Teach GCSE Maths Time Time

Series Series and and Moving Moving

AveragesAverages

Live Births: England and

Wales

Nu

mb

er

of

Bir

ths

( th

ou

san

ds

)

Year1995 2000 2005

600

700

© Christine Crisp

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Time Series Time Series andand

Moving AveragesMoving Averages

Data from the Office for National Statistics which is included in this presentation is reproduced under the terms of the click-use licence.

The data in all the following data sets have been collected at certain intervals of time.

Absence from work ( daily )

Gas bills ( quarterly )Average hourly pay ( yearly )

Quarterly means 4 times a

year.

Average temperatures (monthly )

A graph showing data like these, is called a time series.

Tell your partner at least 2 things this graph tells you ( one each ! )

Ans: Between 1986 and 2004, the average hourly earnings of males was always higher than females. The average hourly earnings of both sexes has increased every year between 1986 and 2004.

Source: Office for National Statistics

MalesFemales

Year

Earn

ing

s(£

per

hou

r)

e.g.Average Hourly Earnings

We have to be very careful when making predictions from data but the consistency of this graph suggests that the pay gap between males and females is very unlikely to change in the next few years.However, some time series are less clear.

Source: Office for National Statistics

MalesFemales

Year

Earn

ing

s(£

per

hou

r)

e.g.Average Hourly Earnings

e.g. Use the graph below to predict the birth rate for 2005.

Live Births ( England and Wales )

Nu

mb

er

of

Bir

ths

( th

ou

san

ds )

YearSource: Office for National Statistics

In 2002, the numbers went up slightly . . .

but the trend since 1992 is downwards.

We do this by averaging the values for several years.

To predict from the graph we need to “smooth” it.

Suppose we average the first 5 values ( for 1992 to 1996 ).

650

648

665

674

690

Births( ‘000s )

5962002

5952001

6042000

6221999

6361998

6431997

1996

1995

1994

1993

1992

YearThis is the data set.

690 + 674 + 665 + 648 + 6505

= 665 ( 3 s.f. )

Decide with your partner which year you would plot this

value against.Ans: Since it is an average, we plot at 1994, the middle of the 5 years.

5962002

5952001

6042000

6221999

6361998

6431997

6501996

6481995

6651994

6741993

6901992

Births( ‘000s )

YearSuppose we average the first 5 values ( for 1992 to 1996 ).

This is the data set.

690 + 674 + 665 + 648 + 6505

= 665 ( 3 s.f. )

Decide with your partner which year you would plot this

value against.Ans: Since it is an average, we plot at 1994, the middle of the 5 years.

665

643

690

5962002

5952001

6042000

6221999

6361998

1997

6501996

6481995

6651994

6741993

1992

Births( ‘000s )

Year

665

656

Moving Averag

esFor the 2nd average, we drop the value for the 1st year ( 1992 ) and include the value for 1997.

= 656

674 + 665 + 648 + 650 + 6435

We continue like this moving the averages forward . . .

596

595

604

622

636

643

650

648

665

674

690

2002

2001

2000

1999

1998

1997

1996

1995

1994

1993

1992

Births( ‘000s )

YearFor the 2nd average, we drop the value for the 1st year ( 1992 ) and include the value for 1997. 665

611

620

631

640

648

656

= 656

674 + 665 + 648 + 650 + 6435

We continue like this moving the averages forward . . . until we no longer have 5 values to average.We can now plot the points on the time series graph.

Moving Averag

es

xx

x

x

x

x

x

We say the trend in the birth rate is downwards.

5-point moving averages

Live Births ( England and Wales )

Nu

mb

er

of

Bir

ths

( th

ou

san

ds )

Year

YearBirths (‘000

s)

Moving averag

e

1992 690

1993 674

1994 665 665

1995 648 656

1996 450 648

1997 643 640

1998 646 631

1999 622 620

2000 604 611

2001 595

2002 596

xx

x

x

x

x

x

5-point moving averages

Live Births ( England and Wales )

Nu

mb

er

of

Bir

ths

( th

ou

san

ds )

Year

YearBirths (‘000

s)

Moving averag

e

1992 690

1993 674

1994 665 665

1995 648 656

1996 450 648

1997 643 640

1998 646 631

1999 622 620

2000 604 611

2001 595

2002 596

600

To predict the birth rate for 2003, we extend the trend line to find the next moving average.

600

To predict the birth rate for 2003, we extend the trend line to find the next moving average.

An average of 600 means that the total for the 5 years from 1999 to 2003 is 5 600 = 3000

2003

5962002

5952001

6116042000

6206221999

6316461998

6406431997

6484501996

6566481995

6656651994

6741993

6901992

Moving averag

e

Births (‘000

s)Year

600

583

To find the 2003 estimate we can subtract the values for the 4 years we know ( 1999 to 2002 ).

Estimate for 2003

= 3000 622 604 595 596

= 583

x

There is no obvious number of points to use for a moving average with yearly (annual) data and 5 was about right for the number of values I had.

This table gives my quarterly gas bills.

The 1st quarter, Q1, covers the gas used from February to April, the 2nd from May to July and so on.

DateBill

(£)

2004 Q1 93

Q2 24

Q3 37

Q4 142

2005 Q1 89

Q2 27

Q3 36

Q4 173

2006 Q1 164

Q2 35

Q3 53

Q4 198

Decide with your partner how many points to use for the moving average. Where would you plot the 1st average ?

DateBill (£)

2004 Q1 93

Q2 24

Q3 37

Q4 142

2005 Q1 89

Q2 27

Q3 44

Q4 173

2006 Q1 164

Q2 35

Q3 53

Q4 198

We need 4-point moving averages so that each one has all 4 seasons of the year.

89

142

37

24

53Q3

198Q4

35Q2

164Q12006

173Q4

44Q3

27Q2

Q12005

Q4

Q3

Q2

93Q12004

Moving

Average

Bill (£)

DateDecide with your partner how many points to use for the moving average. Where would you plot the 1st average ?

We need 4-point moving averages so that each one has all 4 seasons of the year.

74

We must plot at the middle of the 4 values, so halfway between Q2 and Q3.

89

142

37

24

53Q3

198Q4

35Q2

164Q12006

173Q4

44Q3

27Q2

Q12005

Q4

Q3

Q2

93Q12004

Moving

Average

Bill (£)

Date

74

73

The next moving average drops Q1 for 2004 and brings in Q1 for 2005.

(a) Copy and complete the table.

EXERCISE

(c) Using a different colour or symbol, plot the moving averages and again join the points.

(b)Draw the graph for the original data, joining the points.

(d)Use the graph to describe the trend.

Solution:(a

)

89

142

37

24

53Q3

198Q4

35Q2

164Q12006

173Q4

44Q3

27Q2

Q12005

Q4

Q3

Q2

93Q12004

Moving

Average

Bill (£)

Date

74

73

74

76

83

102

104

106

113

Solution:(b

)

Q1 Q4Q2 Q3 Q1 Q4Q2 Q3 Q1 Q4Q2 Q32004 2005 2006

Date

Bill (£)

Quarterly Gas Bills

Q1 Q4Q2 Q3 Q1 Q4Q2 Q3 Q1 Q4Q2 Q32004 2005 2006

Date

Bill (£)

Quarterly Gas Bills

Solution:(c

)

x x xx

x

x

x x x

(d)Charges were steady at the start of the period but then moved upwards.

4-point moving averages

SUMMARY Moving averages are used to show a trend

in a set of data varying in time.

e.g. 3-point moving averages are found by• averaging the first 3 values,• dropping the 1st value and introducing

the 4th, to give the average of the 2nd, 3rd and 4th,• continuing to calculate the average of 3 values by dropping the earliest and including the next point not yet used.

On a graph, moving averages are plotted at the middle of the times used for each calculation. Joining the moving averages gives a trend line.

The number of points gives the number of values in each average

Exercise1. The table shows the change in the population

of farmland birds every 5 years between 1970 and 2000.

* Index numbers are studied in another presentation. Here you just need to know that they show changes in bird numbers, taking 1970 as the starting point.

Source: Office for National Statistics, Social Trends 34.

Year 1970 1975 1980 1985 1990 1995 2000

Index Number*

100 109 100 76 71 62 59

3-point moving averages

103 95

Exercise1. The table shows the change in the population

of farmland birds every 5 years between 1970 and 2000.Year 1970 1975 1980 1985 1990 1995 2000

Index Number

100 109 100 76 71 62 59

3-point moving averages

103 95The first two 3-point moving averages are shown.(a) Complete the table.(b) Draw a time series graph showing the data

and join up the points.(c) On the graph plot the moving averages.

(d) Use the moving averages to describe the trend.

Exercise

Year 1970 1975 1980 1985 1990 1995 2000

Index Number 100 109 100 76 71 62 59

3-point moving averages

103 95

Solution:(a)

82 70 64

(b)

Farmland Birds

Year

Exercise

(c)

3-point moving averages

Index

x

x

x

x

x

(d) Throughout the period shown, the trend is downwards.

Q1 Q4Q2 Q3 Q1 Q4Q2 Q3 Q1 Q4Q2 Q32004 2005 2006

Date

Bill (£)

Quarterly Gas Bills

x x xx

x

x

x x x

4-point moving averages

2. This is the graph we drew earlier showing my gas bills. The data for 2006 is also shown.

An estimate of the next moving average is 116.Use this moving average to help you calculate an estimate of my gas bill for the 1st quarter of 2007.

x 1162006

Bill(£)

Q1 164

Q2 35

Q3 53

Q4 198

Solution: Bill(£)

Moving Average

2006 Q1 164

Q2

Q3

Q4

2007 Q1

Using the estimated moving average, the total for the final 4 quarters, including 2007 Q1, is

4 116 = 464Subtracting the 3 final values for 2006: Estimate for 2007 = 464

We have:

116

35 53 198= £178

35

53

198

x178