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Maths · Interpreting and representing data

Time series

Sales, temperatures and visitor numbers all change over time. A time series graph shows how - the overall trend, and the pattern that repeats every season.

  • 5 key terms
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Last Lesson and Before

Answer each one, then check.

  1. 1

    Last lesson: a sector of \(90^\circ\) on a pie chart of 200 people - how many people?

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    50

  2. 2

    Last lesson: is time discrete or continuous?

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    Continuous

  3. 3

    Work out the mean of 2, 4 and 6.

    Show answerHide answer

    \(4\)

  4. 4

    What are the coordinates of a point 3 across and 5 up?

    Show answerHide answer

    \((3, 5)\)

Learning Objectives

  1. 1Plot a time series graph.
  2. 2Read values from a time series graph.
  3. 3Describe the trend of a time series.
  4. 4Recognise seasonal variation.
  5. 5Make predictions, and say how reliable they are.

What Is a Time Series?

A time series is a set of data recorded at regular intervals of time.

  • The axes

    Time always goes on the horizontal axis, in equal steps. The data goes on the vertical axis.

  • Plotting

    Plot each value above its time.

  • Joining

    Join the points with straight lines. The lines only show the pattern - they do not give real values between the points.

  • Examples

    Monthly rainfall, quarterly sales, a city's population every ten years.

Ice Cream Sales

A seaside ice cream shop's sales, in thousands of pounds, each quarter (three months) for two years.

  • Year 1

    Q1 (Jan to Mar): 12. Q2 (Apr to Jun): 20. Q3 (Jul to Sep): 26. Q4 (Oct to Dec): 10

  • Year 2

    Q1 (Jan to Mar): 14. Q2 (Apr to Jun): 23. Q3 (Jul to Sep): 29. Q4 (Oct to Dec): 12

Trend and Seasonal Variation

Describe a time series in two parts.

  • The trend

    The general direction over the whole time: increasing, decreasing or staying about the same. Ignore the ups and downs.

  • Seasonal variation

    A pattern that repeats at the same time each year, like higher ice cream sales every summer.

  • Compare like with like

    Compare Q3 with the Q3 before it, not with Q4: 29 against 26 shows the growth.

  • Use the context

    Say what the numbers mean: "sales increased", not just "the line goes up".

Reading a Time Series

Using the ice cream sales, (a) how much higher were sales in Q3 of Year 2 than in Q3 of Year 1? (b) Describe the trend. (c) Describe the seasonal pattern.

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  1. 1 (a) Read the two values Year 1 Q3 = 26, Year 2 Q3 = 29
  2. 2 Subtract, and remember the units \(29 - 26 = 3\) thousand pounds
  3. 3 (b) Compare each quarter with the same quarter a year earlier Every quarter is higher in Year 2
  4. 4 (c) Look for the repeating pattern Highest in Q3, lowest in Q4, every year

Answer(a) £3000 higher. (b) Sales are increasing. (c) Sales are highest in summer (Q3) and lowest in Q4.

Making Predictions

A time series can be used to predict what will happen next - with care.

  • Follow both patterns

    Continue the trend AND the seasonal pattern: Year 3 Q3 might be about 32.

  • Near is better

    A prediction for next quarter is more reliable than one for five years' time.

  • Things change

    A new competitor, a wet summer or a price rise could break the pattern.

  • Say so

    In an exam, a comment on reliability usually mentions how far ahead the prediction is.

A Good Time Series Graph

  • Equal time steps

    Each quarter, month or year takes the same space on the axis.

  • Labelled axes

    With units, such as "Sales (£ thousands)".

  • Accurate points

    Each point plotted exactly above its time.

  • Straight lines

    Points joined with a ruler, in order.

Case study

The Central England Temperature Record

The Central England Temperature series is the longest record of measured temperatures in the world. It gives the average temperature of a triangle of central England for every month since 1659, built from old diaries and weather records for the earliest years. Plotted as a time series, it shows a strong seasonal pattern - warm summers and cold winters, every year - and, over the last century, a clear upward trend in temperature.

1659 The year the monthly record begins
360+ Years of monthly temperatures in one time series

Make Your Own Time Series

A cinema's visitors, in thousands, each quarter for two years were: Year 1: 18, 15, 22, 30. Year 2: 20, 16, 25, 34. Draw a time series graph. Describe the trend and the seasonal variation, and predict Year 3 Q4 with a comment on how reliable your prediction is.

1. Draw and label the axes.

2. Plot and join the points.

3. Describe the trend and seasonal pattern.

4. Predict, and comment.

A good answer shows: A graph with equal quarters on the horizontal axis and points joined with straight lines. Trend: increasing (every quarter higher in Year 2). Seasonal: busiest in Q4 (the Christmas holidays), quietest in Q2. Prediction around 37 to 38 thousand, reasonably reliable as it is only one year ahead - but new films or prices could change it.

Can I...?

  1. 1Plot a time series graph.
  2. 2Read values from a time series.
  3. 3Describe the trend.
  4. 4Describe seasonal variation.
  5. 5Compare values a year apart.
  6. 6Make a prediction.
  7. 7Comment on how reliable a prediction is.
  8. 8Answer in the context of the data.

Summary & Exam Focus

  • Time goes on the horizontal axis in equal steps; points are joined with straight lines.
  • The trend is the overall direction; seasonal variation is the pattern that repeats each year.
  • Compare each quarter with the same quarter in the year before.
  • Predictions are less reliable the further ahead they go.

Exam focus

Describe the trend in the number of visitors to the museum. (1 mark) (1 marks)

Always answer in context: "the number of visitors increased" earns the mark; "it goes up" often does not.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Time series
Data recorded at regular intervals of time.
Trend
The general direction of the data over time.
Seasonal variation
A pattern that repeats at the same time each year.
Quarter
A three-month period of the year.
Prediction
An estimate of a future value based on the pattern so far.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Calculator 3 marks

    The time series graph shows the number of visitors, in thousands, to a museum each quarter for three years. (a) How many visitors were there in Q2 of Year 2? (b) Describe the trend in the number of visitors. (c) In which quarter is the museum busiest each year? Suggest a reason.

    A time series of museum visitors each quarter for three years, peaking every Q3, with each year a little higher than the one before.
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    Model answer

    (a) 4900 (b) The number of visitors is increasing. (c) Q3 - July to September, the summer holidays.

    Mark scheme

    • (a) 4900, or 4.9 thousand — B1
    • (b) Increasing, in context — C1
    • (c) Q3, with a sensible reason such as summer holidays — C1
  2. Question 2 Calculator 2 marks

    In Year 3 the museum had 3800, 5200, 7600 and 3400 visitors in the four quarters. Work out the mean number of visitors per quarter in Year 3.

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    Model answer

    \(\dfrac{3800 + 5200 + 7600 + 3400}{4} = \dfrac{20\,000}{4} = 5000\)

    Mark scheme

    • \((3800 + 5200 + 7600 + 3400) \div 4\), or 20 000 seen — M1
    • 5000 — A1
  3. Question 3 Calculator 1 mark

    The museum's manager uses the graph to predict the number of visitors in Year 8. Give one reason why this prediction may not be reliable.

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    Model answer

    Year 8 is a long way beyond the data, and the pattern may not continue - for example the museum could change its prices or opening times.

    Mark scheme

    • A reason referring to predicting far beyond the data, or to the pattern changing — C1

Quick check

  1. In a time series, what goes on the horizontal axis?

    1. ATime
    2. BFrequency
    3. CThe data values
    4. DThe trend
    Show answerHide answer

    A: Time

    Time always goes on the horizontal axis, in equal steps.

  2. A pattern that repeats at the same time every year is called...

    1. AThe trend
    2. BAn outlier
    3. CSeasonal variation
    4. DCorrelation
    Show answerHide answer

    C: Seasonal variation

    Seasonal variation repeats at the same time each year; the trend is the overall direction.

  3. Sales were 26 in Q3 last year and 29 in Q3 this year, but 10 in Q4 last year. What is the best evidence of the trend?

    1. AThe fall from 26 to 10
    2. BThe rise from 26 to 29
    3. CThe fall from 29 to 10
    4. DThe rise from 10 to 29
    Show answerHide answer

    B: The rise from 26 to 29

    Compare the same quarter a year apart: 26 to 29 shows sales increasing. Q3 to Q4 is just the seasonal pattern.

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