Question 18
The table shows the number of visitors to a museum each week for seven weeks. A special event was held in Week 3.
| Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Visitors | 310 | 295 | 480 | 305 | 320 | 300 | 335 |
| 3-point moving median | — | — |
Sort each window and keep the middle value
A three-week window slides along the table and sorts each group of three to find the moving median, then the Week 3 moving mean is set against the moving median.
Slide the window along the weeks
Choose a week to centre the three-week window on it. Each median you find goes into the table.
Work through the solution one mark at a time
Try each step yourself before you reveal it.
Put the three values around each week in order and take the middle one.
$$\begin{aligned}\text{Week 2:}\ &295,\ \mathbf{310},\ 480\\ \text{Week 3:}\ &295,\ \mathbf{305},\ 480\\ \text{Week 4:}\ &305,\ \mathbf{320},\ 480\end{aligned}$$
Each 3-point moving median is the middle value of the week and its two neighbours, once the three values are in order.
Marker: correctly calculates at least three moving medians
The same rule gives the last two windows, which completes the table.
$$\begin{aligned}\text{Week 5:}\ &300,\ \mathbf{305},\ 320\\ \text{Week 6:}\ &300,\ \mathbf{320},\ 335\end{aligned}$$
$$\begin{array}{l|ccccc}\text{Week} & 2 & 3 & 4 & 5 & 6\\\hline \text{3-point moving median} & 310 & 305 & 320 & 305 & 320\end{array}$$
Marker: correctly calculates all five moving medians
Add the visitors for Weeks 2, 3 and 4, then divide by 3.
$$\begin{aligned}&\text{3-point moving mean for Week 3}\\ &= \frac{295 + 480 + 305}{3} = \frac{1080}{3} = 360\end{aligned}$$
The Week 3 moving mean averages Weeks 2, 3 and 4, the same three weeks as the Week 3 moving median.
Marker: calculates the 3-point moving mean for Week 3
Set the Week 3 moving mean against the Week 3 moving median and say which one the special event affects more.
The moving mean for Week 3 (360) is much higher than the moving median (305) because the mean includes the size of the Week 3 value (480). The moving median uses the middle value of each group of three, so the special event has less effect on the moving median than on the moving mean.
The comparison must use both Week 3 results and say which one the special event affects more.
Follow-through applies from an error in earlier working.
An equivalent comparison also earns this mark, for example that the moving median is not affected by the size of an extreme value.
b) The Week 3 moving mean is 360, well above the Week 3 moving median of 305, so the special event affects the moving mean much more than the moving median.
Marker: compares the effect of the special event on the moving mean and the moving median
If I had to guess where you might lose marks…
- Taking the middle value of each window without sorting it first: 295, 480, 305, 320, 300 for Weeks 2 to 6. None of these match, so this loses Mark 1 and Mark 2.
- Calculating 3-point moving means instead of medians: 361.67, 360, 368.33, 308.33, 318.33. This loses Mark 1 and Mark 2.
- Centring the Week 3 moving mean on the wrong weeks, for example Weeks 3 to 5: $\frac{480 + 305 + 320}{3} = 368.33$. This loses Mark 3.
- Stating “the mean is 360 and the median is 305” without saying which one the special event affects more. This loses Mark 4.
- Comparing the moving mean with the actual Week 3 value (480) instead of with the moving median. This loses Mark 4.
Only mark this done when you could do it without help. Reading the solution does not count.
A Tangent Tuition mock exam question. It is not a past QCAA question, and Tangent Tuition is not affiliated with the QCAA.
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