Time series | AQA GCSE Maths Notes

Visual model

Time series show trends over time

timeoverall trend upwards

Gold-standard guide

20 mins

What you will learn

Read patterns and trends over time.

Use a clear step-by-step method for time series.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier statistics skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Moving average smooths out seasonal fluctuations to reveal the trend

Step 1

Calculate the first 4-point moving average

Average of Q1–Q4 (year 1): (12 + 18 + 22 + 16)/4 = $6\frac{8}{4}$ = 17

Step 2

Calculate the second 4-point moving average

Average of Q2(yr1)–Q1(yr2): (18 + 22 + 16 + 14)/4 = $7\frac{0}{4}$ = 17.5

Step 3

State the plotting position

The first average represents the midpoint of the 4 values — plotted between Q2 and Q3 of year 1

Watch out

Students calculate the average correctly but plot it at the first data point rather than the midpoint of the group

Moving average

$moving average = total of chosen bloc\frac{k}{number} in block.$

Trend

Use smoothed values to describe the overall pattern.

Worked example

Quarterly sales (£000s): Q1=12, Q2=18, Q3=22, Q4=16, Q1=14, Q2=20. Calculate the first two 4-point moving averages.

Calculate the first 4-point moving average: Average of Q1–Q4 (year 1): (12 + 18 + 22 + 16)/4 = $6\frac{8}{4}$ = 17.

Calculate the second 4-point moving average: Average of Q2(yr1)–Q1(yr2): (18 + 22 + 16 + 14)/4 = $7\frac{0}{4}$ = 17.5.

State the plotting position: The first average represents the midpoint of the 4 values — plotted between Q2 and Q3 of year 1.

Final answer

First moving average = 17; second = 17.5

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Quarterly sales (£000s): Q1=12, Q2=18, Q3=22, Q4=16, Q1=14, Q2=20. Calculate the first two 4-point moving averages.

3 marks4 minstime-series-worked

Show solution

Worked solution

1.Calculate the first 4-point moving average: Average of Q1–Q4 (year 1): (12 + 18 + 22 + 16)/4 = $6\frac{8}{4}$ = 17.
2.Calculate the second 4-point moving average: Average of Q2(yr1)–Q1(yr2): (18 + 22 + 16 + 14)/4 = $7\frac{0}{4}$ = 17.5.
3.State the plotting position: The first average represents the midpoint of the 4 values — plotted between Q2 and Q3 of year 1.

Final answer

First moving average = 17; second = 17.5

Mark points

M1: calculate the first 4-point moving average
M1: calculate the second 4-point moving average
M1: state the plotting position
A1: First moving average = 17; second = 17.5

Watch out

Students calculate the average correctly but plot it at the first data point rather than the midpoint of the group.5, between data points 2 and 3.

Diagnosticrecall

Calculate 3-point moving averages for: 10, 14, 12, 16, 11, 15.

1 mark2 minstime-series-q1

Show solution

Worked solution

1.Spot the skill: Moving average smooths out seasonal fluctuations to reveal the trend.
2.Use the calculate the first 4-point moving average stage first, then calculate the second 4-point moving average.
3.Keep the final answer visible: 12, 14, 13, 14.

Final answer

12, 14, 13, 14

Mark points

M1: use the correct moving average smooths out seasonal fluctuations to reveal the trend.for a 4-point moving average: average each group of 4 consecutive values.the average is plotted at the middle of its 4 values — for 4 points this falls between the 2nd and 3rd values (centred moving average uses two averages).
A1: 12, 14, 13, 14

Watch out

Students calculate the average correctly but plot it at the first data point rather than the midpoint of the group.5, between data points 2 and 3.

Easyprocedure

What does an upward trend in moving averages indicate?

2 marks3 minstime-series-q2

Show solution

Worked solution

1.Spot the skill: Moving average smooths out seasonal fluctuations to reveal the trend.
2.Use the calculate the second 4-point moving average stage first, then state the plotting position.
3.Keep the final answer visible: Overall sales/data are increasing over time.

Final answer

Overall sales/data are increasing over time

Mark points

M1: use the correct moving average smooths out seasonal fluctuations to reveal the trend.for a 4-point moving average: average each group of 4 consecutive values.the average is plotted at the middle of its 4 values — for 4 points this falls between the 2nd and 3rd values (centred moving average uses two averages).
A1: Overall sales/data are increasing over time

Watch out

Students calculate the average correctly but plot it at the first data point rather than the midpoint of the group.5, between data points 2 and 3.

Mediumreasoning

Actual Q3 = 22, trend at Q3 = 19. Find the seasonal variation.

3 marks4 minstime-series-q3

Show solution

Worked solution

1.Spot the skill: Moving average smooths out seasonal fluctuations to reveal the trend.
2.Use the state the plotting position stage first, then calculate the first 4-point moving average.
3.Keep the final answer visible: 22 − 19 = +3 (Q3 tends to be above the trend).

Final answer

22 − 19 = +3 (Q3 tends to be above the trend)

Mark points

M1: use the correct moving average smooths out seasonal fluctuations to reveal the trend.for a 4-point moving average: average each group of 4 consecutive values.the average is plotted at the middle of its 4 values — for 4 points this falls between the 2nd and 3rd values (centred moving average uses two averages).
A1: 22 − 19 = +3 (Q3 tends to be above the trend)

Watch out

Students calculate the average correctly but plot it at the first data point rather than the midpoint of the group.5, between data points 2 and 3.

Hardproblem solving

Why are moving averages used rather than plotting raw data?

3 marks5 minstime-series-q4

Show solution

Worked solution

1.Spot the skill: Moving average smooths out seasonal fluctuations to reveal the trend.
2.Use the calculate the first 4-point moving average stage first, then calculate the second 4-point moving average.
3.Keep the final answer visible: To smooth seasonal fluctuations and reveal the underlying trend.

Final answer

To smooth seasonal fluctuations and reveal the underlying trend

Mark points

M1: use the correct moving average smooths out seasonal fluctuations to reveal the trend.for a 4-point moving average: average each group of 4 consecutive values.the average is plotted at the middle of its 4 values — for 4 points this falls between the 2nd and 3rd values (centred moving average uses two averages).
A1: To smooth seasonal fluctuations and reveal the underlying trend

Watch out

Students calculate the average correctly but plot it at the first data point rather than the midpoint of the group.5, between data points 2 and 3.

Exam-stylemulti-step

A 4-point moving average uses data from positions 3 to 6. At what position should it be plotted?

4 marks6 minstime-series-q5

Show solution

Worked solution

1.Spot the skill: Moving average smooths out seasonal fluctuations to reveal the trend.
2.Use the calculate the second 4-point moving average stage first, then state the plotting position.
3.Keep the final answer visible: Between positions 4 and 5 (i.e. at position 4.5).

Final answer

Between positions 4 and 5 (i.e. at position 4.5)

Mark points

M1: use the correct moving average smooths out seasonal fluctuations to reveal the trend.for a 4-point moving average: average each group of 4 consecutive values.the average is plotted at the middle of its 4 values — for 4 points this falls between the 2nd and 3rd values (centred moving average uses two averages).
A1: Between positions 4 and 5 (i.e. at position 4.5)

Watch out

Students calculate the average correctly but plot it at the first data point rather than the midpoint of the group.5, between data points 2 and 3.

Grade 9 stretchproblem solving

Calculate the four-point moving average for the values 12, 18, 15 and 23.

4 marks7 minstime-series-g9

Show solution

Worked solution

1.Add the four values.
2.Divide by 4.

Final answer

Mark points

M1: use (12 + 18 + 15 + 23) / 4
A1: 17

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Time series - 2 marksCalculate 3-point moving averages for: 10, 14, 12, 16, 11, 15.Mark answer

Answer

12, 14, 13, 14

2Collecting and sampling data - 2 marksWhy might a questionnaire question be biased?Mark answer

Answer

Leading wording, only offering responses that agree, or not including a 'no' option

3Averages and range - 2 marksThe mean of 5 numbers is 12. Four of them are 8, 14, 10, 15. Find the fifth.Mark answer

Answer

4Grouped data and estimated mean - 3 marksA survey records [10,20): 5 responses and [20,30): 15. Estimate total mean across both groups.Mark answer

Answer

Use midpoints 15 and 25: (5×15 + 15×25)/20 = $45\frac{0}{2}0$ = 22.5

Mastery check

I can explain the method for time series.
I can show clear working without skipping key steps.
I can avoid this mistake: Students calculate the average correctly but plot it at the first data point rather than the midpoint of the group.5, between data points 2 and 3.

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