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GCSE · Statistics

S15 Time Series and Moving Averages

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Statisticians use special words to describe patterns in data collected over time.

(a)State the name given to the smooth, general direction of change shown by a moving average, once the ups and downs within each year have been smoothed out.(1)

(b)State the name given to the regular rise and fall in a time series that repeats in the same way each year, such as ice cream sales rising every summer.(1)

(Total for Question 1 is 2 marks)

The Bay View Ice Cream Kiosk records the number of ice creams it sells each quarter. The time series graph shows sales for Year 1 and Year 2.

(a)Write down the number of ice creams sold in Year 2 Quarter 3.(1)

(b)Write down the quarter in which sales are lowest in both Year 1 and Year 2.(1)

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

(Total for Question 2 is 3 marks)

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

(i)A time series is a set of observations recorded at successive points in time, usually at equal intervals.(1)

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

(iii)Each 4-point moving average value is plotted at the same time point as the last of the 4 values used to calculate it.(1)

(iv)The trend is the underlying long-term pattern that remains once a moving average has smoothed out seasonal variation.(1)

(Total for Question 3 is 4 marks)

A pharmacy records the number of bottles of sun cream it sells each quarter.

Quarter	Y1 Q1	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1	Y2 Q2	Y2 Q3	Y2 Q4
Bottles sold	10	40	70	20	20	50	80	30

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)

A garden centre records the number of bags of compost it sells each quarter.

Quarter	Y1 Q1	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1	Y2 Q2
Bags of compost sold	54	88	112	46	58	94

(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 compost sales over this period?(1)

(Total for Question 5 is 5 marks)

Quarterly data, like the compost sales in 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)

A garden centre's quarterly compost sales are shown again.

Quarter	Y1 Q1	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1	Y2 Q2
Bags of compost sold	54	88	112	46	58	94

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

A) (54 + 88 + 112 + 46) / 4
B) (88 + 112 + 46 + 58) / 4
C) (112 + 46 + 58 + 94) / 4
D) (54 + 88 + 112 + 46 + 58) / 5

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

(Total for Question 7 is 2 marks)

A shop's quarterly sales are recorded for 2 complete years, giving 8 quarters of data in total.

(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 Quarter 1 or at Quarter 8.(1)

(Total for Question 8 is 3 marks)

A shop's quarterly sales produce the following five successive 4-point moving averages, in order: 120, 118, 121, 125, 130.

(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 student claims that, because the trend is increasing, Quarter 3 sales must always be the highest quarter of the year. Is the student correct? Write Yes or No, and give a brief reason.(2)

(Total for Question 9 is 4 marks)

A theme park manager wants to investigate whether monthly visitor numbers show a repeating seasonal pattern and a long-term trend, to help plan how many staff to employ each quarter.

(a)Plan: state an appropriate time interval over which to collect visitor number 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 park's own turnstile records (a secondary data source) rather than carrying out a new visitor survey.(1)

(c)Process: the table shows quarterly visitor numbers, in thousands, for two years.

Quarter	Y1 Q1	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1	Y2 Q2	Y2 Q3	Y2 Q4
Visitors (thousands)	8	22	35	12	10	25	39	14

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

(d)Interpret: three further moving averages, continuing from part (c), are 20.5, 21.5 and 22. Using all five values, describe the trend, and suggest one staffing decision the manager could make based on it, linking back to the original purpose of the investigation.(3)

(Total for Question 10 is 9 marks)

The Bay View Ice Cream Kiosk (see Question 2) has now recorded three full years of quarterly sales.

Quarter	Y1 Q1	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1	Y2 Q2	Y2 Q3	Y2 Q4	Y3 Q1	Y3 Q2	Y3 Q3	Y3 Q4
Ice creams sold	20	50	80	30	30	60	90	40	30	70	100	40

The graph shows this data plotted as a time series.

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

(b)Two further moving averages have already been calculated: the third is 50 and the fourth is 52.5. Plot all four moving averages (your two values from part (a), and these two) on the graph, marking each with a cross (X) at the correct point in time. Use the four dashed guide lines shown to help you.(2)

(Total for Question 11 is 5 marks)

The graph shows the moving-average trend lines for ice cream sales in two different towns over the same 8 quarters.

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

(b)The manager of Town B says: 'Our trend line is flat, so our ice cream sales are never affected by the season.' Comment on whether this is a fair conclusion.(2)

(Total for Question 12 is 4 marks)

A garden centre sells hanging baskets. The table shows the actual number sold each quarter, alongside the trend value (from a moving average) for that quarter.

Quarter	Y1 Q2	Y1 Q3	Y1 Q4	Y2 Q1
Actual hanging baskets sold	64	78	38	32
Trend (from moving average)	50	52	54	56

(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 13 is 8 marks)

The garden centre's hanging basket sales (see Question 13) have a trend value of 56 at Y2 Q1, and this trend is expected to keep increasing by 2 each quarter. The seasonal variation for Quarter 2 was found to be +14.

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

(b)Hence forecast the actual number of hanging baskets the garden centre 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 14 is 6 marks)

The graph shows the Bay View Ice Cream Kiosk's 12 quarters of raw sales data (from Question 11), 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 15 is 6 marks)

Mark scheme · S15 Time Series and Moving Averages

Question 1

(a) B1 Trend cao
(a) Answer: Trend
(b) B1 Seasonal variation cao
(b) Answer: Seasonal variation

Question 2

(a) B1 90 cao
(a) Answer: 90
(b) B1 Quarter 1 (Q1) cao
(b) Answer: Quarter 1
(c) B1 identifies both the repeating seasonal pattern (low in Q1, high in Q3) and that Year 2 is higher than Year 1 overall, oe
(c) Answer: Sales rise and fall in the same seasonal pattern each year (lowest in Q1, highest in Q3), and Year 2 sales are higher than Year 1 in every matching quarter.

Question 3

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

Question 4

B1 a suitable uniform scale chosen on the vertical axis, starting at 0
B1 the first four points plotted correctly: 10, 40, 70, 20 (± half a small square)
B1 the last four points plotted correctly: 20, 50, 80, 30 (± 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 10, 40, 70, 20, 20, 50, 80, 30 joined in order.

Question 5

(a) M1 correct method shown for at least one moving average, e.g. (54+88+112+46)/4
(a) A1 75 cao
(a) A1 76 cao
(a) A1 77.5 cao
(a) Answer: 75, 76, 77.5
(b) B1 the trend is increasing/rising, since the moving averages increase from 75 to 76 to 77.5, 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) (88 + 112 + 46 + 58) / 4
(b) B1 76 cao
(b) Answer: 76

Question 8

(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; quarter 1 and quarter 8 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: Quarter 1 and Quarter 8 can never be the midpoint of a group of 4 consecutive quarters, so no moving average is plotted there.

Question 9

(a) B1 Increasing cao
(a) Answer: Increasing
(b) B1 compares the first and last values (e.g. 130 is greater than 120), or notes the values generally rise overall, oe
(b) Answer: The final moving average (130) is higher than the first (120), 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 sales within a year.

Question 10

(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: Collect visitor numbers 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 visitor exactly / is quicker and cheaper than running a new survey, oe
(b) Answer: Turnstile records already exist and count every visitor exactly, so they are quicker and cheaper to use than collecting new survey data.
(c) M1 correct method shown, e.g. (8+22+35+12)/4
(c) A1 19.25 cao
(c) A1 19.75 cao
(c) Answer: 19.25, 19.75
(d) B1 trend is increasing (visitor numbers are rising from one year to the next), since 19.25, 19.75, 20.5, 21.5, 22 rises steadily
(d) B1 identifies that Quarter 3 remains the seasonal peak each year (raw data 35 then 39)
(d) B1 gives a sensible staffing recommendation linked to the purpose, e.g. schedule extra staff every Quarter 3, and gradually increase baseline staffing over time as the underlying trend rises, oe
(d) Answer: The trend is increasing; Quarter 3 is the seasonal peak each year, so the manager should schedule extra staff every Quarter 3 and gradually increase overall staffing as the rising trend continues.

Question 11

(a) M1 correct method shown, e.g. (20+50+80+30)/4
(a) A1 45 cao
(a) A1 47.5 cao
(a) Answer: 45, 47.5
(b) B1 first two moving averages (45, 47.5) plotted at the correct guide lines, ± half a small square
(b) B1 next two moving averages (50, 52.5) plotted at the correct guide lines, ± half a small square
(b) Answer: Points at (between Y1Q2/Y1Q3, 45), (between Y1Q3/Y1Q4, 47.5), (between Y1Q4/Y2Q1, 50), (between Y2Q1/Y2Q2, 52.5).

Question 12

(a) B1 Town A - increasing trend
(a) B1 Town B - roughly constant/flat trend
(a) Answer: Town A: increasing. Town B: 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 sales can still rise and fall with the season 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 - the raw quarterly sales in Town B could still rise and fall with the seasons; the moving average has simply smoothed that seasonal variation out of the graph.

Question 13

(a) B1 Y1 Q2: +14 cao
(a) B1 Y1 Q3: +26 cao
(a) B1 Y1 Q4: -16 cao
(a) B1 Y2 Q1: -24 cao
(a) Answer: +14, +26, -16, -24
(b) M1 sums all four seasonal variations found: 14+26-16-24
(b) A1 sum = 0, confirming the values in part (a) are consistent with the rule, oe
(b) Answer: 14 + 26 - 16 - 24 = 0, which confirms the seasonal variations are consistent.
(c) B1 Y1 Q3
(c) B1 sales in Quarter 3 are typically about 26 higher than the underlying trend, e.g. because it is the busiest (summer) season for hanging baskets, oe
(c) Answer: Y1 Q3, with seasonal variation +26 - hanging basket sales in Quarter 3 are typically about 26 above the trend, consistent with Quarter 3 (summer) being the busiest season for garden sales.

Question 14

(a) M1 56 + 2
(a) A1 58 cao
(a) Answer: 58
(b) M1 58 + 14
(b) A1 72 cao
(b) Answer: 72
(c) B2 a valid reason that the trend or seasonal pattern might not continue as assumed, e.g. unusual weather that quarter, a new garden centre opening nearby, a change in prices, or any other sensible real-world factor, oe
(c) Answer: For example, unusually poor weather in Quarter 2 could reduce sales below the forecast, or a new competing garden centre opening nearby could take away some customers, so the trend and seasonal pattern might not continue exactly as assumed.

Question 15

(a) B1 a single straight line is drawn (a ruled line, not curved or joining every point exactly)
(a) B1 the line follows the general increasing direction of the moving-average points, passing reasonably close to them
(a) Answer: A straight, rising line drawn through (or close to) the nine moving-average squares, from about 45 up to about 60.
(b) M1 extends the ruled trend line beyond the plotted points as far as the Y4 Q2 guide line
(b) A1 a value consistent with the candidate's own trend line, accept 60 to 75 (ft)
(b) Answer: Approximately 66-68 (accept any value from 60 to 75 consistent with the candidate's own line).
(c) B1 identifies that this is extrapolation - predicting beyond the range of the given data - which assumes the trend continues in exactly the same way
(c) B1 a suitable contextual limitation, e.g. ice cream sales depend on weather which varies year to year, a competitor kiosk could open nearby, or the kiosk's opening hours/location could change, so the prediction may not be reliable, oe
(c) Answer: The prediction is an extrapolation beyond the data collected, and assumes the trend continues exactly as before. In reality, ice cream sales depend heavily on the weather, which varies from year to year, and a new competitor could open nearby, so the prediction may not be reliable.

Time Series and Moving Averages - Worksheets, Questions and Revision