2020 Paper 2, Q4
The following data shows the profits per quarter for a company for the last two years.
| Year | 2018 | 2019 | ||||||
|---|---|---|---|---|---|---|---|---|
| Quarter | \(1\) | \(2\) | \(3\) | \(4\) | \(1\) | \(2\) | \(3\) | \(4\) |
| Profit ($’000s) | \(64\) | \(98\) | \(116\) | \(122\) | \(87\) | \(156\) | \(180\) | \(177\) |
Deseasonalise the data and plot this on the same set of axes as the original data in the graph on the next page.
The paper’s graph runs from Q1 2018 to Q4 2019, with profit from 50 to 190 thousand dollars. The original data is already plotted on it.
Take the seasons out and see the trend
Every year starts low in Q1 and peaks later. Dividing each profit by its seasonal index flattens that pattern, and what is left shows how the business is really going.
Four calculations and a plot
Each quarter is compared with its own year. 2018 averages 100 and 2019 averages 150, so one average for all eight quarters would hide the growth.
Divide each profit by its yearly average, then average the two Q1 ratios, the two Q2 ratios and so on. Check that the four indices add to 4.
\[\text{deseasonalised}=\frac{\text{actual}}{\text{seasonal index}}\] Multiplying puts the seasons back in twice.
One season at a time
Pick a quarter to see where its index comes from. Then switch to multiplying, the slip I see most often, and watch the seasons get worse instead of disappearing.
Q3 2019 made 180 and the Q3 index is 1.18. Deseasonalised?
That is \(180\times1.18\). Q3 is a strong quarter, so its deseasonalised value should come out lower than the actual profit.
Yes. \(180\div1.18=152.5\). An index above 1 means the quarter normally runs above average, so taking the season out brings the value down.
That subtracts the index. A seasonal index is a ratio, so you divide by it.
QCAA marking guide · 5 marks
\[2018:\ \frac{64+98+116+122}{4}=100\]\[2019:\ \frac{87+156+180+177}{4}=150\]
Marker: correctly determines the yearly averages.
| Quarter | Q1 | Q2 | Q3 | Q4 |
|---|---|---|---|---|
| 2018 | \(0.64\) | \(0.98\) | \(1.16\) | \(1.22\) |
| 2019 | \(0.58\) | \(1.04\) | \(1.20\) | \(1.18\) |
Marker: determines the profit ÷ yearly average values. Equivalent forms such as \(\tfrac{64}{100}\) or \(\tfrac{16}{25}\) are accepted.
\[\text{Q1}:\ \frac{0.64+0.58}{2}=0.61\qquad \text{Q2}:\ \frac{0.98+1.04}{2}=1.01\]\[\text{Q3}:\ \frac{1.16+1.20}{2}=1.18\qquad \text{Q4}:\ \frac{1.22+1.18}{2}=1.20\]
Marker: determines the seasonal indices. Follow-through marks are allowed from here on. As a check, \(0.61+1.01+1.18+1.20=4\).
| Quarter | Q1 | Q2 | Q3 | Q4 |
|---|---|---|---|---|
| 2018 | \(104.92\) | \(97.03\) | \(98.31\) | \(101.67\) |
| 2019 | \(142.62\) | \(154.46\) | \(152.54\) | \(147.50\) |
Marker: determines the deseasonalised values, each one actual ÷ seasonal index, for example \(64\div0.61=104.92\).
Plot the eight deseasonalised values at their quarters on the graph provided, and join them.
The result is the black line in the interactive above with "Divide by the index" selected. It sits near 100 through 2018 and near 150 through 2019.
Marker: accurately plots the deseasonalised data on the provided graph. Follow-through is allowed from your values.
What the flat line tells you
Deseasonalised profit barely moves within each year, then jumps by about 50 between Q4 2018 and Q1 2019. The growth happened at the turn of the year, and the raw data hides that behind the seasons.
The vertical axis on the paper’s graph starts at 50. Every deseasonalised value fits between 97 and 155, so plot carefully against the gridlines rather than estimating by eye.
The method is the standard one, but you are not given the seasonal indices or told how to find them. You have to remember the whole chain, from yearly averages through to plotting, without any prompts. Four of the five marks follow through, so one early slip costs you only one mark. I want you to set the working out as tables, one per step, so the marker can follow each number to where it came from.
Question wording and marking-guide steps are from the 2020 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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