Question 6
The table shows the maximum daily temperature (°C) for a week.
| Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|
| 24.4 | 25.2 | 24.6 | 25.2 | 25.6 | 25.7 | 25.9 |
If the simple 5-point moving average for Wednesday is 25.0 °C, what is the simple 5-point moving average (°C) for Friday?
That’s it. Step through the hints below to see why. In 2024, 83.7% of Queensland students chose (A).
Not quite. 25.5 is the mean of Thursday, Friday and Saturday, which is a 3-point moving average. One way to get it is to take one day either side of Friday instead of two. Five points means Wednesday to Sunday. In 2024, 6.6% of Queensland students chose (B).
Not quite. 25.6 is Friday’s own temperature, read straight from the table. The median of Wednesday to Sunday and the mean of Thursday to Sunday also land on 25.6, so a few different slips give this. Friday’s 5-point average is the mean of the five days centred on Friday. In 2024, 2.9% of Queensland students chose (C).
Not quite. One way to get 26.0 is to copy Wednesday’s adjustment: Wednesday’s average is 0.4 above its own temperature, and $25.6+0.4=26.0$. A quick check rules it out. An average can’t be bigger than every value it averages, and the hottest day in Friday’s window is 25.9. In 2024, 6.3% of Queensland students chose (D).
Pick a window size and a centre day. The shaded window shows which temperatures go into the average.
Black dots: the temperatures in the window. Orange: their mean. Blue tick: the 25.0 the question gives for Wednesday.
This is the question: Friday and two days either side of it, Wednesday to Sunday. The mean is option (A).
This is the average the question gives you. Getting 25.0 here proves the window is right: the day itself and two days on each side. Now move the centre to Friday.
Three points, Thursday to Saturday, gives option (B). Right centre, wrong window size.
Slide the window one day and it drops one temperature off one end and picks one up at the other. That’s why moving averages change smoothly.
A simple 5-point moving average for a day is the mean of five values with that day in the middle. That means the day itself plus two days before and two days after.
For Friday, the window is Wednesday, Thursday, Friday, Saturday and Sunday.
Add the five temperatures and divide by 5. Type it in one line, with brackets around the sum.
$$\frac{24.6+25.2+25.6+25.7+25.9}{5}=\frac{127.0}{5}=25.4$$
Sanity check: the answer sits between the coolest (24.6) and hottest (25.9) days in the window, as any mean must.
The QCAA gave you Wednesday’s average so you can test your method. Wednesday’s window is Monday to Friday:
$$\frac{24.4+25.2+24.6+25.2+25.6}{5}=\frac{125.0}{5}=25.0$$
It matches, so the method is right. You can even use it as a shortcut. Moving from Wednesday to Friday drops Monday and Tuesday and adds Saturday and Sunday, so the average changes by the new pair minus the old pair, shared over all 5 points:
$$25.0+\frac{25.7+25.9-24.4-25.2}{5}=25.0+0.4=25.4$$
$$\begin{aligned}\text{Friday’s window}&=\text{Wed, Thu, Fri, Sat, Sun}\\\text{moving average}&=\frac{24.6+25.2+25.6+25.7+25.9}{5}\\&=\frac{127.0}{5}\\&=25.4\ ^\circ\text{C}\end{aligned}$$
Check with the value you were given: Monday to Friday gives $\frac{125.0}{5}=25.0$, which matches Wednesday’s average in the question.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2024 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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