- The lengths, \(x \mathrm {~mm}\), of 50 pebbles are summarised in the table below.
| Length | Frequency |
| \(20 \leqslant x < 30\) | 2 |
| \(30 \leqslant x < 32\) | 16 |
| \(32 \leqslant x < 36\) | 20 |
| \(36 \leqslant x < 40\) | 8 |
| \(40 \leqslant x < 45\) | 3 |
| \(45 \leqslant x < 50\) | 1 |
A histogram is drawn to represent these data.
The bar representing the class \(32 \leqslant x < 36\) is 2.5 cm wide and 7.5 cm tall.
- Calculate the width and the height of the bar representing the class \(30 \leqslant x < 32\)
- Using linear interpolation, estimate the median of \(x\)
The weight, \(w\) grams, of each of the 50 pebbles is coded using \(10 y = w - 20\) These coded data are summarised by
$$\sum y = 104 \quad \sum y ^ { 2 } = 233.54$$
- Show that the mean of \(w\) is 40.8
- Calculate the standard deviation of \(w\)
The weight of a pebble recorded as 40.8 grams is added to the sample.
- Without carrying out any further calculations, state, giving a reason, what effect this would have on the value of
- the mean of \(w\)
- the standard deviation of \(w\)