2. The following table summarises the distances, to the nearest km , that 134 examiners travelled to attend a meeting in London.
| Distance (km) | Number of examiners |
| 41-45 | 4 |
| 46-50 | 19 |
| 51-60 | 53 |
| 61-70 | 37 |
| 71-90 | 15 |
| 91-150 | 6 |
- Give a reason to justify the use of a histogram to represent these data.
- Calculate the frequency densities needed to draw a histogram for these data.
(DO NOT DRAW THE HISTOGRAM) - Use interpolation to estimate the median \(Q _ { 2 }\), the lower quartile \(Q _ { 1 }\), and the upper quartile \(Q _ { 3 }\) of these data.
The mid-point of each class is represented by \(x\) and the corresponding frequency by \(f\). Calculations then give the following values
$$\Sigma f _ { x } = 8379.5 \quad \text { and } \quad \Sigma f _ { x ^ { 2 } } = 557489.75$$
- Calculate an estimate of the mean and an estimate of the standard deviation for these data.
One coefficient of skewness is given by
$$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
- Evaluate this coefficient and comment on the skewness of these data.
- Give another justification of your comment in part (e).