| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2025 |
| Session | January |
| Marks | 9 |
| Topic | Data representation |
| Type | Calculate frequency density from frequency |
| Difficulty | Moderate -0.8 This is a straightforward statistics question testing standard histogram/frequency density calculations and basic summary statistics. Part (i) requires simple proportion using frequency density formula, part (ii) is conceptual understanding of median position, and part (iii) involves routine application of mean/standard deviation formulas from grouped data. All techniques are standard textbook exercises with no problem-solving insight required. |
| Spec | 2.02b Histogram: area represents frequency2.02g Calculate mean and standard deviation |
| Time (minutes) | \(t \leqslant 27\) | \(28 \leqslant t \leqslant 30\) | \(31 \leqslant t \leqslant 35\) | \(36 \leqslant t \leqslant 45\) | \(46 \leqslant t \leqslant 60\) | \(t \geqslant 61\) |
| Number of people | 0 | 4 | 28 | 14 | 4 | 0 |
4. The table shows information about the time, $t$ minutes correct to the nearest minute, taken by 50 people to complete a race.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Time (minutes) & $t \leqslant 27$ & $28 \leqslant t \leqslant 30$ & $31 \leqslant t \leqslant 35$ & $36 \leqslant t \leqslant 45$ & $46 \leqslant t \leqslant 60$ & $t \geqslant 61$ \\
\hline
Number of people & 0 & 4 & 28 & 14 & 4 & 0 \\
\hline
\end{tabular}
\end{center}
(i) In a histogram illustrating the data, the height of the block for the $31 \leqslant t \leqslant 35$ class is 5.6 cm . Find the height of the block for the $28 \leqslant t \leqslant 30$ class. (There is no need to draw the histogram.)\\
(ii) The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.\\
(iii) Calculate estimates of the mean and standard deviation of the data.\\[0pt]
\\
\hfill \mbox{\textit{SPS SPS SM Statistics 2025 Q4 [9]}}