GCSE Statistics · Higher · Chapter 9: Time series

Time series, moving averages and seasonal variation

About 80 minutesCalculator allowed80 marks

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Date

Do Now

The first two are what this chapter needs. The rest bring back earlier chapters so nothing is forgotten.

A1
Needed today
The 18 plants in tray A have a mean height of $48$ cm. The 10 plants in tray B have a mean height of $20$ cm. Work out the mean height of all the plants in the two groups together.
[2 marks]
Answer

A2
Needed today
Find the distance between the points $(- 4, - 2)$ and $(8, - 11)$ .
[2 marks]
Answer
A3
Maths skill
Work out $1.96$ .
[2 marks]
Answer
A4
Maths skill
Given that $27 \times 38 = 1026$ , work out $2.7 \times 0.38$ .
[2 marks]
Answer
A5
Maths skill
A spinner can land on A, B, C or D. $P (A) = 0.2$ , $P (B) = 0.2$ and $P (C) = 0.3$ . It is spun $100$ times. How many times would you expect it to land on D?
[2 marks]
Answer

Examples, then your turn

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

The graph shows a shop's umbrella sales in every quarter of a three-year period. Comment on how sales change over the years, and on the pattern within each year.

Hint The trend is the change from year to year; the seasonal pattern is what happens in the same quarter each year.

B1
The graph shows how many people (in thousands) visited a theme park in every quarter of three years. Comment on the long-term movement and on the pattern inside each year.
[2 marks]

Example 2

The table gives a library's book loans per quarter. Find every 4-point moving average. At what point on the time axis does the second one go?

Hint The second average uses Q2 of year 1 to Q1 of year 2.

B2
The table gives bus passenger numbers (thousands) per quarter. Calculate the 4-point moving averages that start in Q1, Q2 and Q3 of year 1. At what point on the time axis is the second placed?
[3 marks]
Answer

Example 3

A bakery is open Monday to Friday. The loaves it sold over two weeks were $40$ , $52$ , $47$ , $61$ , $75$ , $42$ , $55$ , $49$ , $63$ and $79$ . Work out the first two 5-point moving averages, and say why 5 points are used.

Hint The pattern repeats every week of five days.

B3
A hairdresser is open Tuesday to Saturday. Its customers on seven days in a row were $18$ , $24$ , $21$ , $30$ , $37$ , $20$ and $26$ , starting on a Tuesday. Calculate the first two 5-point moving averages. Against which day is the first average plotted?
[3 marks]
Answer

Example 4

Over three years a cafe's Q2 sales came to $62$ , $70$ and $75$ , against trend values of $55$ , $61$ and $69$ . What is the mean seasonal variation for Q2?

Hint Find each seasonal variation as actual

-

trend, then take their mean.

B4
In three successive years a florist took $118$ , $131$ and $140$ orders in Q4; the trend values for those quarters were $132$ , $141$ and $155$ . Calculate the Q4 mean seasonal variation.
[3 marks]
Answer

Example 5

The first 4-point moving average, $210$ , sits midway between Q2 and Q3 of year 1. The ninth, $234$ , sits midway between Q2 and Q3 of year 3. Q1 has a mean seasonal variation of $- 18$ . Assume the trend is a straight line. Predict the value for Q1 of year 4.

Hint Work out the change in the trend per quarter, then the trend for Q1 of year 4.

B5
Two points on a straight trend line are the moving averages $96$ (centred halfway through year 1, between its Q2 and Q3) and $84$ (centred halfway through year 2). Q4 has a mean seasonal variation of $9$ . Assume the trend is a straight line. Predict the value for Q4 of year 3.
[3 marks]
Answer

Example 6

Explain why a prediction for next quarter is likely to be more reliable than a prediction for three years ahead.

Hint What has to stay the same for the prediction to work?

B6
A shop used its trend line and the mean seasonal variation for Q4 to predict its Q4 sales. The actual sales were much lower. Give two possible reasons.
[2 marks]

Practice

Each question asks for something different. Show your working.

C1
A cinema sold $300$ , $420$ and $360$ tickets on the Friday, Saturday and Sunday of one weekend, and $291$ , $444$ and $369$ on the next. Work out the 3-point moving averages.
[3 marks]
Answer

C2
A 3-point moving average is $45$ . Two of its three values are $38$ and $49$ . Work out the third value.
[2 marks]
Answer
C3
A theatre records its ticket sales each month for five years. The pattern repeats every year. What order of moving average should it use? Give a reason.
[2 marks]
C4
A 4-point moving average uses the values from Q3 of year 1 to Q2 of year 2. Between which two quarters is it plotted?
[1 mark]
Answer
C5
In Q1 a cafe sold $3420$ hot drinks, against a trend value of $2980$ . Find the seasonal variation for that quarter and explain what the number tells you.
[2 marks]
C6
Over four years a farm shop's Q3 seasonal variations came to $- 22$ , $- 18$ , $- 25$ and $- 19$ . What is their mean?
[2 marks]
Answer
C7
A straight trend line is drawn through two moving averages: $140$ , centred in the middle of year 1, and $164$ , centred in the middle of year 3. Estimate where the line is at Q3 of year 3.
[2 marks]
Answer
C8
At Q4 of year 3 a trend line reads $250$ , and it drops $4$ every quarter. Q3 has a mean seasonal variation of $22$ . Predict the value for Q3 of year 4.
[3 marks]
Answer

Exam-style questions

Answer every question in the space given. Show your working: most of the marks are for method.

D1
A shop sold $58$ games consoles in Q2 of year 2. The mean of all twelve quarters of sales is $50$ , and the trend value for Q2 of year 2 is $54$ . Ella says, "The seasonal variation for Q2 of year 2 is $58 - 50 = 8$ ."
Is Ella correct? Give a reason.
[2 marks]

D2
A school's electricity use (MWh) over twelve quarters is in the table. Its nine 4-point moving averages, in order, are $56$ , $55$ , $c$ , $53.5$ , $52$ , $51.25$ , $d$ , $49.5$ and $49$ .
(a)
Calculate $c$ and $d$ .
[2 marks]
Answer
(b)
Where on the time axis does the final average, $49$ , go?
[1 mark]
Answer
(c)
What do the moving averages show about the trend? Use a figure in your answer.
[2 marks]
Total for D2: 5 marks
D3
A firm's boiler repairs over twelve quarters are graphed. Its first eight 4-point moving averages are marked with crosses, and a dashed trend line runs on into year 4. The seasonal variations for Q1 in years 1 and 2 were $14$ and $17$ .
(a)
Calculate the ninth 4-point moving average and say where on the time axis it belongs.
[2 marks]
Answer
(b)
The firm made $82$ repairs in Q1 of year 3. Read the trend line to find that quarter's seasonal variation.
[1 mark]
Answer
(c)
Find the mean seasonal variation for Q1.
[2 marks]
Answer
(d)
Predict the repairs in Q1 of year 4, reading the trend from the dashed line.
[1 mark]
Answer
Total for D3: 6 marks
D4
Tom has a shop's sales for each quarter of four years. He works out 3-point moving averages to show the trend. Explain why this is not a good choice, and say what he should use.
[2 marks]
D5
A hotel's bookings were $70$ , $92$ and $101$ in Q1, Q2 and Q3 of year 1. The 4-point moving average for Q1 to Q4 of year 1 is $85$ , and the next 4-point moving average is $87$ . Work out the bookings in Q4 of year 1 and in Q1 of year 2.
[3 marks]
Answer
D6
A firm's quarterly sales have a straight trend line through the moving averages $136$ (middle of year 1) and $152$ (middle of year 3). Q4 has a mean seasonal variation of $- 16$ .
(a)
Predict the sales for Q4 of year 4.
[3 marks]
Answer
(b)
The sales in Q4 of year 4 were $128$ . Comment on your prediction.
[1 mark]
Total for D6: 4 marks
D7
The mean seasonal variations for an ice cream van's sales are $- 18$ for Q1, $6$ for Q2, $25$ for Q3 and $- 13$ for Q4.
(a)
What does the mean seasonal variation for Q3 tell you?
[1 mark]
(b)
Work out the total of the four mean seasonal variations. Explain why this total is what you would expect.
[2 marks]
Total for D7: 3 marks
D8
The 4-point moving averages for a gym's members are $412$ , $409$ , $405$ , $404$ , $398$ and $395$ .
(a)
Describe the trend.
[1 mark]
(b)
Work out the mean change in the moving averages per quarter.
[2 marks]
Answer
Total for D8: 3 marks

Extension

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Consecutive 4-point moving averages of $68$ and $72$ are plotted half a quarter before and half a quarter after Q3 of year 1. That quarter had $85$ sales. Using a centred moving average, find its seasonal variation.
[3 marks]
Hint 1 · What to try
Neither average is plotted at Q3 itself. The mean of the two is.
Hint 2 · The first line
The centred moving average is $(68 + 72) \div 2 = 70$ , plotted at Q3.
Hint 3 · The full method
The seasonal variation is $85 - 70$ .
Answer

E2
A firm's prediction for Q1 next year is $145$ . The mean seasonal variation for Q1 is $- 23$ , and the trend rises by $4$ each quarter. Work out the trend value for Q3 this year.
[3 marks]
Hint 1 · What to try
Prediction = trend + mean seasonal variation. Find the trend for Q1 next year.
Hint 2 · The first line
The trend for Q1 next year is $145 + 23 = 168$ .
Hint 3 · The full method
Q3 this year is 2 quarters before Q1 next year.
Answer
E3
Three 3-point moving averages in a row are $20$ , $22$ and $25$ . The first two values in the list are $17$ and $21$ . Work out the next three values.
[3 marks]
Hint 1 · What to try
Each moving average times 3 is the total of its three values.
Hint 2 · The first line
The totals are $60$ , $66$ and $75$ . The first total gives the third value.
Hint 3 · The full method
Each new total adds one new value and drops the oldest.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can describe a time series and work out moving averages of the right order, plotted at the middle of their period.
I can work out seasonal variations from the trend, and predict a value from the trend and the mean seasonal variation.

Answers and mark scheme

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