Time Series and Moving Averages: O Level Statistics 4040 and IGCSE 0479 Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs. Use them alongside the official past papers from Cambridge.

Study the method first: time series study guide.

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Questions

1. A school café records its sales (in hundreds of rupees) each term for three years.

	Autumn	Spring	Summer
Year 1	34	25	19
Year 2	38	27	22
Year 3	41	31	25

(a) Explain why a 3-point moving average is appropriate for these data. [1]

(b) Calculate all the 3-point moving averages. [3]

(c) Explain why these moving averages do not need to be centred. [1]

(d) Calculate the seasonal component for each term. [3]

(e) A trend line is drawn through the first and last moving averages. Use it, with your answer to (d), to predict the café’s sales in the Autumn term of Year 4. [3]

(f) State one assumption made in your prediction. [1]

2. A household’s quarterly electricity bills (in hundreds of rupees) are:

	Q1	Q2	Q3	Q4
Year 1	120	95	80	130
Year 2	128	99	86	139

(a) Calculate the five 4-point moving averages. [2]

(b) Calculate the four centred moving averages and state the quarter each belongs to. [3]

(c) For each of these four quarters, calculate the difference between the bill and the centred moving average. [2]

(d) Explain why two years of data are not enough to give a reliable seasonal component for each quarter. [1]

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Answers

1. (a) There are three terms in each yearly cycle, so each moving average covers one full cycle and the seasonal pattern is averaged out [1].

(b) (34 + 25 + 19) ÷ 3 = 26.0; (25 + 19 + 38) ÷ 3 = 27.3; (19 + 38 + 27) ÷ 3 = 28.0; (38 + 27 + 22) ÷ 3 = 29.0; (27 + 22 + 41) ÷ 3 = 30.0; (22 + 41 + 31) ÷ 3 = 31.3; (41 + 31 + 25) ÷ 3 = 32.3 [3] (2 marks for five or six correct, 1 mark for three or four).

(c) With an odd number of points, the middle of each group is an actual term (for example, the first average belongs to Year 1 Spring), so it can be plotted against that term directly [1].

(d) Differences (sales − moving average):

• Autumn: Year 2 38 − 28.0 = 10.0; Year 3 41 − 31.33 = 9.67; mean +9.83
• Spring: Year 1 25 − 26.0 = −1.0; Year 2 27 − 29.0 = −2.0; Year 3 31 − 32.33 = −1.33; mean −1.44
• Summer: Year 1 19 − 27.33 = −8.33; Year 2 22 − 30.0 = −8.0; mean −8.17

[1] for the method (sales − moving average), [1] for the differences, [1] for the three means.

(e) The trend line passes through (Year 1 Spring, 26.0) and (Year 3 Spring, 32.33): a rise of 6.33 over 6 terms, about 1.06 per term [1]. Year 4 Autumn is 2 terms after Year 3 Spring, so the trend value is about 32.33 + 2 × 1.06 = 34.4 [1]. Prediction = 34.4 + 9.83 ≈ 44.3, about Rs 4 430 [1].

(f) Any one: the trend continues in a straight line beyond the data; the seasonal pattern stays the same in Year 4; there are no unusual events (such as a change in prices or school closure) [1].

2. (a) (120 + 95 + 80 + 130) ÷ 4 = 106.25; then 108.25, 109.25, 110.75, 113.0 [2] (1 mark for three or four correct).

(b) (106.25 + 108.25) ÷ 2 = 107.25 (Year 1 Q3); (108.25 + 109.25) ÷ 2 = 108.75 (Year 1 Q4); (109.25 + 110.75) ÷ 2 = 110.0 (Year 2 Q1); (110.75 + 113.0) ÷ 2 = 111.875 (Year 2 Q2) [3] (1 mark for the values, 1 mark for the method, 1 mark for the quarters).

(c) Year 1 Q3: 80 − 107.25 = −27.25; Year 1 Q4: 130 − 108.75 = +21.25; Year 2 Q1: 128 − 110.0 = +18.0; Year 2 Q2: 99 − 111.875 = −12.875 [2] (1 mark for two correct).

(d) Each quarter has only one difference, so its “average” rests on a single year and may be distorted by an unusual year [1].

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Where marks are usually lost

• In 1(b), rounding early. Keep 27.33 and 31.33 in the working even if you quote 27.3 and 31.3.
• In 1(d), subtracting the wrong way round, which flips every sign.
• In 1(e), adding the seasonal component to the last actual Autumn value (41) instead of to the trend value.
• In 2(b), attaching the centred averages to the wrong quarters. The first centred value belongs to the third quarter of the data.

Next steps

• Crude and standardised rates practice
• Index numbers practice
• O Level Statistics 4040 hub · IGCSE Statistics 0479 hub