Calculate seasonal indices from complete cycles practice questions | General Mathematics (Year 12, VCE Units 3 & 4)

Calculating seasonal indices

A seasonal index compares a season with the mean season in its own cycle. With several cycles, calculate each cycle's ratios first, then average the ratios for matching seasons

A_j=\frac{\text{cycle total}}{S},\quad I_s=\frac{1}{m}\sum_{j=1}^{m}\frac{x_{j,s}}{A_j}

Find each cycle mean Add the observations in one complete cycle and divide by the number of seasons \(\displaystyle A=\frac{360+450+540+450}{4}=450\)
Form within-cycle ratios Divide a seasonal observation by the mean of its own cycle \(\displaystyle r=\frac{540}{450}=1.2\)
Average corresponding ratios For several cycles, average the unrounded ratios for the same season \(\displaystyle I_3=\frac{1.20+1.10}{2}=1.15\)
Round only at the end Keep means and ratios exact until the final seasonal indices are rounded \(\displaystyle \frac{1.234+1.238}{2}=1.236\approx1.24\)

Worked example

Quarterly observations for two years are 400, 500, 600, 500 and 600, 660, 600, 540. Calculate the index for the third quarter using each year's own mean, then state whether it is above or below average

x_{1}=(400,500,600,500),\quad x_{2}=(600,660,600,540)

\(\displaystyle A_1=\frac{400+500+600+500}{4}=500\)The first cycle mean is found from all four observations in the first year
\(\displaystyle A_2=\frac{600+660+600+540}{4}=600\)The second cycle has its own mean because its overall level is different
\(\displaystyle r_{1,3}=\frac{600}{500}=1.2,\quad r_{2,3}=\frac{600}{600}=1\)The third-quarter observations are divided by the means of their own cycles
\(\displaystyle I_3=\frac{1.2+1}{2}=1.1>1\Rightarrow10\%\text{ above average}\)A third-quarter index of 1.1 means that season is 10% above the average season

Common mistakes

Dividing by the cycle total

✕\(\displaystyle I_3=\frac{600}{2000}=0.3\)✓\(\displaystyle A_1=\frac{2000}{4}=500,\quad I_3=\frac{600}{500}=1.2\)

A seasonal index compares a season with the mean season, not with the complete cycle total

Pooling cycles before normalising

✕\(\displaystyle I_3=\frac{\frac{600+600}{2}}{\frac{2000+2400}{8}}=\frac{600}{550}=\frac{12}{11}\)✓\(\displaystyle I_3=\frac{\frac{600}{500}+\frac{600}{600}}{2}=\frac{11}{10}\)

Each cycle must be normalised by its own mean before ratios from the same season are averaged

Rounding ratios before averaging

✕\(\displaystyle \frac{1.2+1.2}{2}=1.2\)✓\(\displaystyle \frac{1.234+1.238}{2}=1.236\approx1.24\)

Within-cycle ratios stay unrounded until the final seasonal index is calculated and rounded

Difficulty levels

introOne short quarterly cycle is used to calculate a complete set of indices

standardSeveral quarterly cycles require selected cycle means, ratios and final indices

challengeChanging cycle levels make separate normalisation essential before an interpretation is made

examA compact one-season choice tests the cycle mean, ratio and final-index method

Questions about this topic

Why is a separate mean used for each cycle?+

A cycle can have a different overall level from another cycle. Dividing by each cycle's own mean removes that level difference before seasons are compared

Why are ratios kept unrounded?+

Rounding a ratio before averaging can change the final index, especially when several cycles are combined

What should the seasonal indices sum to?+

For S seasons, the exact indices sum to S and therefore have average 1. For four seasons the exact sum is 4

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 5: Time series

In this chapter

Construct and read a time series plot Identify qualitative features in a time series Calculate moving means Moving median smoothing Select and compare smoothing methods Interpret and complete seasonal indices Deseasonalise and reseasonalise values Calculate seasonal indices from complete cycles Fit a time trend and forecast