than average. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| Time, \(t\) | Number of students |
| 10-19 | 2 |
| 20-29 | 4 |
| 30-39 | 8 |
| 40-49 | 11 |
| 50-69 | 5 |
| 70-79 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Median \(= 32/2 = 16^{\text{th}}\) term (16.5) | ||
| \(\frac{x - 39.5}{49.5 - 39.5} = \frac{16-14}{25-14}\) or \(x = 39.5 + \left(\frac{2}{11} \times 10\right)\) | M1 | For attempt to use interpolation. Condone use of 39 or 40 for 39.5. e.g. allow \(39 + \frac{2}{11} \times 10\) or \(40 + \frac{2}{11} \times 10\) to score M1A0 but must have the 10 |
| Median \(= 41.3\) (use of \(n+1\) gives 41.8) (awrt 41.3) | A1 | For awrt 41.3 (or awrt 41.8 if using \((n+1)\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Mean \(= \frac{1414}{32} = 44.1875\) (awrt 44.2) | B1 | For awrt 44.2. A correct expression for \(E(X)\) or \(E(X^2)\) that is later divided by 4 scores M0 |
| Standard deviation \(= \sqrt{\frac{69378}{32} - \left(\frac{1414}{32}\right)^2}\) | M1 | For correct expression including square root (allow ft of their mean). Mid-points: scores B0 for mean but can score M1 for correct st.dev expression |
| \(= 14.7\) (or \(s = 14.9\)) | A1 | For awrt 14.7 (if using \(s\), awrt 14.9). You may see \(\sum t = 1339 \to \bar{t} = 41.8\) and \(\sum t^2 = 62928 \to \sigma\ 14.7\) or \(s = 14.9\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| mean \(>\) median therefore positive skew | B1ft B1ft | 1st B1ft for correct comparison of their mean and median. 2nd B1ft for correct description of skewness based on their values. Only allow comparison \(\approx 0\) if \( |
# Question 5:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Median $= 32/2 = 16^{\text{th}}$ term (16.5) | | |
| $\frac{x - 39.5}{49.5 - 39.5} = \frac{16-14}{25-14}$ or $x = 39.5 + \left(\frac{2}{11} \times 10\right)$ | M1 | For attempt to use interpolation. Condone use of 39 or 40 for 39.5. e.g. allow $39 + \frac{2}{11} \times 10$ or $40 + \frac{2}{11} \times 10$ to score M1A0 but must have the 10 |
| Median $= 41.3$ (use of $n+1$ gives 41.8) (awrt 41.3) | A1 | For awrt 41.3 (or awrt 41.8 if using $(n+1)$) |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Mean $= \frac{1414}{32} = 44.1875$ (awrt 44.2) | B1 | For awrt 44.2. A correct expression for $E(X)$ or $E(X^2)$ that is later divided by 4 scores M0 |
| Standard deviation $= \sqrt{\frac{69378}{32} - \left(\frac{1414}{32}\right)^2}$ | M1 | For correct expression including square root (allow ft of their mean). Mid-points: scores B0 for mean but can score M1 for correct st.dev expression |
| $= 14.7$ (or $s = 14.9$) | A1 | For awrt 14.7 (if using $s$, awrt 14.9). You may see $\sum t = 1339 \to \bar{t} = 41.8$ and $\sum t^2 = 62928 \to \sigma\ 14.7$ or $s = 14.9$ |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| mean $>$ median therefore positive skew | B1ft B1ft | 1st B1ft for correct comparison of their mean and median. 2nd B1ft for correct description of skewness based on their values. Only allow comparison $\approx 0$ if $|\text{mean} - \text{median}| \leq 0.5$. "Positive correlation" is B0. Quartiles: 1st B1ft if $Q_1$ = awrt 32 and $Q_3$ = awrt 49 seen and correct comparison made |
---5. On a randomly chosen day, each of the 32 students in a class recorded the time, $t$ minutes to the nearest minute, they spent on their homework. The data for the class is summarised in the following table.
\begin{center}
\begin{tabular}{|l|l|}
\hline
Time, $t$ & Number of students \\
\hline
10-19 & 2 \\
\hline
20-29 & 4 \\
\hline
30-39 & 8 \\
\hline
40-49 & 11 \\
\hline
50-69 & 5 \\
\hline
70-79 & 2 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use interpolation to estimate the value of the median.
Given that
$$\sum t = 1414 \quad \text { and } \quad \sum t ^ { 2 } = 69378$$
\item find the mean and the standard deviation of the times spent by the students on their homework.
\item Comment on the skewness of the distribution of the times spent by the students on their homework. Give a reason for your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2011 Q5 [7]}}