| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw histogram then perform other calculations |
| Difficulty | Moderate -0.3 This is a standard S1 histogram question requiring frequency density calculations, quartile identification, and mean estimation from grouped data. While it involves multiple parts and unequal class widths (requiring careful frequency density work), all techniques are routine textbook exercises with no novel problem-solving required. Slightly easier than average due to straightforward application of standard methods. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Number of incorrect notes | \(1 - 5\) | \(6 - 10\) | \(11 - 20\) | \(21 - 40\) | \(41 - 70\) |
| Frequency | 10 | 5 | 26 | 32 | 18 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Class widths: 5, 5, 10, 20, 30 | M1 | At least 3 class widths correct and used in a calculation |
| Frequency density: 2, 1, 2.6, 1.6, 0.6 | M1 | At least 3 correct frequency densities unsimplified – FT their class widths |
| [Histogram drawn] | A1 | All correct heights on a histogram using a linear vertical scale from zero – no FT |
| B1 | Correct upper bar ends (5.5, 10.5, 20.5, 40.5, 70.5) and 4 correct lower bar ends of 5.5, 10.5, 20.5, 40.5. Condone 0 or 1. | |
| B1 | Linear scales with at least 3 values indicated on each axis, vertical scale from 0, axes labelled 'fd' and 'no. of (incorrect) notes', or better | |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| LQ: \(11 - 20\), UQ: \(21 - 40\) | B1 | Both UQ and LQ correct |
| Greatest IQR \(= 40 - 11 = 29\) | B1 FT | Subtract lower end of their LQ interval from upper end of their UQ interval |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Midpoints: 3, 8, 15.5, 30.5, 55.5 | M1 | At least 4 midpoints correct and used |
| \(\text{Mean} = \frac{3\times10 + 8\times5 + 15.5\times26 + 30.5\times32 + 55.5\times18}{91}\) | M1 | Correct formula with their midpoints (not upper boundary, lower boundary, class width, frequency density, frequency or cumulative frequency) |
| \(= \frac{30 + 40 + 403 + 976 + 999}{91} = \frac{2448}{91}\) | ||
| \(26.9\), \(26\frac{82}{91}\) | A1 | Accept 26 or 27 |
| 3 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Class widths: 5, 5, 10, 20, 30 | M1 | At least 3 class widths correct and used in a calculation |
| Frequency density: 2, 1, 2.6, 1.6, 0.6 | M1 | At least 3 correct frequency densities unsimplified – FT their class widths |
| [Histogram drawn] | A1 | All correct heights on a histogram using a linear vertical scale from zero – no FT |
| | B1 | Correct upper bar ends (5.5, 10.5, 20.5, 40.5, 70.5) and 4 correct lower bar ends of 5.5, 10.5, 20.5, 40.5. Condone 0 or 1. |
| | B1 | Linear scales with at least 3 values indicated on each axis, vertical scale from 0, axes labelled 'fd' and 'no. of (incorrect) notes', or better |
| | **5** | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| LQ: $11 - 20$, UQ: $21 - 40$ | B1 | Both UQ and LQ correct |
| Greatest IQR $= 40 - 11 = 29$ | B1 FT | Subtract lower end of their LQ interval from upper end of their UQ interval |
| | **2** | |
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Midpoints: 3, 8, 15.5, 30.5, 55.5 | M1 | At least 4 midpoints correct and used |
| $\text{Mean} = \frac{3\times10 + 8\times5 + 15.5\times26 + 30.5\times32 + 55.5\times18}{91}$ | M1 | Correct formula with their midpoints (not upper boundary, lower boundary, class width, frequency density, frequency or cumulative frequency) |
| $= \frac{30 + 40 + 403 + 976 + 999}{91} = \frac{2448}{91}$ | | |
| $26.9$, $26\frac{82}{91}$ | A1 | Accept 26 or 27 |
| | **3** | |7 A particular piece of music was played by 91 pianists and for each pianist, the number of incorrect notes was recorded. The results are summarised in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Number of incorrect notes & $1 - 5$ & $6 - 10$ & $11 - 20$ & $21 - 40$ & $41 - 70$ \\
\hline
Frequency & 10 & 5 & 26 & 32 & 18 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Draw a histogram to represent this information.\\
\includegraphics[max width=\textwidth, alt={}, center]{9f0f0e3c-7baf-42eb-a4fb-9ce61922c3cd-10_1488_1493_785_365}
\item State which class interval contains the lower quartile and which class interval contains the upper quartile.
Hence find the greatest possible value of the interquartile range.
\item Calculate an estimate for the mean number of incorrect notes.\\
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 [10]}}