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Teach GCSE Maths Time Series and Moving Averages Live Births: England and Wales Number of Births ( thousands ) Year 1995 2000 2005 600 700

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

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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.

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Tell your partner at least 2 things this graph tells you ( one each ! ) Ans:Between 1 986 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 Males Females Year Earnings(£ per hour) e.g. Average Hourly Earnings

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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 Males Females Year Earnings(£ per hour) e.g. Average Hourly Earnings

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e.g.Use the graph below to predict the birth rate for 2005. Live Births ( England and Wales ) Number of Births ( thousands ) Year Source: 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.

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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 Year This is the data set. 690 + 674 + 665 + 648 + 650 5 = 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.

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5962002 5952001 6042000 6221999 6361998 6431997 6501996 6481995 6651994 6741993 6901992 Births ( 000s ) Year Suppose we average the first 5 values ( for 1992 to 1996 ). This is the data set. 690 + 674 + 665 + 648 + 650 5 = 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

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643 690 5962002 5952001 6042000 6221999 6361998 1997 6501996 6481995 6651994 6741993 1992 Births ( 000s ) Year 665 656 Moving Averages For the 2 nd average, we drop the value for the 1 st year ( 1992 ) and include the value for 1997. = 656 674 + 665 + 648 + 650 + 643 5 We continue like this moving the averages forward...

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596 595 604 622 636 643 650 648 665 674 690 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 Births ( 000s ) Year For the 2 nd average, we drop the value for the 1 st year ( 1992 ) and include the value for 1997. 665 611 620 631 640 648 656 = 656 674 + 665 + 648 + 650 + 643 5 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 Averages

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x x x x x x x We say the trend in the birth rate is downwards. 5-point moving averages Live Births ( England and Wales ) Number of Births ( thousands ) Year Births (000s) Moving average 1992690 1993674 1994665 1995648656 1996450648 1997643640 1998646631 1999622620 2000604611 2001595 2002596

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x x x x x x x 5-point moving averages Live Births ( England and Wales ) Number of Births ( thousands ) Year Births (000s) Moving average 1992690 1993674 1994665 1995648656 1996450648 1997643640 1998646631 1999622620 2000604611 2001595 2002596 600 To predict the birth rate for 2003, we extend the trend line to find the next moving average. 600

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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 665 1994 6741993 6901992 Moving average Births (000s) 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

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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 1 st quarter, Q1, covers the gas used from February to April, the 2 nd from May to July and so on. Date Bill (£) 2004Q193 Q224 Q337 Q4142 2005Q189 Q227 Q336 Q4173 2006Q1164 Q235 Q353 Q4198

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Decide with your partner how many points to use for the moving average. Where would you plot the 1 st average ? Date Bill (£) 2004Q1 93 Q2 24 Q3 37 Q4 142 2005Q1 89 Q2 27 Q3 44 Q4 173 2006Q1 164 Q2 35 Q3 53 Q4 198 We need 4 -point moving averages so that each one has all 4 seasons of the year.

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89 142 37 24 53 Q3 198 Q4 35 Q2 164 Q12006 173 Q4 44 Q3 27 Q2 Q12005 Q4 Q3 Q2 93 Q12004 Moving Average Bill (£) Date Decide with your partner how many points to use for the moving average. Where would you plot the 1 st 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.

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89 142 37 24 53 Q3 198 Q4 35 Q2 164 Q12006 173 Q4 44 Q3 27 Q2 Q12005 Q4 Q3 Q2 93 Q12004 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.

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Solution: (a) 89 142 37 24 53 Q3 198 Q4 35 Q2 164 Q12006 173 Q4 44 Q3 27 Q2 Q12005 Q4 Q3 Q2 93 Q12004 Moving Average Bill (£) Date 74 73 74 76 83 102 104 106 113

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Solution: (b) Q1Q4Q2Q3Q1Q4Q2Q3Q1Q4Q2Q3 200420052006 Date Bill (£) Quarterly Gas Bills

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Q1Q4Q2Q3Q1Q4Q2Q3Q1Q4Q2Q3 200420052006 Date Bill (£) Quarterly Gas Bills Solution: (c) xx x x x x x x x (d)Charges were steady at the start of the period but then moved upwards. 4 -point moving averages

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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 1 st value and introducing the 4 th, to give the average of the 2 nd, 3 rd and 4 th, 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

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Exercise 1.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 1970197519801985199019952000 Index Number* 10010910076716259 3 -point moving averages 10395

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Exercise 1.The table shows the change in the population of farmland birds every 5 years between 1970 and 2000. Year 1970197519801985199019952000 Index Number 10010910076716259 3 -point moving averages 10395 The 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.

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Exercise Year 1970197519801985199019952000 Index Number 10010910076716259 3 -point moving averages 10395 Solution: (a) 827064

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(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.

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Q1Q4Q2Q3Q1Q4Q2Q3Q1Q4Q2Q3 200420052006 Date Bill (£) Quarterly Gas Bills xx x x 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 1 st quarter of 2007. x 116 2006 Bill (£) Q1 164 Q2 35 Q3 53 Q4 198

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Solution: Bill (£) Moving Average 2006Q1164 Q2 Q3 Q4 2007Q1 Using the estimated moving average, the total for the final 4 quarters, including 2007 Q1, is 4 116 = 464 Subtracting the 3 final values for 2006 : Estimate for 2007 = 464 We have: 116 35 53 198 = £178 35 53 198 x 178

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