| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2021 |
| Session | September |
| Marks | 6 |
| Topic | Measures of Location and Spread |
| Type | Calculate statistics from grouped frequency table |
| Difficulty | Easy -1.2 This is a routine statistics question testing standard procedures: finding median from cumulative frequency (straightforward interpolation in the 20-29 class), calculating mean and standard deviation using given summations with standard formulas, and explaining right-skew. All techniques are direct textbook applications with no problem-solving or insight required. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Distance (to the nearest mile) | Number of commuters |
| 0-9 | 10 |
| 10-19 | 19 |
| 20-29 | 43 |
| 30-39 | 25 |
| 40-49 | 8 |
| 50-59 | 6 |
| 60-69 | 5 |
| 70-79 | 3 |
| 80-89 | 1 |
\begin{enumerate}
\item A random sample of distances travelled to work for 120 commuters from a train station in Devon is recorded. The distances travelled, to the nearest mile, are summarised below.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|}
\hline
Distance (to the nearest mile) & Number of commuters \\
\hline
0-9 & 10 \\
\hline
10-19 & 19 \\
\hline
20-29 & 43 \\
\hline
30-39 & 25 \\
\hline
40-49 & 8 \\
\hline
50-59 & 6 \\
\hline
60-69 & 5 \\
\hline
70-79 & 3 \\
\hline
80-89 & 1 \\
\hline
\end{tabular}
\end{center}
For this distribution:\\
a estimate the median.
The mid-point of each class was represented by $x$ and its corresponding frequency by $f$. The mid-point of the lowest class was taken to be 4.75 giving:
$$\Sigma f x = 3552.5 \text { and } \Sigma f x ^ { 2 } = 138043.125$$
b Estimate the mean and the standard deviation of this distribution.\\
c Explain why the median is less than the mean for these data.\\[0pt]
\\
\hfill \mbox{\textit{SPS SPS SM Statistics 2021 Q1 [6]}}