Teach GCSE Maths Number of Births thousands 700

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

Time Series and 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. "Certain images and/or photos on this presentation are the copyrighted property of Jupiter. Images and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from Jupiter. Images" © Christine Crisp

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.

Earnings(£ per hour) e. g. Average Hourly Earnings Males Females Year Source: Office for National Statistics 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.

Earnings(£ per hour) e. g. Average Hourly Earnings Males Females Year Source: Office for National Statistics 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.

Number of Births ( thousands ) e. g. Use the graph below to predict the birth rate for 2005. Live Births ( England Wales ) In 2002, the numbers To predict from went up slightly. . . the graph we need to “smooth” it. but the trend since 1992 is downwards. Year Source: Office for National Statistics We do this by averaging the values for several years.

This is the data set. Suppose we average the first 5 values ( for 1992 to 1996 ). 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. Year Births ( ‘ 000 s ) 1992 1993 1994 1995 1996 1997 1998 1999 2000 690 674 665 648 650 643 636 622 604 2001 595 2002 596

This is the data set. Year Births ( ‘ 000 s ) Decide with your partner which year you would plot this value against. 1992 1993 1994 1995 1996 1997 1998 690 674 665 648 650 643 636 Ans: Since it is an average, we plot at 1994, the middle of the 5 years. 1999 2000 2001 622 604 595 2002 596 Suppose we average the first 5 values ( for 1992 to 1996 ). 690 + 674 + 665 + 648 + 650 5 = 665 ( 3 s. f. ) 665

For the 2 nd average, we drop the value for the 1 st year ( 1992 ) and include the value for 1997. 674 + 665 + 648 + 650 + 643 5 = 656 We continue like this moving the averages forward. . . Year Births Moving ( ‘ 000 s ) Averages 1992 1993 1994 1995 1996 1997 1998 690 674 665 648 650 643 636 1999 2000 2001 622 604 595 2002 596 665 656

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

Births Moving (‘ 000 s) average 1992 690 1993 674 1994 665 1995 648 656 1996 450 648 1997 643 640 1998 646 631 1999 622 620 2000 604 611 2001 595 2002 596 Number of Births ( thousands ) Year Live Births ( England Wales ) x x 5 -point moving averages x x x Year We say the trend in the birth rate is downwards.

(‘ 000 s) average 1992 690 1993 674 1994 665 1995 648 656 1996 450 648 1997 643 640 1998 646 631 1999 622 620 2000 604 611 2001 595 600 2002 596 Number of Births ( thousands ) To predict the birth rate for 2003, we extend the trend line to find the next moving average. Live Births ( England Wales ) Births Moving Year x x 5 -point moving averages x Year x x 600

To predict the birth rate for 2003, we extend the trend line to find the next moving average. Year Births Moving (‘ 000 s) average 1992 690 1993 674 1994 665 1995 648 656 1996 450 648 1997 643 640 1998 646 631 1999 622 620 2000 604 611 2001 595 600 2002 596 2003 x 583 An average of 600 means that the total for the 5 years from 1999 to 2003 is 5 600 = 3000 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

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. Date Bill (£) This table gives my quarterly gas bills. The 1 st quarter, Q 1, covers the gas used from February to April, the 2 nd from May to July and so on. 2004 2005 2006 Q 1 93 Q 2 24 Q 3 37 Q 4 142 Q 1 89 Q 2 27 Q 3 36 Q 4 173 Q 1 164 Q 2 35 Q 3 53 Q 4 198

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. Date 2004 2005 2006 Bill (£) Q 1 93 Q 2 24 Q 3 37 Q 4 142 Q 1 89 Q 2 27 Q 3 44 Q 4 173 Q 1 164 Q 2 35 Q 3 53 Q 4 198

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. We must plot at the middle of the 4 values, so halfway between Q 2 and Q 3. Date 2004 2005 2006 Bill (£) Q 1 93 Q 2 24 Q 3 37 Q 4 142 Q 1 89 Q 2 27 Q 3 44 Q 4 173 Q 1 164 Q 2 35 Q 3 53 Q 4 198 Moving Average 74

The next moving average drops Q 1 for 2004 and brings in Q 1 for 2005. Date 2004 EXERCISE (a) Copy and complete the table. (b) Draw the graph for the original data, joining the points. (c) Using a different colour or symbol, plot the moving averages and again join the points. (d) Use the graph to describe the trend. 2005 2006 Bill (£) Q 1 93 Q 2 24 Q 3 37 Q 4 142 Q 1 89 Q 2 27 Q 3 44 Q 4 173 Q 1 164 Q 2 35 Q 3 53 Q 4 198 Moving Average 74 73

Solution: (a) Date 2004 2005 2006 Bill (£) Q 1 93 Q 2 24 Q 3 37 Q 4 142 Q 1 89 Q 2 27 Q 3 44 Q 4 173 Q 1 164 Q 2 35 Q 3 53 Q 4 198 Moving Average 74 73 74 76 83 102 104 106 113

Solution: (b) Bill (£) Quarterly Gas Bills Q 1 Q 2 Q 3 Q 4 Date 2004 2005 2006

Solution: (c) Quarterly Gas Bill (£) x x x x x 4 -point moving averages Q 1 Q 2 Q 3 Q 4 Date 2004 2005 2006 (d) Charges were steady at the start of the period but then moved upwards.

Ø Ø SUMMARY Moving averages are used to show a trend in a set of data varying in time. The number of points gives the number of values in each average 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.

Exercise 1. 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 3 -point moving averages 109 100 103 95 76 71 62 59 Source: Office for National Statistics, Social Trends 34. * 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.

Exercise 1. 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 3 -point moving averages 109 100 103 95 76 71 62 59 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.

Exercise Solution: (a) Year 1970 1975 1980 1985 1990 1995 2000 Index Number 100 3 -point moving averages 109 100 76 71 62 103 95 82 70 64 59

Exercise (c) (b) Farmland Birds Index 3 -point moving x averages x x Year (d) Throughout the period shown, the trend is downwards.

2. This is the graph we drew earlier showing my gas bills. The data for 2006 is also shown. Quarterly Gas Bill (£) 4 -point moving 2006 Bill(£) averages x x x x x 116 Q 1 164 Q 2 35 Q 3 53 Q 4 198 Q 1 Q 2 Q 3 Q 4 Date 2004 2005 2006 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.

Solution: Bill(£) We have: 2006 Q 1 Q 2 Q 3 Q 4 2007 Q 1 Moving Average 164 35 53 198 x 178 116 Using the estimated moving average, the total for the final 4 quarters, including 2007 Q 1, is 4 116 = 464 Subtracting the 3 final values for 2006: Estimate for 2007 = 464 - 35 - 53 - 198 = £ 178