| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw histogram then find median/quartiles from cumulative frequency |
| Difficulty | Moderate -0.8 This is a routine S1 statistics question testing standard procedures: calculating mid-points (trivial arithmetic), drawing histogram bars using frequency density (direct formula application), computing mean/SD from grouped data (calculator work), linear interpolation for median (standard algorithm), and identifying skewness from quartile positions. All parts follow textbook methods with no problem-solving or insight required, making it easier than average A-level questions. |
| Spec | 2.02b Histogram: area represents frequency2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| Hours | \(1 - 10\) | \(11 - 20\) | \(21 - 25\) | \(26 - 30\) | \(31 - 40\) | \(41 - 59\) |
| Frequency | 6 | 15 | 11 | 13 | 8 | 3 |
| Mid-point | 5.5 | 15.5 | 28 | 50 |
**Question 5(c):** 1st M1 for reasonable attempt at $\sum x$ and /56. 2nd M1 for a method for $\sigma$ or $s$, $\sqrt{\quad}$ is required. Typical errors: $\sum(fx)^2 = 354806.3$ M0, $\sum f^2 x = 13922.5$ M0 and $(\sum fx)^2 = 1733172$ M0. Correct answers only, award full marks.
**Question 5(d):** Use of $\sum f(x - \bar{x})^2 = \text{awrt } 6428.75$ for B1. lcb can be 20, 20.5 or 21, width can be 4 or 5 and the fraction part of the formula correct for M1. Allow 28.5 in fraction that gives awrt 23.9 for M1A1.
**Question 5(e):** M1for attempting a test for skewness using quartiles or mean and median. Provided median greater than 22.55 and less than 29.3 award for M1 for $Q_3 - Q_2 < Q_2 - Q_1$ without values as a valid reason. **SC** Accept mean close to median and no skew oe for M1A1.5. A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
Hours & $1 - 10$ & $11 - 20$ & $21 - 25$ & $26 - 30$ & $31 - 40$ & $41 - 59$ \\
\hline
Frequency & 6 & 15 & 11 & 13 & 8 & 3 \\
\hline
Mid-point & 5.5 & 15.5 & & 28 & & 50 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the mid-points of the 21-25 hour and 31-40 hour groups.
A histogram was drawn to represent these data. The $11 - 20$ group was represented by a bar of width 4 cm and height 6 cm .
\item Find the width and height of the 26-30 group.
\item Estimate the mean and standard deviation of the time spent watching television by these students.
\item Use linear interpolation to estimate the median length of time spent watching television by these students.
The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.
\item State, giving a reason, the skewness of these data.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2010 Q5 [14]}}