Hard GCSE Time series Questions

Challenging, exam-style GCSE Time series questions with worked solutions. Stretch yourself on the hardest percentage change, yearly totals, percentage of a total, time series problems.

percentage changeyearly totalspercentage of a totaltime seriesmissing valuecomparing yearly totals
GCSE Foundation34 questionsStep-by-step solutions
Question 1
5 markschallenging
The line graph shows the number of trees a charity planted in each quarter of 2022, 2023 and 2024. The readings are Q1 2022 = 77, Q2 2022 = 57, Q3 2022 = 67, Q4 2022 = 59, Q1 2023 = 74, Q2 2023 = 46, Q3 2023 = 60, Q4 2023 = 52, Q1 2024 = 63, Q2 2024 = 43, Q3 2024 = 49, Q4 2024 = 49. Which of these best describes the trend shown by the graph?
Show worked solution

Worked solution

  1. Look at the whole line, not one step of it

    trend=the overall direction over time\text{trend} = \text{the overall direction over time}

    A trend is the general direction of travel once the ups and downs are ignored. It is never decided by a single pair of points.

  2. Work out the change from each point to the next

    −20, +10, −8, +15, −28, +14, −8, +11, −20, +6, 0-20, \ +10, \ -8, \ +15, \ -28, \ +14, \ -8, \ +11, \ -20, \ +6, \ 0

    Some of these are positive and some are negative, so the line goes up and down. Those swings are the seasonal pattern, not the trend.

  3. Compare the first half of the period with the second half

    380 > 316380 \ > \ 316

    The first 66 readings total 380380 and the last 66 total 316316. Comparing equal-sized blocks like this cancels out the seasonal swing.

  4. Draw a trend line through the points

    line of best fit through all 12 points\text{line of best fit through all 12 points}

    One straight line drawn through the middle of the points shows the overall direction at a glance.

  5. Look at the gradient of that trend line

    gradient < 0\text{gradient} \ < \ 0

    The trend line falls from left to right, so its gradient is negative.

  6. Say what that gradient means

    gradient negative⇒downward trend\text{gradient negative} \Rightarrow \text{downward trend}

    A negative gradient means the readings decrease over the period as a whole.

  7. Rule out "it increases every single time"

    Q1 2022→Q2 2022:−20\text{Q1 2022} \rightarrow \text{Q2 2022}: -20

    The line falls from Q1 2022 to Q2 2022, so it is simply not true that the readings rise every time.

  8. Rule out "it decreases every single time"

    Q2 2022→Q3 2022:+10\text{Q2 2022} \rightarrow \text{Q3 2022}: +10

    The line rises from Q2 2022 to Q3 2022, so it does not fall every time either.

  9. Rule out the opposite trend

    316 < 380316 \ < \ 380

    The second half of the period is clearly lower than the first, so a upward trend is impossible.

  10. Rule out "no trend"

    316−380=−64316 - 380 = -64

    The two halves differ by −64-64, which is far too big to call the readings "about the same".

  11. Keep the season and the trend apart

    season repeats,trend downward\text{season repeats}, \quad \text{trend downward}

    A time series can have a strong seasonal pattern AND a clear trend at the same time. Describing one does not describe the other.

  12. Compare the same point one cycle apart

    77, 6077, \ 60

    Comparing like with like — the same quarter a year apart, say — is another way to see the trend without the season getting in the way.

  13. Say how far a trend can be trusted

    a trend describes the data shown, not the future\text{a trend describes the data shown, not the future}

    The trend describes what happened over the period on the graph. Carrying it far into the future is guesswork, not mathematics.

  14. Put the description into words

    readings decrease overall, with a seasonal swing on top\text{readings decrease overall, with a seasonal swing on top}

    Overall the readings decrease over time, with a seasonal rise and fall on top of that.

  15. State the answer

    Downward trend\text{Downward trend}

    So the correct description is: There is a downward trend: overall the readings decrease over time.

Answer
Downward trend\text{Downward trend}
Question 2
5 markschallenging
The line graph shows the number of houses an estate agent sold in each quarter of 2022, 2023 and 2024. The readings are Q1 2022 = 37, Q2 2022 = 21, Q3 2022 = 31, Q4 2022 = 51, Q1 2023 = 41, Q2 2023 = 33, Q3 2023 = 43, Q4 2023 = 55, Q1 2024 = 53, Q2 2024 = 37, Q3 2024 = 51, Q4 2024 = 63. Which of these best describes the trend shown by the graph?
Show worked solution

Worked solution

  1. Look at the whole line, not one step of it

    trend=the overall direction over time\text{trend} = \text{the overall direction over time}

    A trend is the general direction of travel once the ups and downs are ignored. It is never decided by a single pair of points.

  2. Work out the change from each point to the next

    −16, +10, +20, −10, −8, +10, +12, −2, −16, +14, +12-16, \ +10, \ +20, \ -10, \ -8, \ +10, \ +12, \ -2, \ -16, \ +14, \ +12

    Some of these are positive and some are negative, so the line goes up and down. Those swings are the seasonal pattern, not the trend.

  3. Compare the first half of the period with the second half

    214 < 302214 \ < \ 302

    The first 66 readings total 214214 and the last 66 total 302302. Comparing equal-sized blocks like this cancels out the seasonal swing.

  4. Draw a trend line through the points

    line of best fit through all 12 points\text{line of best fit through all 12 points}

    One straight line drawn through the middle of the points shows the overall direction at a glance.

  5. Look at the gradient of that trend line

    gradient > 0\text{gradient} \ > \ 0

    The trend line rises from left to right, so its gradient is positive.

  6. Say what that gradient means

    gradient positive⇒upward trend\text{gradient positive} \Rightarrow \text{upward trend}

    A positive gradient means the readings increase over the period as a whole.

  7. Rule out "it increases every single time"

    Q1 2022→Q2 2022:−16\text{Q1 2022} \rightarrow \text{Q2 2022}: -16

    The line falls from Q1 2022 to Q2 2022, so it is simply not true that the readings rise every time.

  8. Rule out "it decreases every single time"

    Q2 2022→Q3 2022:+10\text{Q2 2022} \rightarrow \text{Q3 2022}: +10

    The line rises from Q2 2022 to Q3 2022, so it does not fall every time either.

  9. Rule out the opposite trend

    302 > 214302 \ > \ 214

    The second half of the period is clearly higher than the first, so a downward trend is impossible.

  10. Rule out "no trend"

    302−214=88302 - 214 = 88

    The two halves differ by 8888, which is far too big to call the readings "about the same".

  11. Keep the season and the trend apart

    season repeats,trend upward\text{season repeats}, \quad \text{trend upward}

    A time series can have a strong seasonal pattern AND a clear trend at the same time. Describing one does not describe the other.

  12. Compare the same point one cycle apart

    37, 4337, \ 43

    Comparing like with like — the same quarter a year apart, say — is another way to see the trend without the season getting in the way.

  13. Say how far a trend can be trusted

    a trend describes the data shown, not the future\text{a trend describes the data shown, not the future}

    The trend describes what happened over the period on the graph. Carrying it far into the future is guesswork, not mathematics.

  14. Put the description into words

    readings increase overall, with a seasonal swing on top\text{readings increase overall, with a seasonal swing on top}

    Overall the readings increase over time, with a seasonal rise and fall on top of that.

  15. State the answer

    Upward trend\text{Upward trend}

    So the correct description is: There is an upward trend: overall the readings increase over time.

Answer
Upward trend\text{Upward trend}
Question 3
6 markschallenging
The line graph shows the number of calls a helpline answered in each quarter of 2022 and 2023. The readings are Q1 2022 = 49, Q2 2022 = 33, Q3 2022 = 57, Q4 2022 = 41, Q1 2023 = 49, Q2 2023 = 33, Q3 2023 = 57, Q4 2023 = 41. Which of these best describes the pattern shown by the graph?
Show worked solution

Worked solution

  1. Realise the description has two separate parts

    pattern=season+trend\text{pattern} = \text{season} + \text{trend}

    Describing a time series means saying WHEN the peaks come (the season) and which way the whole line is drifting (the trend). Both parts must be right.

  2. Split the readings into years

    2022:49,33,57,412023:49,33,57,412022: 49, 33, 57, 41 \quad 2023: 49, 33, 57, 41

    Take the years one at a time to find the seasonal part.

  3. Find the peak quarter in each year

    2022:Q32023:Q32022: \text{Q3} \quad 2023: \text{Q3}

    In every year the readings are highest in Quarter 3.

  4. Confirm the peak falls in the same quarter every year

    Q3=Q3\text{Q3} = \text{Q3}

    The peak is in Quarter 3 each time, so the seasonal part of the description is "peaks in Quarter 3".

  5. Now look at the trend: work out each step

    −16, +24, −16, +8, −16, +24, −16-16, \ +24, \ -16, \ +8, \ -16, \ +24, \ -16

    The line goes up and down within each year — that is the season. The trend is what is left once those swings are ignored.

  6. Compare the two halves of the period

    180 = 180180 \ = \ 180

    The first half totals 180180 and the second half totals 180180. Equal-sized blocks cancel the season out.

  7. Draw a trend line through the points

    gradient = 0\text{gradient} \ = \ 0

    A line of best fit through all 88 points stays level from left to right, so the trend is level.

  8. Put the two parts together

    peak Q3+level trend\text{peak Q3} + \text{level trend}

    The readings peak in Quarter 3 every year and the whole line is drifting level.

  9. Rule out the wrong peak quarters

    2022: Q1=49, Q2=33, Q3=57, Q4=412022: \ \text{Q1} = 49, \ \text{Q2} = 33, \ \text{Q3} = 57, \ \text{Q4} = 41

    In 2022 the biggest of these is Quarter 3, so any option naming a different peak quarter is wrong.

  10. Rule out the wrong trend

    180−180=0180 - 180 = 0

    The second half is exactly the same size than the first, so the trend cannot be upward or downward.

  11. Check the line really does rise and fall inside each year

    3 steps up,4 steps down3 \ \text{steps up}, \quad 4 \ \text{steps down}

    The line is not simply climbing all the time; it climbs and dips in a repeating shape. That is exactly what a seasonal pattern is.

  12. Compare the same quarter across the years

    2022:57, 2023:572022: 57, \ 2023: 57

    Looking at the peak quarter year by year shows the trend again, this time with the season held fixed.

  13. Say what the description is for

    season tells you when; trend tells you which way\text{season tells you when; trend tells you which way}

    A shop that knows both can order extra stock for the peak quarter and plan for the way the whole business is moving.

  14. Check both halves of the chosen description

    peak Q3 correctlevel trend correct\text{peak Q3} \ \text{correct} \quad \text{level trend} \ \text{correct}

    The option has to get the peak quarter AND the trend right. An option that gets only one of them right is still wrong.

  15. State the answer

    Peak in Quarter 3, level trend\text{Peak in Quarter 3, level trend}

    So the correct description is: The readings peak in Quarter 3 each year, and the overall trend is level.

Answer
Peak in Quarter 3, level trend\text{Peak in Quarter 3, level trend}
Question 4
5 markschallenging
The line graph shows the number of new library cards a library issued in each quarter of 2022, 2023 and 2024. The readings are Q1 2022 = 47, Q2 2022 = 59, Q3 2022 = 65, Q4 2022 = 49, Q1 2023 = 38, Q2 2023 = 50, Q3 2023 = 64, Q4 2023 = 48, Q1 2024 = 37, Q2 2024 = 45, Q3 2024 = 55, Q4 2024 = 43. Which of these best describes the pattern shown by the graph?
Show worked solution

Worked solution

  1. Realise the description has two separate parts

    pattern=season+trend\text{pattern} = \text{season} + \text{trend}

    Describing a time series means saying WHEN the peaks come (the season) and which way the whole line is drifting (the trend). Both parts must be right.

  2. Split the readings into years

    2022:47,59,65,492023:38,50,64,482024:37,45,55,432022: 47, 59, 65, 49 \quad 2023: 38, 50, 64, 48 \quad 2024: 37, 45, 55, 43

    Take the years one at a time to find the seasonal part.

  3. Find the peak quarter in each year

    2022:Q32023:Q32024:Q32022: \text{Q3} \quad 2023: \text{Q3} \quad 2024: \text{Q3}

    In every year the readings are highest in Quarter 3.

  4. Confirm the peak falls in the same quarter every year

    Q3=Q3=Q3\text{Q3} = \text{Q3} = \text{Q3}

    The peak is in Quarter 3 each time, so the seasonal part of the description is "peaks in Quarter 3".

  5. Now look at the trend: work out each step

    +12, +6, −16, −11, +12, +14, −16, −11, +8, +10, −12+12, \ +6, \ -16, \ -11, \ +12, \ +14, \ -16, \ -11, \ +8, \ +10, \ -12

    The line goes up and down within each year — that is the season. The trend is what is left once those swings are ignored.

  6. Compare the two halves of the period

    308 > 292308 \ > \ 292

    The first half totals 308308 and the second half totals 292292. Equal-sized blocks cancel the season out.

  7. Draw a trend line through the points

    gradient < 0\text{gradient} \ < \ 0

    A line of best fit through all 1212 points falls from left to right, so the trend is downward.

  8. Put the two parts together

    peak Q3+downward trend\text{peak Q3} + \text{downward trend}

    The readings peak in Quarter 3 every year and the whole line is drifting downward.

  9. Rule out the wrong peak quarters

    2022: Q1=47, Q2=59, Q3=65, Q4=492022: \ \text{Q1} = 47, \ \text{Q2} = 59, \ \text{Q3} = 65, \ \text{Q4} = 49

    In 2022 the biggest of these is Quarter 3, so any option naming a different peak quarter is wrong.

  10. Rule out the wrong trend

    292−308=−16292 - 308 = -16

    The second half is smaller than the first, so the trend cannot be upward.

  11. Check the line really does rise and fall inside each year

    6 steps up,5 steps down6 \ \text{steps up}, \quad 5 \ \text{steps down}

    The line is not simply climbing all the time; it climbs and dips in a repeating shape. That is exactly what a seasonal pattern is.

  12. Compare the same quarter across the years

    2022:65, 2023:64, 2024:552022: 65, \ 2023: 64, \ 2024: 55

    Looking at the peak quarter year by year shows the trend again, this time with the season held fixed.

  13. Say what the description is for

    season tells you when; trend tells you which way\text{season tells you when; trend tells you which way}

    A shop that knows both can order extra stock for the peak quarter and plan for the way the whole business is moving.

  14. Check both halves of the chosen description

    peak Q3 correctdownward trend correct\text{peak Q3} \ \text{correct} \quad \text{downward trend} \ \text{correct}

    The option has to get the peak quarter AND the trend right. An option that gets only one of them right is still wrong.

  15. State the answer

    Peak in Quarter 3, downward trend\text{Peak in Quarter 3, downward trend}

    So the correct description is: The readings peak in Quarter 3 each year, and the overall trend is downward.

Answer
Peak in Quarter 3, downward trend\text{Peak in Quarter 3, downward trend}
Question 5
5 markschallenging
The line graph shows the number of cinema tickets a cinema sold in each quarter of 2022 and 2023. The readings are Q1 2022 = 40, Q2 2022 = 54, Q3 2022 = 34, Q4 2022 = 32, Q1 2023 = 56, Q2 2023 = 66, Q3 2023 = 46, Q4 2023 = 40. Which of these best describes the pattern shown by the graph?
Show worked solution

Worked solution

  1. Realise the description has two separate parts

    pattern=season+trend\text{pattern} = \text{season} + \text{trend}

    Describing a time series means saying WHEN the peaks come (the season) and which way the whole line is drifting (the trend). Both parts must be right.

  2. Split the readings into years

    2022:40,54,34,322023:56,66,46,402022: 40, 54, 34, 32 \quad 2023: 56, 66, 46, 40

    Take the years one at a time to find the seasonal part.

  3. Find the peak quarter in each year

    2022:Q22023:Q22022: \text{Q2} \quad 2023: \text{Q2}

    In every year the readings are highest in Quarter 2.

  4. Confirm the peak falls in the same quarter every year

    Q2=Q2\text{Q2} = \text{Q2}

    The peak is in Quarter 2 each time, so the seasonal part of the description is "peaks in Quarter 2".

  5. Now look at the trend: work out each step

    +14, −20, −2, +24, +10, −20, −6+14, \ -20, \ -2, \ +24, \ +10, \ -20, \ -6

    The line goes up and down within each year — that is the season. The trend is what is left once those swings are ignored.

  6. Compare the two halves of the period

    160 < 208160 \ < \ 208

    The first half totals 160160 and the second half totals 208208. Equal-sized blocks cancel the season out.

  7. Draw a trend line through the points

    gradient > 0\text{gradient} \ > \ 0

    A line of best fit through all 88 points rises from left to right, so the trend is upward.

  8. Put the two parts together

    peak Q2+upward trend\text{peak Q2} + \text{upward trend}

    The readings peak in Quarter 2 every year and the whole line is drifting upward.

  9. Rule out the wrong peak quarters

    2022: Q1=40, Q2=54, Q3=34, Q4=322022: \ \text{Q1} = 40, \ \text{Q2} = 54, \ \text{Q3} = 34, \ \text{Q4} = 32

    In 2022 the biggest of these is Quarter 2, so any option naming a different peak quarter is wrong.

  10. Rule out the wrong trend

    208−160=48208 - 160 = 48

    The second half is bigger than the first, so the trend cannot be downward.

  11. Check the line really does rise and fall inside each year

    3 steps up,4 steps down3 \ \text{steps up}, \quad 4 \ \text{steps down}

    The line is not simply climbing all the time; it climbs and dips in a repeating shape. That is exactly what a seasonal pattern is.

  12. Compare the same quarter across the years

    2022:54, 2023:662022: 54, \ 2023: 66

    Looking at the peak quarter year by year shows the trend again, this time with the season held fixed.

  13. Say what the description is for

    season tells you when; trend tells you which way\text{season tells you when; trend tells you which way}

    A shop that knows both can order extra stock for the peak quarter and plan for the way the whole business is moving.

  14. Check both halves of the chosen description

    peak Q2 correctupward trend correct\text{peak Q2} \ \text{correct} \quad \text{upward trend} \ \text{correct}

    The option has to get the peak quarter AND the trend right. An option that gets only one of them right is still wrong.

  15. State the answer

    Peak in Quarter 2, upward trend\text{Peak in Quarter 2, upward trend}

    So the correct description is: The readings peak in Quarter 2 each year, and the overall trend is upward.

Answer
Peak in Quarter 2, upward trend\text{Peak in Quarter 2, upward trend}

Unlock 29 more Time series questions

Create a free account to work through every GCSE Time series question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Time series practice

Practice QuestionsWorked Solutions

Related Statistics topics

Averages and spreadCharts and tablesComparing distributionsHistograms and cumulative frequencyQuartiles and box plotsSampling