| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Histogram from continuous grouped data |
| Difficulty | Moderate -0.8 This is a straightforward grouped data question requiring standard techniques: drawing a histogram with unequal class widths (calculating frequency densities), finding the IQR boundary, and computing mean/standard deviation from grouped data using midpoints. All are routine S1 procedures with no conceptual challenges or novel problem-solving required. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Number of chocolate bars sold | \(1 - 10\) | \(11 - 15\) | \(16 - 30\) | \(31 - 50\) | \(51 - 60\) |
| Number of days | 18 | 24 | 30 | 20 | 8 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Class widths: 10, 5, 15, 20, 10 | M1 | |
| Frequency density = frequency/*their* class width: 1.8, 4.8, 2, 1, 0.8 | M1 | |
| All heights correct on diagram (using a linear scale) | A1 | |
| Correct bar ends | B1 | |
| Bar ends: 10.5, 15.5, 30.5, 50.5, 60.5 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(11-15\) and \(31-50\) | B1 | |
| Greatest IQR \(= 50 - 11 = 39\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Mean} = \frac{18\times5.5+24\times13+30\times23+20\times40.5+8\times55.5}{100} = \frac{2355}{100} = 23.6\) | B1 | |
| \(\text{Var} = \frac{18\times5.5^2+24\times13^2+30\times23^2+20\times40.5^2+8\times55.5^2}{100} - \text{mean}^2\) | M1 | |
| \(\frac{77917.5}{100} - \text{mean}^2 = 224.57\) | A1 | |
| Standard deviation \(= 15.0\) | A1 FT | FT *their* variance |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Class widths: 10, 5, 15, 20, 10 | M1 | |
| Frequency density = frequency/*their* class width: 1.8, 4.8, 2, 1, 0.8 | M1 | |
| All heights correct on diagram (using a linear scale) | A1 | |
| Correct bar ends | B1 | |
| Bar ends: 10.5, 15.5, 30.5, 50.5, 60.5 | B1 | |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $11-15$ and $31-50$ | B1 | |
| Greatest IQR $= 50 - 11 = 39$ | B1 | |
---
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Mean} = \frac{18\times5.5+24\times13+30\times23+20\times40.5+8\times55.5}{100} = \frac{2355}{100} = 23.6$ | B1 | |
| $\text{Var} = \frac{18\times5.5^2+24\times13^2+30\times23^2+20\times40.5^2+8\times55.5^2}{100} - \text{mean}^2$ | M1 | |
| $\frac{77917.5}{100} - \text{mean}^2 = 224.57$ | A1 | |
| Standard deviation $= 15.0$ | A1 FT | FT *their* variance |7 The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Number of chocolate bars sold & $1 - 10$ & $11 - 15$ & $16 - 30$ & $31 - 50$ & $51 - 60$ \\
\hline
Number of days & 18 & 24 & 30 & 20 & 8 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Draw a histogram to represent this information.\\
\includegraphics[max width=\textwidth, alt={}, center]{3ada18de-c4f7-4049-9032-46b796be83c3-12_1203_1399_833_415}
\item What is the greatest possible value of the interquartile range for the data?
\item Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2020 Q7 [11]}}