| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Calculate using histogram bar dimensions |
| Difficulty | Moderate -0.3 This is a standard S1 histogram and boxplot question requiring routine application of frequency density calculations, linear interpolation for quartiles, and boxplot construction. While multi-part with several steps, all techniques are textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread |
| Number of patients | 10 - 19 | 20 - 29 | 30 - 34 | 35 - 39 | 40 - 44 | 45 - 49 | 50 - 69 |
| Frequency | 2 | 18 | 24 | 30 | 27 | 14 | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| 20 - 29: class width 10 \(\rightarrow\) 2 cm \(\therefore\) class width 5 \(\rightarrow\) 1 cm | M1 | |
| freq. den. = \(\frac{18}{10} = 1.8 \rightarrow 7.2\) cm \(\therefore\) freq. den. 1 \(\rightarrow\) 4 cm | M1 | |
| (i) 30 - 34: class width 5 \(\therefore\) width 1 cm | A1 | |
| freq. den. = \(\frac{24}{5} = 4.8 \therefore\) height 19.2 cm | A1 | |
| (ii) 50 - 69: class width 20 \(\therefore\) width 4 cm | A1 | |
| freq. den. = \(\frac{5}{20} = 0.25 \therefore\) height 1 cm | A1 | |
| cum. freqs: 2, 20, 44, 74, 101, 115, 120 | M1 | |
| \(Q_1 = 30.25^{\text{th}} = 29.5 + 5(\frac{10.25}{22}) = 31.61\) [30th \(\rightarrow\) 31.6] | M2 A3 | |
| \(Q_2 = 60.5^{\text{th}} = 34.5 + 5(\frac{16.5}{30}) = 37.3\) [60th \(\rightarrow\) 37.2] | ||
| \(Q_3 = 90.75^{\text{th}} = 39.5 + 5(\frac{16.75}{27}) = 42.6\) [90th \(\rightarrow\) 42.5] | ||
| [Boxplot shown] | B4 | |
| symmetrical (or slight +ve skew) | A1 | (17) |
| 20 - 29: class width 10 $\rightarrow$ 2 cm $\therefore$ class width 5 $\rightarrow$ 1 cm | M1 | |
| freq. den. = $\frac{18}{10} = 1.8 \rightarrow 7.2$ cm $\therefore$ freq. den. 1 $\rightarrow$ 4 cm | M1 | |
| (i) 30 - 34: class width 5 $\therefore$ width 1 cm | A1 | |
| freq. den. = $\frac{24}{5} = 4.8 \therefore$ height 19.2 cm | A1 | |
| (ii) 50 - 69: class width 20 $\therefore$ width 4 cm | A1 | |
| freq. den. = $\frac{5}{20} = 0.25 \therefore$ height 1 cm | A1 | |
| cum. freqs: 2, 20, 44, 74, 101, 115, 120 | M1 | |
| $Q_1 = 30.25^{\text{th}} = 29.5 + 5(\frac{10.25}{22}) = 31.61$ [30th $\rightarrow$ 31.6] | M2 A3 | |
| $Q_2 = 60.5^{\text{th}} = 34.5 + 5(\frac{16.5}{30}) = 37.3$ [60th $\rightarrow$ 37.2] | | |
| $Q_3 = 90.75^{\text{th}} = 39.5 + 5(\frac{16.75}{27}) = 42.6$ [90th $\rightarrow$ 42.5] | | |
| [Boxplot shown] | B4 | |
| symmetrical (or slight +ve skew) | A1 | (17) |The number of patients attending a hospital trauma clinic each day was recorded over several months, giving the data in the table below.
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Number of patients & 10 - 19 & 20 - 29 & 30 - 34 & 35 - 39 & 40 - 44 & 45 - 49 & 50 - 69 \\
\hline
Frequency & 2 & 18 & 24 & 30 & 27 & 14 & 5 \\
\hline
\end{tabular}
These data are represented by a histogram.
Given that the bar representing the 20 - 29 group is 2 cm wide and 7.2 cm high,
\begin{enumerate}[label=(\alph*)]
\item calculate the dimensions of the bars representing the groups
\begin{enumerate}[label=(\roman*)]
\item 30 - 34
\item 50 - 69
\end{enumerate} [6]
\item Use linear interpolation to estimate the median and quartiles of these data. [6]
\end{enumerate}
The lowest and highest numbers of patients recorded were 14 and 67 respectively.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Represent these data with a boxplot drawn on graph paper and describe the skewness of the distribution. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q5 [17]}}