QCE General Mathematics - Unit 3 - Time series analysis

Analysing Time Series Data | QCE General Mathematics

Learn moving averages, seasonal indices, deseasonalising and trend lines for QCE General Mathematics time series analysis.

Updated 2026-05-18 - 5 min read

Time series analysis turns a noisy sequence of values into a clearer description of trend and seasonal behaviour. General Mathematics focuses on moving averages, seasonal indices, deseasonalising and fitting a trend line.

Moving averages

A moving average smooths short-term variation by replacing each value with the mean or median of nearby values. For an odd number of data values, the smoothed value sits naturally on the middle time point.

For a 3-point moving average:

| Month | Value | |---|---:| | Jan | 42 | | Feb | 45 | | Mar | 51 |

The smoothed value for February is:

$ \frac{42+45+51}{3}=46 $

Moving averages make the underlying trend easier to see, but they also remove detail. The larger the moving-average window, the smoother the series becomes.

Seasonal indices and deseasonalising

A seasonal index measures how a season compares with the average level. An index of $120$ means the season is about $20\%$ above average. An index of $85$ means it is about $15\%$ below average.

To deseasonalise a value:

$ \text{deseasonalised value}=\frac{\text{actual value}}{\text{seasonal index}/100} $

Worked example

Relating back to least squares

Once data are deseasonalised, a least-squares line can model the long-term trend. Predictions often use two stages: predict the deseasonalised value from the trend line, then reseasonalise using the relevant seasonal index.

Odd and even moving averages

For an odd moving average, such as a 3-point or 5-point moving average, the smoothed value lines up with the middle time point. For an even moving average, such as a 4-point moving average, the first average sits between two time points, so a centred moving average is needed.

For a 4-point moving average:

1. Average four consecutive values.
2. Repeat for the next four consecutive values.
3. Average pairs of consecutive 4-point moving averages to centre them.

This is why even moving-average tables often have more blank cells at the start and end.

Seasonal-index procedure

The average percentage method can be organised as:

1. Find the average value for each complete year or cycle.
2. Divide each seasonal value by its cycle average.
3. Average the same season's index across cycles.
4. Use the overall seasonal indices to deseasonalise.

If indices are written as decimals, their average should be close to $1$. If there are four seasons, their sum should be close to $4$. If indices are written as percentages, the average should be close to $100\%$.

Reseasonalising predictions

After fitting a trend line to deseasonalised data, the predicted value is also deseasonalised. To convert it back to an actual seasonal prediction:

$ \text{actual prediction}=\text{deseasonalised prediction}\times\frac{\text{seasonal index}}{100} $

Depth: moving averages

Moving averages smooth short-term fluctuations so that the underlying trend is easier to see. A $3$-point moving average averages each group of three consecutive observations. A $5$-point moving average averages each group of five.

For an even number of observations, such as a $4$-point moving average, the average sits between two time points. A centred moving average is then used so the smoothed values align with actual time periods.

Seasonal indices

A seasonal index compares a season's typical value with the overall average level. An index above $1$ means that season is usually above average. An index below $1$ means it is usually below average.

| Seasonal index | Interpretation | |---:|---| | $1.20$ | about $20\%$ above average | | $1.00$ | about average | | $0.85$ | about $15\%$ below average |

For quarterly data, the four seasonal indices should usually average to $1$ or sum to $4$. For monthly data, the twelve indices should usually sum to $12$.

Deseasonalising and reseasonalising

Deseasonalising removes the seasonal effect:

$ \text{deseasonalised value}=\frac{\text{actual value}}{\text{seasonal index}} $

Reseasonalising adds it back when making a seasonal prediction:

$ \text{seasonal forecast}=\text{trend forecast}\times\text{seasonal index} $

Choosing a smoothing period

The moving-average period should match the seasonal cycle when possible. Use a $4$-point moving average for quarterly data with annual seasonality, or a $12$-point moving average for monthly data with annual seasonality. A poor choice can smooth the data without properly removing the seasonal pattern.

Quick check

Sources

• QCAA General Mathematics subject page
• QCAA General Mathematics 2025 syllabus
• QCAA General Mathematics 2025 formula book
• NIST Engineering Statistics Handbook: Smoothing