Time Series, Moving Averages and Seasonal Variation: O Level Statistics 4040 and IGCSE 0479 Study Guide

This guide covers time series: Topic 10 of Cambridge O Level Statistics (4040, syllabus for examination in 2025, 2026 and 2027) and Topic 12 of Cambridge IGCSE Statistics (0479, syllabus for examination in 2027). Practise afterwards with the time series practice questions.

What the syllabuses ask for

4040 Topic 10	0479 Topic 12 (learning outcomes)
	12.1 Plot data to form a time series graph
10.1 Understanding of trend, “including determination by calculation of moving averages, with centring, where appropriate”	12.2 Understand the reasons for finding moving averages
	12.3 Calculate and plot moving average values (may include centring; understand when centring is appropriate)
	12.4 Understand trend and draw a trend line by eye using moving average values
10.2 Understanding of seasonal variation, “including calculation of mean seasonal variation. Use of a trend line and seasonal component in prediction.”	12.5 Calculate seasonal components (average seasonal variation) from tables of values and moving averages, and from time series graphs and trend lines
	12.6 Use a trend line and seasonal component to make predictions; understand the assumptions and limitations

Sources: 4040 syllabus PDF, 0479 syllabus PDF.

1. Why moving averages?

Data recorded at regular intervals (monthly, quarterly, by school term) often rises and falls in a repeating pattern: ice-cream sales peak every summer, electricity use peaks every winter. That repeating pattern is seasonal variation. The trend is the general direction underneath it.

A moving average smooths out the seasonal pattern so the trend can be seen. The number of values in each average should equal the number of seasons in one cycle: a 4-point moving average for quarterly data, a 3-point moving average for data by school term, a 12-point average for monthly data.

2. Calculating moving averages

Average the first cycle of values, then move along one place and repeat. Each average is plotted at the middle of the values it covers.

• Odd number of points (3-point). The middle is one of the data points, so each average is plotted against an actual time period. No centring is needed.
• Even number of points (4-point). The middle falls between two time periods (for quarterly data, between Q2 and Q3). To line the averages up with actual quarters, centre them: average each pair of neighbouring moving averages. This is the “centring, where appropriate” in the syllabus.

Worked example (quarterly data, with centring)

A shop’s sales (in thousands of units) over three years:

	Q1	Q2	Q3	Q4
2023	52	40	36	64
2024	56	44	40	68
2025	62	48	44	74

Step 1: 4-point moving averages

• (52 + 40 + 36 + 64) ÷ 4 = 48.0 (between 2023 Q2 and Q3)
• (40 + 36 + 64 + 56) ÷ 4 = 49.0 (between 2023 Q3 and Q4)
• then 50.0, 51.0, 52.0, 53.5, 54.5, 55.5, 57.0

Step 2: centre them (average neighbouring pairs)

Quarter	Sales	Centred moving average	Sales − moving average
2023 Q3	36	(48.0 + 49.0) ÷ 2 = 48.5	−12.5
2023 Q4	64	49.5	+14.5
2024 Q1	56	50.5	+5.5
2024 Q2	44	51.5	−7.5
2024 Q3	40	52.75	−12.75
2024 Q4	68	54.0	+14.0
2025 Q1	62	55.0	+7.0
2025 Q2	48	56.25	−8.25

There are no centred moving averages for the first two and last two quarters: a 4-point average needs two quarters on each side.

Step 3: trend line. Plot the centred moving averages on the time series graph and draw a straight line through them by eye. Here they rise steadily from 48.5 to 56.25 over seven quarters, about 1.1 per quarter, so sales are increasing.

Step 4: seasonal components. The difference sales − moving average shows how far each quarter sits above or below the trend. Average these differences for each quarter to get the mean seasonal variation (the seasonal component):

• Q1: (5.5 + 7.0) ÷ 2 = +6.25
• Q2: (−7.5 + −8.25) ÷ 2 = −7.875
• Q3: (−12.5 + −12.75) ÷ 2 = −12.625
• Q4: (14.5 + 14.0) ÷ 2 = +14.25

Q4 is the strong quarter: on average, sales are about 14 thousand above the trend.

Step 5: prediction. Read the trend value for the quarter you want from the extended trend line, then add the seasonal component.

Suppose the trend line drawn by eye passes through (2023 Q3, 48.5) and (2025 Q2, 56.25). It rises by 7.75 over 7 quarters, about 1.107 per quarter. 2026 Q1 is 10 quarters after 2023 Q3, so the trend value there is about 48.5 + 10 × 1.107 = 59.6.

Predicted sales for 2026 Q1 = 59.6 + 6.25 ≈ 66 thousand units.

(An answer read from your own graph will differ slightly, because the trend line is drawn by eye. Show the trend value you read and the seasonal component you add.)

3. Assumptions and limitations of the prediction

• It assumes the trend carries on in a straight line beyond the data (extrapolation), which becomes less reliable the further ahead you go.
• It assumes the seasonal pattern stays the same size each year.
• The trend line is drawn by eye, so different people get slightly different predictions.
• One-off events (a new competitor, a price change) are not captured.

Where marks are usually lost

• Using the wrong number of points: the moving average must span one full cycle (4 for quarters, 3 for terms).
• Plotting 4-point moving averages against actual quarters without centring them.
• Working out “moving average − sales” instead of “sales − moving average”, which reverses every seasonal component’s sign.
• Adding the seasonal component to the actual value instead of to the trend value when predicting.
• Drawing the trend line through the original data points instead of through the moving averages.

Related pages

• Time series: practice questions with worked answers
• Index numbers (4040 Topic 8)
• Crude and standardised rates (4040 Topic 7)
• Cambridge O Level Statistics 4040 syllabus hub and topic checklist
• Cambridge IGCSE Statistics 0479 syllabus hub