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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
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
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
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