GCSE Statistics · Higher · Chapter 9: Time series · Lesson 1

Time series and moving averages

About 50 minutesCalculator allowed42 marks

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Date

Do Now

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
The ages of 7 people in a choir were 35, 130, 30, 50, 44, 49, 33. Which average best represents these values? Give its value and a reason.
[2 marks]
A2
From chapter 4
A choropleth map shades hospital visits per 1000 people. District G had 171 hospital visits, which is 4.5 per 1000 people. How many more hospital visits would have put District G in the band "10 or more per 1000"?
[2 marks]
Answer
A3
From chapter 7
For the masses, in grams, of some parcels at a depot, the mode is 38, the median is 32 and the mean is 25. Describe the skew of the distribution and say what it means in context.
[2 marks]
A4
Last lesson
A machine fills bags of rice. The target is $2000$ g and the standard deviation of the sample means is $1.2$ g. Work out the warning limits and the action limits for a control chart of sample means.
[2 marks]
Answer

Examples, then your turn

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

Example 1

The graph shows a shop's ice cream sales each quarter for three years. Describe the trend and the seasonal pattern.

Hint The trend is what happens over the years. The seasonal pattern is what happens in the same quarter each year.

B1
The graph shows the mean household electricity bill each quarter for three years. Describe the trend and the seasonal pattern.
[2 marks]

Example 2

A museum had $42$ , $51$ , $48$ , $57$ , $54$ and $63$ visitors on six days, Monday to Saturday. Work out the 3-point moving averages. Which day is the first one plotted at?

Hint Average the first three days, then move along one day at a time.

B2
A radio show was sent $36$ , $45$ , $39$ , $48$ , $42$ , $51$ and $45$ texts on days 1 to 7. Work out the first two 3-point moving averages. Which day is the first one plotted at?
[3 marks]
Answer

Example 3

The table shows a hotel's bookings each quarter. Work out the 4-point moving averages. Where is the first one plotted?

Hint Average Q1 to Q4 of year 1, then Q2 of year 1 to Q1 of year 2, and so on.

B3
The table shows a shop's coat sales each quarter. Work out the first three 4-point moving averages. Where is the first one plotted?
[3 marks]
Answer

Example 4

Explain why a 4-point moving average is used for quarterly data.

Hint How many quarters does it take for the pattern to repeat?

B4
A school shop records its uniform sales in each of the three terms of the year. Which moving average should it use to show the trend? Give a reason.
[2 marks]

Example 5

A 4-point moving average of quarterly sales is $62$ . Three of the four quarters had sales of $58$ , $70$ and $51$ . Work out the sales in the fourth quarter.

Hint The four values must add up to

4 \times 62

B5
A 4-point moving average of a gym's quarterly new members is $47.5$ . Three of the four quarters had $40$ , $52$ and $61$ new members. Work out the number in the fourth quarter.
[2 marks]
Answer

Practice

C1
A cafe sold $64$ , $70$ , $58$ , $73$ and $61$ cups of tea on five days, Monday to Friday. Work out the 3-point moving averages.
[3 marks]
Answer
C2
The table shows a chemist's sunscreen sales each quarter. Work out the three 4-point moving averages, and say where the last one is plotted.
[3 marks]
Answer

C3
Ellie records how many pupils buy lunch each school day (Monday to Friday) for four weeks. Which moving average should she use? Give a reason.
[2 marks]
C4
A 4-point moving average of a cinema's quarterly ticket sales (thousands) is $112$ . Three of the quarters had $96$ , $130$ and $118$ . Work out the fourth.
[2 marks]
Answer

Exam-style questions

D1
A train company carried $312$ , $285$ , $340$ and $299$ thousand passengers in the four quarters of one year. Zara says, "The 4-point moving average is $309$ , and I plot it at Q4."
Is Zara correct? Show your working.
[3 marks]

D2
The table shows the number of rail passengers (thousands) at a station each quarter for three years. The 4-point moving averages are $52$ , $53.25$ , $54.25$ , $a$ , $56.5$ , $57.5$ , $58.75$ , $b$ and $61$ .
(a)
Work out the values of $a$ and $b$ .
[2 marks]
Answer
(b)
Between which two quarters is the moving average $52$ plotted?
[1 mark]
Answer
(c)
Describe the seasonal pattern.
[1 mark]
(d)
Use the moving averages to describe the trend. Give a value in your answer.
[2 marks]
Total for D2: 6 marks

Extension

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
For one firm's quarterly sales, the 4-point moving average of quarters 1 to 4 is $40$ , and the 4-point moving average of quarters 2 to 5 is $43$ . Sales in quarter 1 were $34$ . Work out the sales in quarter 5.
[3 marks]
Hint 1 · What to try
Turn each moving average into the total of its four quarters.
Hint 2 · The first line
The totals are $160$ and $172$ . They share quarters 2, 3 and 4.
Hint 3 · The full method
So quarter 5 is $172 - 160 = 12$ more than quarter 1.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can describe a time series by its trend and its seasonal pattern.
I can work out 3-point and 4-point moving averages and plot each at the middle of its period.
I can choose the order of a moving average from the repeating pattern.

Answers and mark scheme

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