moving averages objectives: d gradeinterpret a time series graph b grade construct a time series...

Moving AveragesMoving Averages

Objectives:D Grade Interpret a Time Series Graph

B Grade Construct a Time Series Graph and Plot the Moving Averages


A line graph is a series of points joined with straight lines

e.g. a patient’s temperature was recorded every 2 hours after receiving medication:

Temperature over time

35

36

37

38

39

40

41

42

0 2 4 6 8 10 12

Time after medication (hrs)

Tem

per

atu

re o

C

Note the features:

Title

Axis Labels

The points are plotted as coordinates

The points are joined with straight lines

A second set (or more) of data may be added to the graph.

What does this line represent?

Normal body temperature


Because the line graph of temperature was plotted against timeit can also be called a ‘Time Series Graph’

Time Series Graphs are often used by people to identify trends overa period of time where they may be natural fluctuations.

For example, after installing loft and cavity wall insulation ahouseholder wants to review his gas bills to see if he is saving money.


A graph of gas bills plotted against time.

Why is the x axis labelled with years and 1st, 2nd, 3rd, 4th?Why is there so much fluctuation in the graph – is there a pattern?

Cost of Gas Bill

0

50

100

150

200

250

300

1st 2nd 3rd 4th 1st 2nd 3rd 4th

Year / Quarter

Co

st (

£)

2006 2007

Utility (gas, electric, water, etc.,) bills are sent out every 3 months (quarter of a year)The 1st quarter is winter when you would expect to use a lot of gas for heating,the 3rd quarter is the summer when you don’t use the heating.


A different number of points for a moving average can be selecteddepending upon the data being analysed. Different types of datasuit different moving averages

Electricity Bill Three-point moving average

Gas Bill Four-point moving average

Half-term test results Five-point moving average

Phone Bill Six-point moving average

Daily Superstore Sales Seven-point moving average

Termly test results

Which type of moving average which suit which data set?


Time Series GraphYear 2006 2007 2008Quarter 1st 2nd 3rd 4th 1st 2nd 3rd 4th 1st 2nd 3rd 4th

Ice Cream Sales (£) 340 525 965 470 355 510 1100 375 410 555 1150 485

Plot the data as a time series graph:This is essentially a line graph. The time is always plotted on the x-axis

200

1200

1000

800

600

400

1 st Qtr2 nd Qtr3 rd Qtr4thQtr1 st Qtr

3 rd Qtr4 th Qtr1 st Qtr2 nd Qtr3 rd Qtr

2 nd Qtr

4 th Qtr

2006 2007 2008

Ice

Cre

am S

ales

(£) x

xx x

x

xx

x

xx

x x

Representing DataRepresenting DataMoving AveragesMoving Averages are a way of looking at a trend over a period of time.For a four-point the mean is calculated from the first four pieces of data, then the next point is calculated by moving the selection one place across to calculate the mean of 4 points again

Year 2006 2007 2008Quarter 1st 2nd 3rd 4th 1st 2nd 3rd 4th 1st 2nd 3rd 4th

Ice Cream Sales (£) 340 525 965 470 355 510 1100 375 410 555 1150 485Moving Average

1

The moving average point is plotted in the middle of the 4 points the average is taken from

575 609 585 599 610 623

200

1200

1000

800

600

400

1 st Qtr2 nd Qtr3 rd Qtr4thQtr1 st Qtr

3 rd Qtr4 th Qtr1 st Qtr2 nd Qtr3 rd Qtr

2 nd Qtr

4 th Qtr

2006 2007 2008

Ice

Cre

am S

ales

(£)

xx

x

x

xx

x

xx

x x

x

x x x x x x x x You can see there is a general increase in sales over time

For a 3 point moving averagetake the mean over 3 points,for a 5 point moving averagetake the mean over 5 points,etc.

575

Ice Cream Sales (£) 340 525 965 470 355 510 1100 375 410 555 1150 485Moving Average 575

2579

Calculate the mean of the first 4 points i.e.340+525+965+470 = 575

4

Calculate the mean of the second 4 points i.e.525+965+470 +355= 579

4

Moving AveragesMoving AveragesP20 Q 5

P21 Q 6

Year 2006 2006 2006 2006 2007 2007 2007 2007Quarter 1st 2nd 3rd 4th 1st 2nd 3rd 4thCost £230 £120 £50 £80 £215 £120 £25 £55Four-point moving

average£120 £116

Day Mon Mon Tue Tue Wed Wed Thu Thu Fri FriSession a.m. p.m. a.m. p.m. a.m. p.m. a.m. p.m. a.m. p.m.Number 220 210 243 215 254 218 251 201 185 152

Two point moving average

215 226.5 229 234.5


P21 Q 8

Quarter Jul-04 Oct-04 Jan-05 Apr-05 Jul-05 Oct-05 Jan-05 Apr-06Shop Sales

Jul-0

4

Oct-04

Jan-

05

Apr-0

5

Jul-0

5

Oct-05

Jan-

05

Apr-0

6

120

100

80

60

40

20

0

Sal

es (

£ th

ousa

nds)

x

x

x

x

x

x

x

Year 2005 2006 2006 2006 2007 2007 2007Session Autumn Spring Summer Autumn Spring Summer AutumnExam Result (%) 86 93 70 83 93 67 77

x


Year 2006 2006 2006 2006 2007 2007 2007 2007Quarter 1st 2nd 3rd 4th 1st 2nd 3rd 4thCost £230 £120 £50 £80 £215 £120 £25 £55Four-point moving

average£120 £116

1st 2nd 3rd 4th 1st 2nd 3rd 4th 2006 2007

Time (year, quarter)

250

200

150

100

50

0

Cos

t (£)x

x

xx

x

x

xx

£116.25 £110 £103.75

x x x x

The trend is moving downwards which means that the electricity costs are falling


P21 Q 6

Day Mon Mon Tue Tue Wed Wed Thu Thu Fri FriSession a.m. p.m. a.m. p.m. a.m. p.m. a.m. p.m. a.m. p.m.Number 220 210 243 215 254 218 251 201 185 152

Two point moving average

215 226.5 229 234.5 M

on a.

m.

Mon

p.m

.Tue

a.m

.Tue

p.m

.W

ed a

.m.

Wed

p.m

.

Thu p

.m.

Thu a

.m.

Fri

a.m.

Fri

p.m

.

270

250

230

210

190

170

150

Num

ber

Pre

sent

xx

x

x

x

x

x

x

x

x

236 234.5 226 193 168.5

xx

x x x xx

x

x

The trend is downwards – starting low on Monday, then rising to Wednesday, butFalling sharply towards Friday.


P21 Q 8

Jul-0

4

Oct-04

Jan-

05

Apr-0

5

Jul-0

5

Oct-05

Jan-

05

Apr-0

6

120

100

80

60

40

20

0

Sal

es (

£ th

ousa

nds)

x

x

x

x

x

x

x

Year 2005 2006 2006 2006 2007 2007 2007Session Autumn Spring Summer Autumn Spring Summer AutumnExam Result (%) 86 93 70 83 93 67 77

83 82 82 81 79

Quarter Jul-04 Oct-04 Jan-05 Apr-05 Jul-05 Oct-05 Jan-05 Apr-06Shop Sales 100 40 10 80 110 50 12 86

x x x x x x

57.5 60 62.5 63 64.5

c. The trend is slightly upwards

d. The next point would probably lie at 6767= 50+12+86 +x

467 × 4 = 50 + 10 + 86 + x

268 = 146 + x

x = 122 So the next quarter sales would be about £122 000

100

90

80

70

60

Aut-0

5

Spr-0

6

Sum

-06

Aut-0

6

Spr-0

7

Sum

-07

Aut-0

7

x

x

x

x

x

x

xx x x x x

x

moving averages objectives: d gradeinterpret a time series graph b grade construct a time series...

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