Time Series - Worksheets, Questions and Revision

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GCSE · Statistics

S32 Time Series

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Statisticians use particular words to describe patterns in data recorded over time. For each statement, write down whether it is True or False.

(i)A time series is a set of data values recorded at successive points in time, usually at equal intervals such as every month or every quarter.(1)

(ii)In a time series, the trend describes the short pattern that repeats within each year, while seasonal variation describes the long-term direction of the data.(1)

(iii)A time series graph is drawn by plotting each value in time order along the horizontal axis and joining consecutive points with straight lines.(1)

(Total for Question 1 is 3 marks)

Northfell Dry Ski Slope records the number of ski and snowboard lessons it sells each quarter. The graph shows lesson bookings for Year 1 and Year 2.

(a)Write down the number of lessons booked in Year 2 Quarter 3.(1)

(b)Write down the quarter in which the fewest lessons were booked in both Year 1 and Year 2.(1)

(c)Describe, in one sentence, the pattern shown by the graph over the two years.(1)

(d)State whether the name for this repeating winter-high, summer-low pattern is the trend or the seasonal variation.(1)

(Total for Question 2 is 4 marks)

For each statement about time series and moving averages, write down whether it is True or False.

(i)A 4-point moving average is calculated by adding 4 consecutive data values and dividing the total by 4.(1)

(ii)A moving average removes the trend from a time series, leaving only the seasonal variation.(1)

(iii)Each 4-point moving average value should be plotted at the midpoint in time of the 4 values used to calculate it.(1)

(iv)A time series graph can only be used to show data that was collected once a year, never every month or every quarter.(1)

(Total for Question 3 is 4 marks)

Riverside Bike Hire records the number of bikes it hires out each quarter.

Quarter	Y1 Q1	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1	Y2 Q2	Y2 Q3	Y2 Q4
Bikes hired	30	70	90	50	40	80	100	60

On the grid provided, draw a time series graph to show this information. Plot each point and join them, in order, with straight lines.

(Total for Question 4 is 5 marks)

Riverside Bike Hire's quarterly data is shown again.

Quarter	Y1 Q1	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1	Y2 Q2
Bikes hired	30	70	90	50	40	80

(a)Calculate the first three 4-point moving averages for this data, giving each answer to 1 decimal place where needed.(4)

(b)What do these three moving averages tell you about the trend in bike hire over this period?(1)

(Total for Question 5 is 5 marks)

Riverside Bike Hire's quarterly data (see Question 5) is smoothed using a 4-point moving average rather than a 3-point or 5-point moving average.

(Total for Question 6 is 2 marks)

Riverside Bike Hire's quarterly data is shown again.

Quarter	Y1 Q1	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1	Y2 Q2
Bikes hired	30	70	90	50	40	80

(a)Which of these calculations gives the second 4-point moving average for this data?(1)

A) (30 + 70 + 90 + 50) / 4
B) (70 + 90 + 50 + 40) / 4
C) (90 + 50 + 40 + 80) / 4
D) (30 + 70 + 90 + 50 + 40) / 5

(b)Hence write down the value of this moving average.(1)

(Total for Question 7 is 2 marks)

Mill Lane Market Stall opens on Wednesday, Friday and Saturday each week. The table shows the stall's takings, in pounds, on each trading day.

Day	Week 1 Wed	Week 1 Fri	Week 1 Sat	Week 2 Wed
Takings (£)	120	150	210	135

(a)Since the stall trades 3 times a week, calculate the first two 3-point moving averages for this data.(3)

(b)What do these two moving averages suggest about the trend in the stall's takings?(1)

(Total for Question 8 is 4 marks)

Riverside Bike Hire has now recorded 8 complete quarters of data (see Question 4).

(a)Calculate how many 4-point moving averages can be found from these 8 quarters of data. Show how you worked this out.(2)

(b)Explain why no moving average value can be plotted at Y1 Q1 or at Y2 Q4.(1)

(Total for Question 9 is 3 marks)

A different branch of the same company, Elm Grove Bike Hire, produces five successive 4-point moving averages, in order: 58, 61, 60, 64, 67.

(a)State whether the trend is increasing, decreasing, or roughly constant.(1)

(b)Give a reason for your answer, using values from the list.(1)

(c)A customer claims that, because the trend is increasing, Quarter 3 rentals must always be the highest quarter of the year at every branch. Is the customer correct? Write Yes or No, and give a brief reason.(2)

(Total for Question 10 is 4 marks)

Elmwood Community Library wants to investigate whether the number of books borrowed each quarter shows both a repeating seasonal pattern and a long-term trend, to help decide how many staff to schedule on the enquiries desk each quarter.

(a)Plan: state an appropriate time interval over which to collect loan data, and explain why at least two full years of data should be collected before drawing any conclusion about seasonality.(2)

(b)Collect: state one advantage of using the library's own computerised loan-return records (a secondary data source) rather than carrying out a new survey of members.(1)

(c)Process: the table shows quarterly book loans, in hundreds, for two years.

Quarter	Y1 Q1	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1	Y2 Q2
Books borrowed (hundreds)	42	30	26	50	46	34

Calculate the first two 4-point moving averages for this data.(3)

(d)Interpret: three further moving averages, continuing from part (c), are 39.5, 41 and 42.5. Using all five values, describe the trend, and suggest one staffing decision the library could make based on it, linking back to the original purpose of the investigation.(3)

(Total for Question 11 is 9 marks)

Riverside Bike Hire's 8 quarters of data (see Question 4) are shown again on the graph below.

(a)Calculate the 4th and 5th 4-point moving averages for this data (continuing on from the three you found in Question 5).(3)

(b)Plot all five moving averages (your three from Question 5, and these two) on the graph, marking each with a cross (X) at the correct point in time. Use the five dashed guide lines shown to help you.(2)

(Total for Question 12 is 5 marks)

The graph shows the moving-average trend lines for membership numbers at two branches of BrightFit Gym over the same 8 quarters.

(a)Identify which branch shows an increasing trend, and which branch shows a roughly constant (flat) trend.(2)

(b)The manager of BrightFit South says: 'Our trend line is flat, so our gym is never busier in some quarters than in others.' Comment on whether this is a fair conclusion.(2)

(Total for Question 13 is 4 marks)

Coastview Pier arcade records the number of tokens sold each quarter, along with the trend value (from a moving average) for that quarter.

Quarter	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1
Actual tokens sold (hundreds)	70	93	51	44
Trend (from moving average)	60	63	66	69

(a)Calculate the seasonal variation (actual value minus trend value) for each of the four quarters shown.(4)

(b)Seasonal variations for a full year should sum to (approximately) zero. Use this fact to check your answers to part (a).(2)

(c)State which quarter has the largest positive seasonal variation, and explain what this means in context.(2)

(Total for Question 14 is 8 marks)

Coastview Pier arcade's trend value at Y2 Q1 is 69 (see Question 14), and this trend is expected to keep increasing by 3 each quarter. The seasonal variation for Quarter 2 was found to be +10.

(a)Predict the trend value for Y2 Q2, assuming the trend keeps increasing by 3 each quarter.(2)

(b)Hence forecast the actual number of tokens (in hundreds) the arcade will sell in Y2 Q2, using the trend value from part (a) and the seasonal variation for Quarter 2.(2)

(c)Give one reason why this forecast might not turn out to be accurate.(2)

(Total for Question 15 is 6 marks)

Riverside Bike Hire (see Questions 4, 5 and 12) has now recorded three full years of quarterly data. The graph shows all 12 quarters of raw bike hire figures, together with all nine 4-point moving averages, already plotted as squares. The horizontal axis has been extended to include Year 4 Quarter 1 and Year 4 Quarter 2.

(a)By eye, draw a single straight trend line through the nine moving-average points.(2)

(b)Use your trend line to estimate the trend value at Year 4 Quarter 2, by extending the line to that point on the graph.(2)

(c)Comment on the reliability of this prediction.(2)

(Total for Question 16 is 6 marks)

Mark scheme · S32 Time Series

Question 1

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 True cao
(iii) Answer: True

Question 2

(a) B1 20 cao
(a) Answer: 20
(b) B1 Quarter 3 (Q3) cao
(b) Answer: Quarter 3
(c) B1 identifies both the repeating seasonal pattern (high in Q4 and Q1, low in Q2 and Q3) and that Year 2 is higher than Year 1 overall, oe
(c) Answer: Bookings rise and fall in the same seasonal pattern each year (highest in Q4 and Q1, lowest in Q2 and Q3), and Year 2 bookings are higher than Year 1 in every matching quarter.
(d) B1 Seasonal variation cao
(d) Answer: Seasonal variation

Question 3

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 True cao
(iii) Answer: True
(iv) B1 False cao
(iv) Answer: False

Question 4

B1 a suitable uniform scale chosen on the vertical axis, starting at 0
B1 the first four points plotted correctly: 30, 70, 90, 50 (± half a small square)
B1 the last four points plotted correctly: 40, 80, 100, 60 (± half a small square)
B1 all 8 points joined, in time order, with straight line segments
B1 graph fully labelled, with both axes labelled
Answer: Time series graph with points at 30, 70, 90, 50, 40, 80, 100, 60 joined in order.

Question 5

(a) M1 correct method shown for at least one moving average, e.g. (30+70+90+50)/4
(a) A1 60 cao
(a) A1 62.5 cao
(a) A1 65 cao
(a) Answer: 60, 62.5, 65
(b) B1 the trend is increasing/rising, since the moving averages increase from 60 to 62.5 to 65, oe
(b) Answer: The trend is increasing.

Question 6

B1 there are 4 quarters in a year
B1 so each group of 4 consecutive values includes exactly one value from each quarter/season, so the seasonal effect cancels out and only the trend remains, oe
Answer: A 4-point moving average is used because there are 4 quarters in a year, so every group of 4 consecutive values contains exactly one value from each season - this cancels out the seasonal variation and leaves only the trend.

Question 7

(a) B1 B cao
(a) Answer: B) (70 + 90 + 50 + 40) / 4
(b) B1 62.5 cao
(b) Answer: 62.5

Question 8

(a) M1 correct method shown, e.g. (120+150+210)/3
(a) A1 160 cao
(a) A1 165 cao
(a) Answer: 160, 165
(b) B1 the trend is increasing, since the moving average rises from 160 to 165, oe
(b) Answer: The trend is increasing.

Question 9

(a) M1 correct method, e.g. 8 - 4 + 1, or lists the valid groups (quarters 1-4, 2-5, 3-6, 4-7, 5-8)
(a) A1 5 cao
(a) Answer: 5
(b) B1 each moving average is plotted at the midpoint in time of the 4 values used; Y1 Q1 and Y2 Q4 are always at the very start or end of a group of 4, never at the middle, so no moving average is centred there, oe
(b) Answer: Y1 Q1 and Y2 Q4 can never be the midpoint of a group of 4 consecutive quarters, so no moving average is plotted there.

Question 10

(a) B1 Increasing cao
(a) Answer: Increasing
(b) B1 compares the first and last values (e.g. 67 is greater than 58), or notes the values generally rise overall, oe
(b) Answer: The final moving average (67) is higher than the first (58), and the values generally rise across the list.
(c) B1 No
(c) B1 moving averages smooth out/remove the seasonal variation between quarters, so the trend on its own cannot show which individual quarter is highest within a year, oe
(c) Answer: No - the moving average removes the seasonal variation between quarters, so it cannot show which individual quarter has the highest raw rentals within a year.

Question 11

(a) B1 states a suitable interval, e.g. quarterly (or monthly)
(a) B1 explains that at least two full cycles/years are needed to check that a pattern genuinely repeats, rather than being a one-off event in a single year, oe
(a) Answer: Record the number of books borrowed each quarter, for at least two full years, so any pattern can be checked for repeating rather than being a one-off.
(b) B1 e.g. the data already exists / records every loan exactly / is quicker and cheaper than running a new survey, oe
(b) Answer: Loan-return records already exist and count every loan exactly, so they are quicker and cheaper to use than collecting new survey data.
(c) M1 correct method shown, e.g. (42+30+26+50)/4
(c) A1 37 cao
(c) A1 38 cao
(c) Answer: 37, 38
(d) B1 trend is increasing (book loans are rising from one year to the next), since 37, 38, 39.5, 41, 42.5 rises steadily
(d) B1 identifies from the raw data that Quarter 4 (and Quarter 1) tend to be busier within the year than Quarter 2 or Quarter 3 (e.g. Y1 Q4 = 50, Y1 Q1 = 42, compared with Q2 = 30 and Q3 = 26)
(d) B1 gives a sensible staffing recommendation linked to the purpose, e.g. schedule extra enquiries-desk staff every Quarter 4 and Quarter 1, and gradually increase baseline staffing over time as the underlying trend rises, oe
(d) Answer: The trend is increasing; Quarter 4 and Quarter 1 are the busier quarters each year, so the library should schedule extra enquiries-desk staff in those quarters and gradually increase overall staffing as the rising trend continues.

Question 12

(a) M1 correct method shown, e.g. (50+40+80+100)/4
(a) A1 67.5 cao
(a) A1 70 cao
(a) Answer: 67.5, 70
(b) B1 first three moving averages (60, 62.5, 65) plotted at the correct guide lines, ± half a small square
(b) B1 next two moving averages (67.5, 70) plotted at the correct guide lines, ± half a small square
(b) Answer: Points at (between Y1Q2/Y1Q3, 60), (between Y1Q3/Y1Q4, 62.5), (between Y1Q4/Y2Q1, 65), (between Y2Q1/Y2Q2, 67.5), (between Y2Q2/Y2Q3, 70).

Question 13

(a) B1 BrightFit North - increasing trend
(a) B1 BrightFit South - roughly constant/flat trend
(a) Answer: BrightFit North: increasing. BrightFit South: roughly constant.
(b) B1 not fair/not correct - a flat trend only means the underlying long-term average is not rising or falling
(b) B1 the raw quarterly membership figures can still rise and fall with the season (e.g. a January new year rush) around a flat trend; the moving average simply smooths/removes that seasonal variation from view, oe
(b) Answer: Not a fair conclusion. A flat trend line only shows that the long-term average is not changing - BrightFit South's raw quarterly numbers could still rise and fall with the seasons (for example, a January rush); the moving average has simply smoothed that seasonal variation out of the graph.

Question 14

(a) B1 Y1 Q2: +10 cao
(a) B1 Y1 Q3: +30 cao
(a) B1 Y1 Q4: -15 cao
(a) B1 Y2 Q1: -25 cao
(a) Answer: +10, +30, -15, -25
(b) M1 sums all four seasonal variations found: 10+30-15-25
(b) A1 sum = 0, confirming the values in part (a) are consistent with the rule, oe
(b) Answer: 10 + 30 - 15 - 25 = 0, which confirms the seasonal variations are consistent.
(c) B1 Y1 Q3
(c) B1 token sales in Quarter 3 are typically about 30 (hundred) higher than the underlying trend, e.g. because it is the busiest (summer) season for a pier arcade, oe
(c) Answer: Y1 Q3, with seasonal variation +30 - arcade token sales in Quarter 3 are typically about 3000 above the trend, consistent with Quarter 3 (summer) being the busiest season for a seaside pier.

Question 15

(a) M1 69 + 3
(a) A1 72 cao
(a) Answer: 72
(b) M1 72 + 10
(b) A1 82 cao
(b) Answer: 82
(c) B2 a valid reason that the trend or seasonal pattern might not continue as assumed, e.g. unusually poor weather that quarter, a new arcade opening nearby, a change in ticket prices, or any other sensible real-world factor, oe
(c) Answer: For example, unusually poor weather in Quarter 2 could reduce visitor numbers below the forecast, or a new arcade opening nearby could take away some customers, so the trend and seasonal pattern might not continue exactly as assumed.