| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw histogram from frequency table |
| Difficulty | Moderate -0.8 This is a straightforward statistics question requiring standard histogram construction with unequal class widths (calculating frequency densities), mean estimation from grouped data using midpoints, and finding median/quartile positions. All techniques are routine S1 procedures with no problem-solving or novel insight required, making it easier than average but not trivial due to the multi-part nature and need for careful calculation. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread |
| Capacity | \(3000 - 7000\) | \(8000 - 12000\) | \(13000 - 22000\) | \(23000 - 42000\) | \(43000 - 82000\) |
| Number of stadiums | 40 | 30 | 18 | 34 | 8 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| cw: 5, 5, 10, 20, 40 | M1 | cw either 4 or 5 etc |
| fd: 8, 6, 1.8, 1.7, 0.2 | M1 | fd or scaled freq [f/their cw attempt]; fd may be \(\div 1000\) |
| Correct heights on diagram | A1 | Correct heights seen accurately on diagram |
| Correct bar ends on axis | B1 | Correct bar ends, accurately plotted on axis |
| Labels fd and capacity (thousands); correct horizontal scale; vertical scale linear from 0 | B1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((5 \times 40 + 10 \times 30 + 17.5 \times 18 + 32.5 \times 34 + 62.5 \times 8)/130\) | M1 | \(\Sigma fx/130\) where \(x\) is mid point attempt (value within class, not end pt or cw) |
| \(= 2420/130 = 18.6\) thousand | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| median group \(= 8 - 12\) thousand | B1 | Thousands not needed |
| LQ group \(= 3 - 7\) thousand | B1 [2] |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| cw: 5, 5, 10, 20, 40 | M1 | cw either 4 or 5 etc |
| fd: 8, 6, 1.8, 1.7, 0.2 | M1 | fd or scaled freq [f/their cw attempt]; fd may be $\div 1000$ |
| Correct heights on diagram | A1 | Correct heights seen accurately on diagram |
| Correct bar ends on axis | B1 | Correct bar ends, accurately plotted on axis |
| Labels fd and capacity (thousands); correct horizontal scale; vertical scale linear from 0 | B1 [5] | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(5 \times 40 + 10 \times 30 + 17.5 \times 18 + 32.5 \times 34 + 62.5 \times 8)/130$ | M1 | $\Sigma fx/130$ where $x$ is mid point attempt (value within class, not end pt or cw) |
| $= 2420/130 = 18.6$ thousand | A1 [2] | |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| median group $= 8 - 12$ thousand | B1 | Thousands not needed |
| LQ group $= 3 - 7$ thousand | B1 [2] | |
---5 The number of people a football stadium can hold is called the 'capacity'. The capacities of 130 football stadiums in the UK, to the nearest thousand, are summarised in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Capacity & $3000 - 7000$ & $8000 - 12000$ & $13000 - 22000$ & $23000 - 42000$ & $43000 - 82000$ \\
\hline
Number of stadiums & 40 & 30 & 18 & 34 & 8 \\
\hline
\end{tabular}
\end{center}
(i) On graph paper, draw a histogram to represent this information. Use a scale of 2 cm for a capacity of 10000 on the horizontal axis.\\
(ii) Calculate an estimate of the mean capacity of these 130 stadiums.\\
(iii) Find which class in the table contains the median and which contains the lower quartile.
\hfill \mbox{\textit{CAIE S1 2016 Q5 [9]}}