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153Chapter 4 Time series

4A Time series and trend lines 4B Fitting trend lines and forecasting 4C Smoothing time series 4D Smoothing with an even number

of points 4E Median smoothing 4F Seasonal adjustment

Qualitative analysis of time series; recognition of • trend, seasonal, cyclic and random patternsSeasonal adjustments; seasonal effects and indices, • deseasonalisation of the data using yearly averagesMedian smoothing (as a graphical technique) • and smoothing using a moving average with

consideration of the number of terms required and centring where requiredFitting a trend line to data by eye, by 3-median fi t • (as a graphical technique) or by the least-squares methodForecasting using trend lines (with the data • deseasonalised where necessary).

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Time series

Time series and trend linesIn previous chapters we looked at bivariate, or (x, y), data where both x and y could vary independently. In this chapter we shall consider cases where the x-variable is time and, generally, where time goes up in even increments such as hours, days, weeks or years. In these cases we have what is called a time series. The main purpose of a time series is to see how some quantity varies with time. For example, a company may wish to record its daily sales fi gures over a 10-day period.

Time Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10

Sales ($) 5200 5600 6100 6200 7000 7100 7500 7700 7700 8000

We could also make a graph of this time series as shown below.As can be seen from this graph, there seems

to be a trend upwards — clearly, this company is increasing its revenues!

Types of trendAlthough many types of trend exist, in Further Mathematics we shall be looking at trends that are classifi ed as secular, seasonal, cyclic and random.

Secular trendsIf over a reasonably long period of time a trend appears to be either increasing or decreasing steadily, with no major changes of direction, then it is called a secular trend. It is important to look at the data over a long period. If the trend in the previous fi gure continued for, say, 30 days,

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then we could safely conclude that the company was indeed becoming more profitable. What appears to be a steady increase over a short term — say, stock market share prices — can turn out to be something quite different over the long run.

Seasonal trendsCertain data seem to fluctuate during the year, as the seasons change. Consequently, this is termed a seasonal trend. The most obvious example of a seasonal trend would be total rainfall during summer, autumn, winter and spring in a year.

The name seasonal is not specific to the seasons of a year. It could also be related to other constant periods of highs and lows. For example, sales figures at a fast-food store could be consistently higher on Saturdays and Sundays and drop off during the weekdays. Here the seasons are days of the week and repeat once every week.

A key feature of seasonal trends is that the seasons occur at the same time each cycle. Here are some common seasonal periods

Seasons Cycle Example

Seasons Winter, spring, summer, autumn

Four seasons in a year Rainfall

Months Jan., Feb., Mar., . . ., Nov., Dec.

12 months in a year Grocery store monthly sales figures

Quarters 1st quarter (Q1), 2nd quarter (Q2), 3rd quarter (Q3), 4th quarter (Q4)

Four quarters in a year Quarterly expenditure figures of a company

Days Monday to Friday Five days in a week Daily sales for a store open from Monday to Friday only

Days Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday

Seven days in a week Number of hamburgers sold at a takeaway store daily

Cyclic trendsLike seasonal trends, cyclic trends show fluctuations upwards and downwards, but not according to season. Businesses often have cycles where at times profits increase, then decline, then increase again. A good example of this would be the sales of a new major software product. At first, sales are slow; then they pick up as the product becomes popular. When enough people have bought the product, sales may fall off until a new version of the product comes on the market, causing

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Maths Quest 12 Further Mathematics for the Casio ClassPad


sales to increase again. This cycle can be repeated many times, which is why there are many versions of some software products.

Random trendsTrends may seem to occur at random. This can be caused by external events such as floods, wars, new technologies or inventions, or anything else that results from random causes. There is no obvious way to predict the direction of the trend or even when it changes direction.

In the figure at right, there are a couple of minor fluctuations at t = 4 and t = 8, and a major one at t = 13. The major fluctuation could have been caused by a change in government which positively affected profits.

WoRked exaMPle 1

State the type of trend and fit a straight line to the time series data at right, which represent the body temperature of a patient with appendicitis, taken every hour.

Think WRiTe

1 Attempt to fit a line using your eye. By trial and error, a line such as the one at right could be the trend line.

Hours

0 1 2 3 4 5 6 7 8 9 10 t

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37.237.437.637.838.038.238.4

2 Evaluate the trend. The trend is secular and upward.It is unlikely that the temperature will continue to rise indefinitely, but the line may be significant over the short term.

Plotting time seriesTo make better judgments about the type of time series, data in tabular form need to be plotted on a time-series plot. This is similar to a scatterplot with some notable differences.1. The independent variable is always time. This may be in days, days of the week, time of the

day, weeks, months, quarters, years and so on. Thus, the x-axis variable is time.2. As the periods are often labels and not numerical, the x-axis may be scaled using these period

labels.3. The points are connected. As they occur in chronological order, joining the points assists in

identifying the type of trend or time series pattern that may exist.

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So that we can enter the data into a CAS calculator, time periods that are labels (and not numerical) need to be converted to numerals. For this, an association table is needed. An association table summarises how the time periods are to be converted to numerical values. The first point is converted to 1 and so on, until the series is fully converted. Here are two examples.

example 1

Week 1 Mon.

Week 1 Tues.

Week 1 Wed.

Week 1 Thurs.

Week 1 Fri.

Week 1 Sat.

Week 1 Sun.

Week 2 Mon.

Week 2 Tues.

1 2 3 4 5 6 7 8 9

example 2

Jan. 2009

Feb. 2009

Mar. 2009

Apr. 2009

May 2009

June 2009

July 2009

Aug. 2009

1 2 3 4 5 6 7 8

WoRked exaMPle 2

The following table displays the school fees collected over a 10-week period. Plot the data and decide on the type of time-series pattern. If there is a secular trend, fit a straight line.

Week beginning

8 Jan.

15 Jan.

22 Jan.

29 Jan.

5 Feb.

12 Feb.

19 Feb.

26 Feb.

5 Mar.

12 Mar.

$ × 1000 1.5 2.5 14.0 4.5 13.0 4.5 8.5 0.5 5.0 1.0

Think WRiTe/diSPlaY

1 Set up an association table. One method is to add another row and enter the numerical time code for each of the ten weeks starting at 1, 2, …, through to 10.

Week beginning

8 Jan.

15 Jan.

22 Jan.

29 Jan.

5 Feb.

12 Feb.

19 Feb.

26 Feb.

5 Mar.

12 Mar.

$ × 1000 1.5 2.5 14.0 4.5 13.0 4.5 8.5 0.5 5.0 1.0

Time Code

1 2 3 4 5 6 7 8 9 10

2


On the Statistics screen, enter the time code (week) values into list1 and school fees values into list2. Label the lists accordingly.


3

4

5 Identify the pattern as either seasonal, cyclical or random. If there is a secular trend, is it upwards or downwards?

The school fees can be classified as cyclical or random with a secular downward trend. This is evident by the reducing totals in school fees collected.

Time series are a set of measurements taken over (usually) equally spaced time 1. intervals, such as hourly, daily, weekly, monthly or annually.

There are 4 basic types of trend:2. (a) secular: increasing or decreasing steadily(b) seasonal: varying from season to season(c) cyclic: similar to seasonal but not tied to a calendar cycle(d) random: varying from external causes happening at random.

ReMeMBeR

On the Statistics screen, draw an xyLine of the data. To do this, tap:• SetGraph• SettingSet: Type: xyLine XList: main\week YList: main\feesTap:• Set• yNote: The time-series plot has a random pattern with a downward secular trend.

To fit a least-squares regression line, tap to the Statistic screen and then tap:• Calc• Linear RegSet: XList: main\week YList: main\feesThen tap OK twice.

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Time series and trend linesFor questions 1 to 5, identify whether the trends are likely to be secular, seasonal, cyclic or random for:

1 the amount of rainfall, per month, in Western Victoria

2 the number of soldiers in the United States army, measured annually

3 the number of people living in Australia, measured annually

4 the share price of BHP Billiton, measured monthly

5 the number of seats held by the Liberal Party in Federal Parliament.

6 Fit a trend line by eye to the data in the graph at right.

7 We 1 A wildlife park ranger is travelling on safari towards the centre of a wildlife park. Each day (t), he records the number of sightings (y) of zebra that he notes. He draws up the table below right.

t 1 2 3 4 5 6

y 6 9 13 8 9 14

t 7 8 9 10 11 12

y 15 17 14 11 15 19

Fit a trend line to the data. What type of trend is best refl ected by these data?

8 We2 The monthly share prices of a recently privatised telephone company were recorded as follows.

Date Jan. 09 Feb. 09 Mar. 09 Apr. 09 May 09 June 09 July 09 Aug. 09

Price ($) 2.50 2.70 3.00 3.20 3.60 3.70 3.90 4.20

Graph the data (let 1 = Jan., 2 = Feb. . . . and so on) and fi t a trend line to the data. Comment on the feasibility of predicting share prices for the following year.

9 Plot the following monthly sales data for umbrellas. Fit a trend line. Discuss the type of trend best reflected by the data and the limitations of your trend line.

Month Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.

Sales 5 10 15 40 70 95 100 90 60 35 20 10

10 Consider the data in the figure at right, which represent the price of oranges over a 19-week period.a Fit a straight trend line to the data.b From the graph, predict the price in

week 25.

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11 The following table represents the quarterly sales figures (in thousands) of a popular software product. Plot the data and fit a trend line using the best fit by eye method. Discuss the type of trend best reflected by these data.

Quarter Q1-07 Q2-07 Q3-07 Q4-07 Q1-08 Q2-08 Q3-08 Q4-08 Q1-09 Q2-09 Q3-09 Q4-09

Sales 120 135 150 145 140 120 100 110 120 140 190 220

12 The number of employees at the Comnatpac Bank was recorded over a 10-month period. Plot and fit a trend line to the data. What would you say about the trend?

Month Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.

Employees 6100 5700 5400 5200 4800 4400 4200 4000 3700 3300

Fitting trend lines and forecastingUsing our eyes to fit a straight line to a set of data or to predict values can be an inadequate mathematical technique (as we saw in chapter 3). In this section we shall look at using either the 3-median or least-squares regression techniques to calculate the equation of a trend line.

Secular and random trendsIt is important to note that the techniques for fitting trend lines are used on the original data where the trend is clearly linear; that is, random or secular. These techniques cannot be applied effectively to cyclical or seasonal trends.

Always plot time-series data, so that the type of pattern or trend can be easily seen.

association tables and forecastingAn association table is often required to convert period labels to a numerical value, so that a straight-line equation can be calculated. It is best to set up an extra row if data are in tabular form, or to change the labels shown on the axis of a time-series plot to numerical values. Here are three examples.

example 1

Year 2006 2007 2008 2009

Time code 1 2 3 4

example 2 example 3 1st Quarter 2008 12nd Quarter 2008 23rd Quarter 2008 34th Quarter 2008 41st Quarter 2009 52nd Quarter 2009 6

For forecasting, use the association table to devise a time code for any period in the future. This time code will then be used in the straight-line equation.

From the three examples we can calculate that for:Example 1: 2013 would have a time code of 8Example 2: 1st Quarter 2010 would have a time code of 9Example 3: Monday week 4 would have a time code of 22.

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WoRked exaMPle 3

A new tanning salon has opened in a shopping centre, with customer numbers for its first days shown in the following table. Fit a straight line to the data set using the least-squares regression method.

Week 1 Week 2

Period Mon. Tues. Wed. Thurs. Fri. Sat. Sun. Mon. Tues. Wed.

Number of customers

9 9 11 13 16 18 19 20 23 27

Use the equation of the straight line to predict the number of customers for:a Monday week 4b Thursday week 2.

Think WRiTe/diSPlaY

1 Complete an association table, where Monday week 1 is 1, Tuesday week 1 is 2, Wednesday week 2 is 10.

Week 1 Week 2

Period Mon. Tues. Wed. Thurs. Fri. Sat. Sun. Mon. Tues. Wed.

Number of customers

9 9 11 13 16 18 19 20 23 27

Time code 1 2 3 4 5 6 7 8 9 10

2

3


On the Statistics screen, enter the time code values into list1 and the number of customers values into list2. Label the lists accordingly.

To find the correlation coefficient and to save the least-squares regression equation, tap:• SetGraph• SettingSet: Type: Scatter XList: main\timecode YList: main\customerTap: • Set • Calc • Linear RegSet the same XList and YList and set Copy Formula: y1. Then tap OK.


4

5

a For Monday week 4, the time code is 22. Substitute t = 22 into the equation where the y-variable is the number of customers and the x-variable is the time code.

a y = 5.6667 + 1.9697xNumber of customers = 5.67 + 1.97 × time code = 5.67 + 1.97 × 22Number of customers = 49

b b Number of customers = 5.67 + 1.97 × time code = 5.67 + 1.97 × 11Number of customers = 27 (rounded)

Note: Remember that forecasting is an extrapolation and if going too far into the future, the prediction is not reliable, as the trend may change.

From the list generated, the r value of 0.9871 suggests a very strong linear relationship. Thus it is appropriate to use the least-squares regression line for performing predictions. Tap OK.

Note: The time series plot shows a strong linear relationship with an upward secular trend.

For Thursday week 2, the time code is 11. Substitute t = 11 into the equation. Note: An alternative to substituting into the equation is to use the Main screen and the saved equation in y1.Tap the Main screen and complete the entry lines as: y1(22) y1(11)Press E after each entry.

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Once an equation has been determined for a time series, it can be used to analyse the situation.For the period given in the previous worked example, the equation is:

Number of customers = 5.67 + 1.97 × time codeThe y-intercept (5.67) has no real meaning, as it represents the time code of zero, which is the day before the opening of the salon. The gradient or rate of change is of more importance. It indicates that the number of customers is changing; in this instance, growing by approximately 2 customers per day (gradient of +1.97).

A trend line is a straight line that can be used to represent the entire time series and 1. could be used for predicting the future values of the time series.(a) The gradient of the line indicates the rate at which the trend is increasing (positive)

or decreasing (negative).(b) The y-intercept is the predicted starting value of the time series; that is, the value

before the original data used (time code t = 0).A straight line of time series equations can be found using three techniques.2. (a) Best-fit-by-eye method(b) 3-median method(c) Least-squares regression methodThese three techniques must be used on the original data where the trend is clearly 3. linear; that is, random or secular. They cannot be applied effectively to cyclical or seasonal trends.Always plot the time-series data, so that type of pattern or trend can be easily seen.4. An association table is often required to convert period labels to a numerical value, so 5. that a straight-line equation can be calculated.(a) It is best to set up an extra row if data are in tabular form, or to change the labels

shown on the axis of a time-series plot to numerical values.(b) For predicting into the future, use the same association table to determine the

appropriate time code.

ReMeMBeR

WoRked exaMPle 4

The forecast equation to calculate share prices, y, in a sugar company was calculated from data of the share values over 5 years. The equation is y = 0.42t + 1.56, where t = 1 represents the year 2001, t = 2 represents the year 2002 and so on. a Rewrite the equation putting it in the context of the question.b Interpret the numerical values given in the relationship.c Predict the share value in 2010.

Think WRiTe

a The x-variable represents the time codes and the y-variable represents the share price in dollars.

a Share price = $0.42 × time code + $1.56Time code t = 1 is 2001, t = 2 is 2002 and so on.

b The y-intercept of 1.56 represents the starting value; that is, when t = 0. The gradient of 0.42 represents the rate of change in share price with respect to time. That is, it will grow as it has a positive gradient.

b The y-intercept of $1.56 represents the approximate value of the shares in 2000. The gradient of +$0.42 means that the share value will grow by $0.42 (42 cents) each year.

c If t = 1 is 2001, then for 2010, the time code will be t =10. Substitute into the equation given.

c Share price = $0.42 × time code + $1.56 = $0.42 × 10 + $1.56 = $4.20 + $1.56 = $5.76



Fitting trend lines and forecasting 1 We3 The following table represents the number of cars remaining to be

completed on an assembly line. Fit a straight line to the following data using the least-squares regression method.

Time (hours) 1 2 3 4 5 6 7 8 9

Cars remaining 32 26 27 23 16 17 13 10 9

a Predict the number of cars remaining to be completed after 11 hours.

b At what rate is the numbers of cars on the assembly line being reduced by?

2 From the equation of the trend line, it should be possible to predict when there are no cars left on the assembly line. This is done by finding the value of t which makes y = 0. Using the equation from question 1, find the time when there will be no cars left on the assembly line.

3 When the MicroHard Company first started, it employed only one person. Each month the company has grown, so that after 12 months there are 14 people working there. The time-series data are shown by the graph at right.a Fit a 3-median line to the data.b Predict the number of employees after a

further 12 months.

4 The table below shows the share price of MicroHard during a volatile period in the stock market. Using your CAS calculator:a fi t a least-squares regression lineb fi t a 3-median line.Comment on your result. What type of trend is this?

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Price ($) 2.75 3.30 3.15 2.25 2.10 1.80 1.50 2.70 4.10 4.20 3.55 1.65 2.60 2.95 3.25 3.70

5 The following time series shows the number of internet websites on a webring over a 9-month period. Plot the data and fit a 3-median trend line. Comment on this line as a predictor of further growth.

Time (months) 1 2 3 4 5 6 7 8 9

Sites (millions) 2.00 2.20 2.50 3.10 3.60 4.70 6.10 7.20 8.50

6 We4 The forecast equation to calculate prices, y, of shares in a steel company was calculated from data of the share values of the past 6 years. The equation is

y = 0.72t + 2.56where t = 1 represents the year 2010, t = 2 represents the year 2011 and so on. a Rewrite the equation putting it in the context of the question.b Interpret the numerical values given in the relationship.c Predict the share value in 2020.

7 The Teeny-Tiny-Tot Company has started to make prams. Its sales figures for the first 8 months are given in the table below.

Date Jan. Feb. Mar. Apr. May June July Aug.

Sales 65 95 130 115 145 170 190 220

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a Using the sequence Jan. = 1, Feb. = 2, . . ., calculate the trend line using the least-squares method.

b Plot the data points and the trend line on the same set of axes.c Use the trend line equation to predict the company’s sales for December.d Comment on the suitability of the trend line as a predictor of future trends, supporting

your arguments with mathematical statements.

8 The sales figures of Harold Courtenay’s latest novel (in thousands of units) are given in the table below. The book was released a week before the first figures were collected.

Time (weeks) 1 2 3 4 5 6 7 8 9

Sales (×1000) 1 3 5 17 21 25 28 27 26

a Calculate the trend line for these data using the least-squares method.b Plot the data points and the trend line on the same set of axes.c Use the trend line equation to predict the sales for weeks 10, 12 and 14.d Comment on the suitability of the trend line as a predictor of future trends, supporting


9 The average quarterly price of coffee (per 100 kg) has been recorded for 3 years.


Price ($) 358 323 316 336 369 333 328 351 389 387 393 402

a Calculate the trend line for these data using the least-squares method.b Plot the data points and the trend line on the same set of axes.c Use the trend line equation to predict the price for the next quarter.d Comment on the suitability of the trend line as a predictor of future trends, supporting


10 A mathematics teacher gives her students a test each month for 10 months, and the class average is recorded. The tests are carefully designed to be of similar difficulty.

Test Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov.

Mark (%) 57 63 62 67 65 68 70 72 74 77

a Calculate the trend line for these data using the least-squares method.b Plot the data points and the trend line on the same set of axes.c Use the trend line equation to predict the results for the

last exam in December.d Comment on the suitability of the trend line as a predictor of

future trends, supporting your arguments with mathematical statements.

Smoothing time seriesBy now, you should appreciate the fact that fi tting linear trend lines to time series that are not really linear is both bad mathematics and bad policy — it doesn’t work! So how can we have a method that generates trend lines for such time series? If the non-linear nature of the data is random we can use a technique called smoothing. If the non-linear nature is seasonal, we use a method called seasonal adjustment.

There are two basic techniques for smoothing random or cyclical time series patterns. They are median smoothing and moving-average smoothing.

Median smoothing is preferred where there are small data sets, as it can be done graphically on a time-series plot. Also, for data sets with many outliers due to the volatile random or cyclical trend, median smoothing is preferred. We have seen earlier that outliers do not affect the median, while the average (mean) is affected by outliers.

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Moving-average smoothing is an option that is preferred for data sets with few random fluctuations.

Moving-average smoothingThis technique relies on the principle that averages of data can be used to represent the original data. When applied to time series, a number of data points are averaged, then we move on to another group of data points in a systematic fashion and average them, and so on. It is generally quite simple. Consider the following example:

Notice how the third column in the table at left is computed from the first two.1. Take the first three t points (1, 2, 3) and find their average (2);

take the first three y points in the table (12, 10, 15) and find their average (12.3).

2. Take the next three t points (2, 3, 4) and find their average (3); take the next three y points in the table (10, 15, 13) and find their average (12.7).

3. Repeat until you reach the last three t points.4. Take the last three t points (7, 8, 9) and find their average (8);

take the last three y points in the table (18, 21, 19) and find their average (19.3).As we use three points to average, moving down the table

from top to bottom, this is called a 3-point moving-average smoothing.

We are free to choose any number of points for our smoothed graph; we could have a 4-point smoothing, a 5-point smoothing or even an 11-point smoothing. Although it is preferable to choose an odd number, such as 3 or 5, it is possible to choose even numbers as well, with a slight change in the method. In either case it does not matter how many points are in our time series.

Moving-average smoothing with odd numbers of pointsAs seen above, the method for smoothing with an odd number (3, 5, . . .) is quite simple, and can be done in a vertical tabular form. It is crucial that the time values be equally spaced, but they don’t have to be in the sequence 1, 2, 3.Note: There are fewer smoothed points than original ones. For a 3-point smooth, 1 point at either end is ‘lost’; while for a 5-point smooth, 2 points at either end are ‘lost’.

The main reason for using a smoothing technique is to remove irregularities or wild variations in our time series.

The temperature of a sick patient is measured every 2 hours and the results are recorded.a Create a 3-point moving-average smoothing of the data.b Plot both original and smoothed data.c Predict the temperature for 18 hours using the last smoothed value.

Time (hours) 2 4 6 8 10 12 14 16Temp. (°C) 36.5 37.2 36.9 37.1 37.3 37.2 37.5 37.8

WoRked exaMPle 5

Time (t) Data (y) Moving average

1 12

2 1012 10 15

3+ + = 12.3

3 1510 15 13

3+ + = 12.7

4 1315 13 16

3+ + = 14.7

5 1613 16 13

3+ + = 14.0

6 1316 13 18

3+ + = 15.7

7 1813 18 21

3+ + = 17.3

8 2118 21 19

3+ +

= 19.3

9 19

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Think WRiTe/diSPlaY

a 1 Put the data in a table. a Time (h) Temp. (°C) Smoothed temp. (°C)

2 36.5

4 37.2 13(36.5 + 37.2 + 36.9) = 36.87

6 36.9 13(37.2 + 36.9 + 37.1) = 37.07

8 37.1 13(36.9 + 37.1 + 37.3) = 37.10

10 37.3 13(37.1 + 37.3 + 37.2) = 37.20

12 37.2 13(37.3 + 37.2 + 37.5) = 37.33

14 37.5 13(37.2 + 37.5 + 37.8) = 37.50

16 37.8

2 Calculate a 3-point moving- average for each data point. Note: The ‘lost’ values are at t = 2 and t = 16. Therefore, the first point plotted is (4, 36.87).

b 1 Plot the data. The smoothed line is the thicker, red one.Note: The smoothed data start at the 2nd time point and finish at the 7th point.

b

0 4 8 12Number of hours

16

38

37.5

37

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2 Interpret the result. The smoothed line has removed much of the fluctuation of the original time series and, in fact, clearly exposes the secular trend (upwards) in temperature.

c Last smoothed data point is 37.50. c The temperature at 18 hours is predicted to be 37.5 °C.

Prediction using moving averagesBecause the moving average does not generate a single linear equation, there are limited possibilities for using the resultant smoothed data for prediction. However, there are two things that can be done.1. Predict the next value — use the last smoothed value to predict the next time point. In

the previous example, our prediction for t = 18 would be temperature = 37.50. This is not necessarily an accurate prediction but it is the best we can do without a linear trend equation.

2. Fit a single straight line to the smoothed data — using either the 3-median or least-squares techniques, one could find a single equation for the smoothed data points. This is often the preferred technique.

how many points should be in the moving average?When smoothing data, it is important to decide on the number of points to be used. Should a 3-point, 5-point or even an 11-point moving average be used? This is an important but complicated question. Here are some basic hints. (Let p = 3 for 3 points, p = 5 for 5 points . . . and let n = the number of points in the time series.)1. For small data sets, the value of p should be considerably smaller than n. For example, if

n = 7, p should be no more than about 4.



2. The most common practice if there is a cyclic or seasonal variation that we want to remove, is to let p ≈ length of cycle. Otherwise p should be less than the length of the cycle. However, moving-average smoothing may not be effective at removing the variation. For example, if there are quarterly data with seasonal variation, use p = 4. Other preferred choices for number of points used in the smoothing procedures include:

Monthly sales figures use 12 point centred Daily sales for a store open each day of the week use 7 point Daily sales for a store open from Monday to Friday only use 5 point Quarterly electricity consumption figures use 4 point centred3. It is always preferable to use an odd value of p, regardless of whether n is even or odd.4. The larger the value of p, the smoother the trend line of the resulting data becomes. More of

the fluctuations will be removed. However, you can go too far.

Moving-average smoothing using a spreadsheetA spreadsheet can be devised to calculate the average data values and then the new set of smoothed points plotted on a graph. At right is a section of the spreadsheet for Worked example 5. The graph is shown in Worked example 6.

Below right are the formulas used. Note the row and column numbers carefully. There is no need to calculate the first (C1) and last (C8) average, as these are the ‘lost’ values. It should be clear how to turn this into a 5-point, or 7-point smooth. Why wouldn’t we go any further?

A B C D

time temp. smooth

1 2 36.5

2 4 37.2 =SUM(B1:B3)/3

3 6 36.9 =SUM(B2:B4)/3

4 8 37.1 =SUM(B3:B5)/3

5 10 37.3 =SUM(B4:B6)/3

6 12 37.2 =SUM(B5:B7)/3

7 14 37.5 =SUM(B6:B8)/3

8 16 37.8

9

168

Use the data from Worked example 5 to display time-series plots using a CAS calculator.

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Worked example 6


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On the Spreadsheet screen, enter the time values into column A and the temperature values into column B. Label each accordingly.

Start at cell C3 and complete the entry line as:= (B2 + B3 + B4) / 3.Instead of typing ‘B2’, you can tap on the cell.Highlight the cells to be fi lled down and tap:• Edit• Fill Range• OK

Highlight the ‘time’ and ‘temp’ columns and tap Graph. Ensure ‘Column Series’ is checked. Then tap:• Scatter• ViewEnsure both ‘Lines’ and ‘Markers’ are checked.

To zoom in on the graph, tap:• View• Zoom BoxWith the stylus, draw a box that fi ts closely around the graph. To view the graph at full screen, tap r.


5

Smoothing involves replacing the original time series with another one where most 1. of the variation has been removed to see if there is a secular trend. Points are ’lost‘ at the start and end of the time series. Refer to the text for detailed descriptions of the techniques involved.Median smoothing is best for cyclical time series, as the median is not affected by 2. unusual outliers.Moving-average smoothing is preferred for random time series with few outliers.3. The number of points taken is sometimes dictated by the nature of the data. For example:4. (a) 12-point centred for monthly fi gures (see Exercise 4D)(b) 7-point for daily fi gures(c) 4-point centred for quarterly fi gures.The more points taken, the more smoothing of the original data occurs.5.

ReMeMBeR

Smoothing time series 1 We5, 6 The following table represents sales of a textbook.

Year (t) 2002 2003 2004 2005 2006 2007 2008 2009

Sales (y) 2250 2600 2400 2750 2900 2450 3100 3400

a Create a 3-point moving average of the data.b Plot both the original and smoothed data.c Predict the sales for 2010 using the last smoothed value.

2 The sales of a certain car seem to have been declining in recent months. The management wishes to find out if this is the case.


Sales 120 70 100 110 90 80 70 90 80 100 60 60

a Using a 3-point moving average, smooth the data and comment on the result. Use Jan. = 1, Feb. = 2 . . .

b Using the least-squares method, fi nd the equation for the smoothed data.c Use the equation to predict the number of sales for March next year. Comment on the

predictions.

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Moving average

To display both time series on the same axes, highlight the ‘time’, ‘temp’ and ‘3-p-smooth’ columns and tap Graph. Ensure ‘Column Series’ is checked. Then tap:• Scatter• ViewEnsure both ‘Lines’ and ‘Markers’ are checked. To zoom in, tap:• View• Zoom BoxUse the stylus to draw a box and then tap r.

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3 Management is still not satisfied — perform a 5-point moving average on the data from question 2 and discuss the result.

4 Consider the quarterly rainfall data below. Rainfall has been measured over a 3-year period. Since the data are seasonal, perform a 3-point moving average and comment on whether there is a trend other than the seasonal one.

Time (t)Spring2006

Summer2006

Autumn2007

Winter2007

Spring2007

Summer2007

Autumn2008

Winter2008

Spring2008

Summer2008

Autumn2009

Winter2009

Rainfall (mm)

100 50 65 120 90 50 60 110 85 40 50 100

5 The attendance at Bendigo Football Club games was recorded over 10 years. Management wishes to see if there is a trend.

Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Attendance (× 1000)

75 72 69 74 66 72 61 64 69 65

a Perform a 3-point moving average on the data and comment on the result.b Using the 3-median line of best fit on the smoothed data, state the equation.c Use the equation from b to predict the attendance in 2011. Comment on the prediction.

6 Use a spreadsheet solution to complete a 3-point moving average on the following data which represent sales figures for a 21-week period.

Week Sales Smoothed data Week Sales Smoothed data

1 34 12 44

2 27 13 47

3 31 14 49

4 37 15 41

5 41 16 52

6 29 17 48

7 32 18 44

8 37 19 49

9 47 20 56

10 38 21 54

11 41

7 Coffee price data are reproduced below. To remove the seasonal variation, perform a 3-point moving average to smooth the data. Plot the smoothed and original data and comment on your result.


Price ($) 358 323 316 336 369 333 328 351 389 387 393 402



8 The sales of a new car can vary due to the effect of advertising and promotion. The sales figures for Nassin Motor Company’s new sedan are shown in the table. Perform a 5-point moving average to smooth the data. Plot the data, and use the last smoothed value to predict sales for the next month.

Month Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.

Sales 141 270 234 357 267 387 288 303 367 465 398

9 A large building site requires varying numbers of workers. The weekly employment figures over the last 7 weeks have been recorded. By performing a 3-point moving average, predict the number of people required for the next week.

Week Employees

1 67

2 78

3 54

4 82

5 69

6 88

7 94

Smoothing with an even number of pointsAs mentioned in the previous section, it is usually preferable to use an odd number of points. However, there are times when an even number of points can be used — that is, a 4-point, 6-point or even 12-point moving average. When we used an odd number of points, the result was automatically centred; that is, the y-data had the same t-values as the original (except at the first and last ‘lost’ points). This does not occur with an even-point smoothing, as shown in the following example of a 4-point moving average.

Time y-value

4-point average (smoothed value)

Calculation Result

2003 6

2004 10

(6 + 10 + 14 + 12) ÷ 4 10.5

2005 14

(10 + 14 + 12 + 11) ÷ 4 11.75

2006 12

(14 + 12 + 11 + 15) ÷ 4 13

2007 11

(12 + 11 + 15 + 16) ÷ 4 13.5

2008 15

2009 16

4d

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Observe that the fi rst average (10.5) is not aligned with any particular year — it is aligned with 2004.5! Also note that there are now three ‘lost’ values (the seven original records reduced to four). In other words, the moving average is not centred properly. To align the data correctly, an additional step needs to be performed; this is called centring.

Use the following procedure to centre the data:Step 1. Find the average of the fi rst two smoothed points and align it with the 3rd time point.Step 2. Find the average of the next two smoothed points and align it with the 4th time point.Step 3. Repeat, leaving two blank entries at both top and bottom of the table.

This is demonstrated in the following table, using the data from the previous table.

Time y-value

4-point average (smoothed value) 4-point average after centring

Calculation Result Calculation Result

2003 6

2004 10

(6 + 10 + 14 + 12) ÷ 4 10.5

2005 14 (10.5 + 11.75) ÷ 2 11.125

(10 + 14 + 12 + 11) ÷ 4 11.75

2006 12 (11.75 + 13) ÷ 2 12.375

(14 + 12 + 11 + 15) ÷ 4 13

2007 11 (13 + 13.5) ÷ 2 13.25

(12 + 11 + 15 + 16) ÷ 4 13.5

2008 15

2009 16

The fi rst average (11.125) is now aligned with 2005, the second (12.375) aligned with 2006 and so on. This process not only introduces an extra step, but an extra averaging (or smoothing) as well. It is usually preferable to stick with an odd-point smoothing to reduce these diffi culties.

The quarterly sales fi gures for a dress shop (in thousands of dollars) were recorded over a 2-year period. Perform a centred 4-point moving average on the data and plot the result. Comment on any trends you fi nd.

Time Summer Autumn Winter Spring Summer Autumn Winter Spring

Sales (× $1000) 27 22 19 25 31 25 22 29

Think WRiTe/dRaW

1 Arrange the data in a table.Note: Code the time column.

Time Sales 4-point moving average4-point centred moving average

1 27

2 22

(27 + 22 + 19 + 25) ÷ 4 = 23.25

Note: Table continues . . .

2 Calculate a 4-point moving average in column 3.


Tutorialint-0442

Worked example 7



3 Average the pairs of averages to find the 4-point centred data. This is done in column 4.

Time Sales 4-point moving average4-point centred moving average

3 19 (23.25 + 24.25) ÷ 2 = 23.75

(22 + 19 + 25 + 31) ÷ 4 = 24.25

4 25 (24.25 + 25.00) ÷ 2 = 24.63

(19 + 25 + 31 + 25) ÷ 4 = 25.00

5 31 (25.00 + 25.75) ÷ 2 = 25.38

(25 + 31 + 25 + 22) ÷ 4 = 25.75

6 25 (25.75 + 26.75) ÷ 2 = 26.25

(31 + 25 + 22 + 29) ÷ 4 = 26.75

7 22

8 29

4 Plot the data. The smoothed line is the red one.Note: The smoothed data start at the 3rd time point and finish at the 6th point.

0 2 4 6Time

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5 Interpret the results.

Observe the steadily increasing trend (even with only four smoothed points) that was not obvious from the original data.

even point smoothing with spreadsheetsThe spreadsheet for the 4-point moving average of Worked example 7 is shown below.

174

The formulas are shown below. Note the cell row and column labels.

A B C D E

time sales fourpoint centred

1 1 27

2 2 22 =SUM(B1:B4)/4

3 3 19 =SUM(B2:B5)/4 =SUM(C2:C3)/2

4 4 25 =SUM(B3:B6)/4 =SUM(C3:C4)/2

5 5 31 =SUM(B4:B7)/4 =SUM(C4:C5)/2

6 6 25 =SUM(B5:B8)/4 =SUM(C5:C6)/2

7 7 22

8 8 29

9

10

There is little difference between this and a 3-point moving average spreadsheet, except that the SUMs are located (columns C and D) to correspond to the appropriate term in the time series (columns A and B).

Smoothing involves replacing the original time series with another one where most of 1. the variation has been removed, to see if there is a secular trend.Moving averages work best with an odd number of points. For a 3-point moving 2. average, one point is lost at either end of the time series.Moving average smoothing with an even number of points is a 2-step process. First, 3. we perform a 4-point moving average, then centre by averaging pairs of the 4-point averages. For 4-point averages, two points are lost at each end of the time series.A 4-point moving average is mainly used for quarterly data and a 12-point moving 4. average is used for monthly data.

ReMeMBeR

Smoothing with an even number of points 1 We 7 Perform a 4-point centred moving average to smooth the following data and plot the

result. Comment on any trends that you find.

t 1 2 3 4 5 6 7 8 9 10

y 75 54 62 60 70 45 54 59 62 64

2 The price of oranges fluctuates from season to season. Data have been recorded for 3 years. Perform a 4-point centred moving average, plot the data and comment on any trends.

t Autumn2007

Winter2007

Spring2007

Summer2007

Autumn2008

Winter2008

Spring2008

Summer2008

Autumn2009

Winter2009

Spring2009

Summer2009

Price 45 67 51 44 52 76 63 48 58 80 66 52

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3 a Use a spreadsheet to complete the following table. The time series represents the temperature of a hospital patient over 15 days.

Day Temperature4-point moving

average4-point centredmoving average

1 36.6

2 36.4 36.75

3 36.8 36.825

4 37.2 36.85

5 36.9 36.95

6 36.5 37

7 37.2 37.05

8 37.4 37.275

9 37.1 37.375

10 37.4 37.25

11 37.6 37.275

12 36.9 37.325

13 37.2 37.15

14 37.6

15 36.9

b Using the smoothed data, find the equation of the least-squares trend line.c Use the trend line to predict the temperature of the patient on day 16.

4 The sales of summer clothing vary according to the season. The following table gives seasonal sales data (in thousands of dollars) for 3 years at a Darryl Jones department store.

Season Q3-06 Q4-06 Q1-07 Q2-07 Q3-07 Q4-07 Q1-08 Q2-08 Q3-08 Q4-08 Q1-09 Q2-09Sales 78 92 90 73 62 85 83 70 61 78 74 59

a Calculate a 4-point centred moving average.b Plot the original and smoothed data.c Determine if there is an underlying trend upwards or downwards.

5 Calculate a 6-point centred moving average on the data from question 3.

6 An athlete wishes to measure her performance in running a 1 km race. She records her times over the last 10 days.

Day 1 2 3 4 5 6 7 8 9 10Time (s) 188 179 183 180 173 171 182 168 171 166

a Calculate a 4-point centred moving average.b Plot the original and smoothed data.c Determine if there is a significant

improvement in her times.d Fit a 3-median trend line to the

smoothed data and predict her expected time for day 11.

7 The following table shows the share price index of Industrial Companies during an unstable fortnight’s trading. By calculating a 4-point centred moving average, determine if there seems to be an upward or downward trend.

Day 1 2 3 4 5 6 7 8 9 10Index 678 762 692 714 689 687 772 685 688 712

176

8 MC The table below displays the total monthly rainfall (in mm) in a reservoir catchment area over a 1-year period.

Month Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

Rainfall 9 65 35 99 75 90 133 196 106 56 76 76

Using 3-point moving-average smoothing, the smoothed value for the total rainfall in April is closest to:A 65 B 66 C 70 D 75 E 88

[VCAA 2006]

Median smoothingAn alternative to moving-average smoothing is to replace the averaging of a group of points with the median of each group. Although no particular mathematical advantage is gained, it is a faster technique requiring no calculations (provided you use odd-point median smoothing). Often it can be done directly on a graph of a time series.

Median smoothing from a tableBy placing the data in a table, median smoothing can be performed simply and quickly. Look at each group of three points (for smoothing with 3-point medians) and choose the middle value. Progress through the table one point at a time. As with other methods, points will be lost at the beginning and end of the table.

Perform a 3-point median smoothing on the data in the table below. The table shows the cost of an airline ticket between Perth and Melbourne over an 8-month period. Construct a time-series plot from the data.

Time 1 2 3 4 5 6 7 8

Cost ($) 340 350 320 340 300 330 350 310

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On the Spreadsheet screen, enter time values into column A, cost values into column B and the 3-point median smoothed cost into column C. Label each column accordingly.

Find the median of each group of 3 data points. Again, note the ‘lost’ values at t = 1 and t = 8.


3

Generally, the effect of median smoothing is to remove some random fluctuations. It performs poorly on cyclical or seasonal fluctuations — unless the size of the range being used (3, 5, 7, . . . points) is chosen carefully.

Median smoothing from a graphProvided the graph has clearly marked data points, it is possible to find a median smooth directly from it.

Perform a 3-point median smoothing on the graph of a time series at right.

Think WRiTe/dRaW

1 Read the data values and compute the median.

The 1st data points are 12, 18, 16 — so median = 16.The 2nd data points are 18, 16, 8 — so median = 16.The 3rd data points are 16, 8, 12 — so median = 12.The 4th data points are 8, 12, 16 — so median = 12.The 5th data points are 12, 16, 12 — so median = 12.The 6th data points are 16, 12, 8 — so median = 12.The 7th data points are 12, 8, 10 — so median = 10.The 8th data points are 8, 10, 14 — so median = 10.

2 Plot the medians on the graph. Note: Median smoothing has indicated a downward trend that is probably not in the real time series. This indicates that moving-average smoothing would be the preferred option.

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Highlight columns A, B and C and tap:• Graph• Scatter• O (to adjust the window size)To connect the points, tap View and ensure ‘Lines’ is checked.

178

Smoothing involves replacing the original time series with another one where most 1. of the variation has been removed, in order to see if there is a secular trend. There are three basic smoothing techniques.(a) Moving-average smoothing works best with an odd number of points. For a 3-point

smooth, one point is lost at either end of the time series.(b) Moving-average smoothing with an even number of points is a 2-step process. First

perform a 4-point moving average, then centre by averaging pairs of the 4-point smooth. For a 4-point centred smooth, two points are lost at each end of the time series.

(c) Median smoothing is usually done with an odd number of points. The number of points lost is the same as for moving-average smoothing.

In all cases, points are ‘lost’ at the start and end of the time series. Refer to the text for 2. detailed descriptions of the techniques involved.

ReMeMBeR

Median smoothing 1 We8 Perform a 3-point median smoothing on the following data and plot the result.

Comment on any trends that you find. These are the same data as in question 1, Exercise 4D, so compare the graphs of the median smooth with the moving-average smooth.

t 1 2 3 4 5 6 7 8 9 10

y 75 54 62 60 70 45 54 59 62 64

2 The maximum daily temperatures for a year were recorded as a monthly average. Perform a 3-point median smoothing on the data. Comment on your result.


Temp. (°C) 31 29 27 24 21 20 22 21 23 25 27 26

3 We9 Perform a 3-point median smoothing on the graphical time series shown at right. Comment on the effectiveness of the result.

4 Perform a 3-point median smoothing on the graphical time series shown at right. Comment on the effectiveness of the result.

5 Perform a 3-point median smoothing on the data in the following table, which represent the share price of the HAL computer company over the last 15 days.

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Price 1.45 1.67 1.56 1.72 1.58 1.71 1.67 1.82 1.56 1.78 1.88 1.56 1.67 1.71 1.82

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6 Perform a 5-point median smoothing on the data in the following table, which represent the share price of the Pear-Shaped Computer Company over an 8-week trading period.

Day Price Day Price Day Price Day Price

1 0.87 11 1.04 21 1.01 31 1.89

2 1.34 12 1.19 22 0.98 32 1.75

3 1.14 13 1.09 23 1.12 33 1.55

4 1.08 14 1.10 24 1.07 34 1.35

5 0.89 15 1.04 25 1.23 35 1.15

6 0.67 16 1.02 26 1.32 36 1.30

7 0.98 17 0.94 27 1.45 37 1.20

8 1.23 18 0.98 28 1.56 38 1.17

9 1.06 19 0.89 29 1.67 39 1.07

10 1.08 20 1.00 30 1.78 40 0.87

Seasonal adjustmentA seasonal trend is similar to a cyclical trend where there are defi ned peaks and troughs in the time-series data, except for one notable difference.

Seasonal trends have a fi xed and regular period of time between one peak and the next peak in the data values. Conversely, there is a fi xed and regular period of time between one trough and the next trough.

As we have seen in the sections on fi tting a straight line to a time series, it is diffi cult to fi nd an effective linear equation for such data. As well, the sections on smoothing indicated that seasonal data may not lend themselves to the techniques of moving-average or median smoothing. We may just have to accept that the data vary from season to season and treat each record individually.

For example, the unemployment rate in Australia is often quoted as ‘6.8% — seasonally adjusted’. The Government has accepted that each season has its own time

series, more or less independent of the other seasons. How can we remove the effect of the season on our time series? The technique of seasonally adjusting, or ‘deseasonalising’, will modify the original time series, hopefully removing the seasonal variation, and exposing any other trends (secular, cyclic, random) which may be ‘hidden’ by seasonal variation.

deseasonalising time seriesThe method of deseasonalising time series is best demonstrated with an example. Observe carefully the various steps, which must be performed in the order shown.

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Interactivityint-0185

Seasonal adjustment

tNum

ber

of h

ambu

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ld Joe’s Fast Food – daily hamburger sales

120

100

80

60

40

20

0 5 10 15 20 25

Tues. Tues. Tues.

Sat. Sat. Sat.

Day of the week

180

Unemployment fi gures have been collected over a 5-year period and presented in this table. It is diffi cult to see any trends, other than seasonal ones.

a Calculate the seasonal indices.b Deseasonalise the data using the seasonal

indices.c Plot the original and deseasonalised data.d Comment on your results, supporting your

statements with mathematical evidence.

Think WRiTe/diSPlaY

a 1 Find the yearly averages over the four seasons for each year and put them in a table.

a 2005: (6.2 + 8.1 + 8.0 + 7.2) ÷ 4 = 7.37502006: (6.5 + 7.9 + 8.2 + 7.7) ÷ 4 = 7.57502007: (6.4 + 8.3 + 7.9 + 7.5) ÷ 4 = 7.52502008: (6.7 + 8.5 + 8.2 + 7.7) ÷ 4 = 7.77502009: (6.9 + 8.1 + 8.3 + 7.6) ÷ 4 = 7.7250

Year 2005 2006 2007 2008 2009

Average 7.3750 7.5750 7.5250 7.7750 7.7250

2 Divide each term in the original time series by its yearly average.

Summer 2005: 6.2 ÷ 7.3750 = 0.8407Autumn 2005: 8.1 ÷ 7.3750 = 1.0983Winter 2005: 8.0 ÷ 7.3750 = 1.0847Spring 2005: 7.2 ÷ 7.3750 = 0.9763Summer 2006: 6.5 ÷ 7.5750 = 0.8581...Spring 2009: 7.6 ÷ 7.7250 = 0.9838

Season 2005 2006 2007 2008 2009

Summer 0.8407 0.8581 0.8505 0.8617 0.8932

Autumn 1.0983 1.0429 1.1030 1.0932 1.0485

Winter 1.0847 1.0825 1.0498 1.0547 1.0744

Spring 0.9763 1.0165 0.9967 0.9904 0.9838

3 Determine the seasonal averages from this second table. These are called seasonal indices.

Summer: (0.8407 + 0.8581 + 0.8505 + 0.8617 + 0.8932) ÷ 5 = 0.8608Autumn:(1.0983 + 1.0429 + 1.1030 + 1.0932 + 1.0485) ÷ 5 = 1.0772Winter:(1.0847 + 1.0825 + 1.0498 + 1.0547 + 1.0744) ÷ 5 = 1.0692Spring:(0.9763 + 1.0165 + 0.9967 + 0.9904 + 0.9838) ÷ 5 = 0.9927

Season Summer Autumn Winter Spring

Seasonal index 0.8608 1.0772 1.0692 0.9927

Season 2005 2006 2007 2008 2009

Summer 6.2 6.5 6.4 6.7 6.9

Autumn 8.1 7.9 8.3 8.5 8.1

Winter 8.0 8.2 7.9 8.2 8.3

Spring 7.2 7.7 7.5 7.7 7.6


Tutorialint-0443

Worked example 10



Method 1: Using the rule

b Divide each term in the original series by its seasonal index. This is the seasonally adjusted or deseasonalised time series.Note: Your answers may vary a little, depending upon how and when you rounded your calculations.

b Summer 05: 6.2 ÷ 0.8608 = 7.2023Autumn 05: 8.1 ÷ 1.0772 = 7.5195...Spring 09: 7.6 ÷ 0.9927 = 7.6557

2005 2006 2007 2008 2009

Summer 7.202 7.551 7.435 7.783 8.015

Autumn 7.520 7.334 7.705 7.891 7.520

Winter 7.482 7.669 7.388 7.669 7.763

Spring 7.253 7.756 7.555 7.756 7.656

c Graph the original and the seasonally adjusted (deseasonalised) time series.

c

0 4 8 12Time period

Une

mpl

oym

ent f

igur

es

16 20

8.58.07.57.0

6.06.5

d Note that most, but not all, of the seasonal variation has been removed. However, by using least-squares, we could more confidently fit a straight line to the deseasonalised data.

d There appears to be a slight upward trend in unemployment.

Method 2: Using a CAS calculator

b 1 b

2 Enter the actual unemployment rates data into column C.

On the Spreadsheet screen, enter the seasons values into column A and the time code values into column B.

182

3

c c

d Note that most, but not all, of the seasonal variation has been removed. However, by using least-squares, we could more confidently fit a straight line to the deseasonalised data.

d There appears to be a slight upward trend in unemployment.

Spreadsheet solutionAlthough a CAS calculator can be used to solve some parts of Worked example 10, a spreadsheet can be used to solve the entire problem. Such a spreadsheet has been constructed on the following page.Note: The input data are in the table below. They should also appear at the top of your spreadsheet.Step 1. Yearly averages are calculated just below the data table.Step 2. Each term is divided by the appropriate yearly average.Step 3. Seasonal indices are calculated (to the right of step 2).Step 4. ‘Deseasonalised’ data are calculated (below step 2).

Season 2005 2006 2007 2008 2009

SummerAutumnWinterSpring

6.28.187.2

6.57.98.27.7

6.48.37.97.5

6.78.58.27.7

6.98.18.37.6


Using the seasonal indices calculated in part a, enter the following formula into column D:

Deseasonalised = actual valueseasonal index

Complete the entry line in cell D1 as:

= C10.8608

The next summer value will use C5, then C9 and so on. Set up similar formulas for each of the other three seasons.

Highlight columns A, B and C and tap Graph. Ensure ‘Column Series’ is checked. Then tap:• Scatter• ViewEnsure both ‘Lines’ and ‘Markers’ are checked. To zoom in, tap:• View• Zoom BoxUse the stylus to draw a box and then tap r.


Step 1 Yearly ave. 7.375 7.575 7.525 7.775 7.725

Step 3

Step 2 Season 2005 2006 2007 2008 2009 Seasonal indices


0.840 6781.098 3051.084 7460.976 271

0.858 0861.042 9041.082 5081.016 502

0.850 4981.102 991.049 8340.996 678

0.861 7361.093 2481.054 6620.990 354

0.893 2041.048 5441.074 4340.983 819

0.860 841.077 1981.069 2370.992 725

4.000 000

Step 4 Season 2005 2006 2007 2008 2009


7.202 2647.519 5087.481 9727.252 767

7.550 767.333 8417.669 0227.756 431

7.434 5957.705 1757.388 4487.554 965

7.783 0917.890 8427.669 0227.756 431

8.015 4227.519 5087.762 5467.655 698

The formulas corresponding to the spreadsheet follow. Note carefully the row and column addresses.

B C D E F G H I

3

4 Season 2005 2006 2007 2008 2009

5678


6.28.187.2

6.57.98.27.7

6.48.37.97.5

6.78.58.27.7

6.98.18.37.6

9

10 Step 1 Yearly ave.

=SUM (D5:D8)/4

=SUM (E5:E8)/4

=SUM (F5:F8)/4

=SUM (G5:G8)/4

=SUM (H5:H8)/4

11 Step 3

12 Step 2 Season 2005 2006 2007 2008 2009 Seasonal indices

13141516


=D5/D$10=D6/D$10=D7/D$10=D8/D$10

=E5/E$10=E6/E$10=E7/E$10=E8/E$10

=F5/F$10=F6/F$10=F7/F$10=F8/F$10

=G5/G$10=G6/G$10=G7/G$10=G8/G$10

=H5/H$10=H6/H$10=H7/H$10=H8/H$10

=SUM(D13:H13)/5=SUM(D14:H14)/5=SUM(D15:H15)/5=SUM(D16:H16)/5

17 =SUM(I13:I16)

18 Step 4 Season 2005 2006 2007 2008 2009

19202122


=D5/$I$13=D6/$I$14=D7/$I$15=D8/$I$16

=E5/$I$13=E6/$I$14=E7/$I$15=E8/$I$16

=F5/$I$13=F6/$I$14=F7/$I$15=F8/$I$16

=G5/$I$13=G6/$I$14=G7/$I$15=G8/$I$16

=H5/$I$13=H6/$I$14=H7/$I$15=H8/$I$16

184

Notes1. By adding/deleting columns between columns D and H, you could increase/decrease the

number of years. Remember to change the denominator in the seasonal indices (I13 . . . I16)2. By adding/deleting more rows between Rows 5 and 8, you could increase/decrease the

number of seasons (see Exercise 4F, question 5). Do not forget to change the denominator in row 10.

3. The sum of the seasonal indices always equals the number of seasons (4 in this example).

4. Deseasonalised fi gure or value =

Forecasting with seasonal time seriesIn the previous section we smoothed out the seasonal variation and are now able to see any secular trend more clearly. If there is an upward or downward secular trend, then a straight line equation can be calculated and used for making predictions into the future. Using either the 3-median or least-squares regression methods of the deseasonalised data is always preferred.

The seasonal index measures by what factor a particular season is above or below the average of all seasons for the cycle. For example: Seasonal index = 1.3 means that season is 1.3 times the average season or a peak or high

season.Seasonal index = 0.7 means that season is 0.7 times the average season or a trough or low

season.Seasonal index = 1.0 means that season is same as the average season or neither a peak nor

trough.

However, the prediction obtained using the deseasonalised data means the prediction has also been deseasonalised or smoothed out to the average season. But as we have the relevant seasonal indices, we should be able to use it to remove the smoothing; that is, to re-seasonalise the predicted value.

The formula for re-seasonalising is:Seasonalised fi gure or value = deseasonalised fi gure or value × seasonal index

Use the data from Worked example 10 and fi nd the straight line for the deseasonalised data using the least-squares regression method. Predict the unemployment fi gure for summer in 2010. The original data are reproduced below.

Season 2005 2006 2007 2008 2009

Summer 6.2 6.5 6.4 6.7 6.9

Autumn 8.1 7.9 8.3 8.5 8.1

Winter 8.0 8.2 7.9 8.2 8.3

Spring 7.2 7.7 7.5 7.7 7.6


Tutorial

Worked example 11

actual original fi gure or valueseasonal index


int-0846


Think WRiTe/diSPlaY

1

2

3 Write the deseasonalised time series equation. Deseasonalised unemployment (%) = 0.0227 × time code + 7.357

4 Using the association table, the summer of 2010 will be represented by t = 21. Substitute into the equation.

Deseasonalised unemployment (%) = 0.0227 × 21 + 7.357 = 7.834%

5 The predicted value is very high for summer. Re-seasonalise by using the seasonal index for summer, which was 0.8608. That is, this was a season or period of low unemployment.

Seasonalised value = deseasonalised value × seasonal index = 7.834 × 0.8608 = 6.74%This is a lower unemployment figure as expected.

On the Spreadsheet screen, insert a column, then copy and paste so the ‘timecode’ column is immediately to the left of the ‘deseason’ column.Highlight the columns and tap:• Graph• Scatter

To adjust the window to a more appropriate size, tap:• O• View WindowSet the window as shown at right and tap OK.

To fit a linear trend line, tap:• Series• Trend• LinearTo find the equation of the linear regression line, tap the bottom left of the line; three squares should then appear on the line. Note: The linear equation appears at the bottom of the screen.

186

Quarterly sales fi gures for a pool chemical supplier between 2004 and 2009 were used to determine the following seasonal indices.

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

Seasonal index 1.8 1.2 0.2 0.8

Using the seasonal indices provided in the table, calculate the following.a Find the deseasonalised fi gure if the actual sales fi gure for the second quarter in 2008 was $4680.b Find the deseasonalised fi gure if the actual sales fi gure for the third quarter in 2008 was $800.c Find the predicted value if the deseasonalised predicted value for the fi rst quarter in 2010 is

expected to be $4000.

Think WRiTe

Use the formula for deseasonalising.

a Use the 2nd quarter seasonal index. a Deseasonalised fi gure =

= 46801.2

= $3900

b Use the 3rd quarter seasonal index. b Deseasonalised fi gure =

= 8000.2

= $4000

c Use the seasonalising formula and select the 1st quarter seasonal index.

c Seasonalised fi gure = deseasonalised fi gure × seasonal index = 4000 × 1.8 = $7200The forecast sales fi gure for the fi rst quarter in 2010 is $7200.

actual figureseasonal index

actual figureseasonal index


Tutorialint-0445

Worked example 12

Seasonal indicesFinally, it should be noted that the sum of all the seasonal indices gives a specifi c result, which can be used to answer certain types of queries.

The sum of the seasonal indices is equal to the number of seasons.

This can be summarised as follows.

Type of dataNumber of

seasons CycleSum of all the

seasonal indices

Monthly fi gures 12 A year 12

Quarterly fi gures 4 A year 4

Fortnightly fi gures 26 A year 26

Daily fi gures for data from Monday to Friday only

5 A week 5

Daily fi gures for data from Monday to Sunday

7 A week 7



Deseasonalising a time series involves replacing the original time series with another 1. one where most or all of the seasonal variation is removed. To deseasonalise data:(a) Average over all seasons for each year — these are the yearly averages.(b) Divide each point in the original time series by its corresponding yearly average.(c) Using this new series, average over all years for each season — these are the

seasonal indices.(d) Divide each point in the original time series by its corresponding seasonal index.To deseasonalise fi gures:2.

Deseasonalised fi gure or value =actual original fi gure or value

seasonal indexTo seasonalise (predicted) fi gures:3.

Seasonalised fi gure or value = deseasonalised fi gure or value × seasonal indexThe sum of the seasonal indices is equal to the number of seasons.4.

ReMeMBeR

Seasonal adjustmentNote: Your answers may vary slightly, depending upon rounding. Try to round to 4 decimal places for all intermediate calculations.

1 We 10 The price of sugar ($/kg) has been recorded over 3 years on a seasonal basis.a Compute the seasonal indices.b Deseasonalise the data using the seasonal

indices.c Plot the original and deseasonalised data.d Comment on your results, supporting your

statements with mathematical evidence.

WoRked exaMPle 13

A fast food store that is open seven days a week has the following seasonal indices.

Season Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Index 0.5 0.2 0.5 0.6 2.2 1.1

The index for Friday has not been recorded. Calculate the missing index.

Think WRiTe

1 The sum of the seasonal indices is equal to the number of seasons.

There are 7 seasons (Monday to Sunday), therefore the sum of indices is 7.

2 The missing index is the sum of all the other seasons subtracted from the total.

Friday index = 7 - (sum of all the other indices) = 7 - (0.5 + 0.2 + 0.5 + 0.6 + 2.2 + 1.1) = 7 - 5.1 = 1.9

exeRCiSe

4F eBookpluseBookplus


Seasonal adjustment

Season 2007 2008 2009

Summer 1.03 0.98 0.95

Autumn 1.26 1.25 1.21

Winter 1.36 1.34 1.29

Spring 1.14 1.07 1.04

188

2 Data on the total seasonal rainfall (in mm) have been accumulated over a 6-year period.

Season 2004 2005 2006 2007 2008 2009

Summer 103 97 95 117 118 120

Autumn 93 84 82 100 99 98

Winter 143 124 121 156 155 151

Spring 123 109 107 125 122 124

a Compute the seasonal indices.b Deseasonalise the original time series.c Plot the original and deseasonalised

time series.d Comment on your result, supporting

your statements with mathematical evidence.

3 It is known that young people (18–25) have problems in finding work; these problems are different from those facing older people. The youth unemployment statistics are recorded separately from the overall data. Using the youth unemployment figures for five years shown above right:a Compute the seasonal indices.b Deseasonalise the time series.c Plot the original and deseasonalised time series.d Comment on your result, supporting your statements with mathematical evidence.

4 The unemployment rate in a successful European economy is given in the table at right as a percentage.a Compute the seasonal indices.b Deseasonalise the time series.c Plot the original and deseasonalised

time series.d Find the line-of-best-fit for the deseasonalised data using the least-squares method.e Use the equation of the line from part d to predict the unemployment rate for: i quarter 1 in 2010 ii quarter 3 in 2014.

Comment on each of the predictions.

Season 2005 2006 2007 2008 2009

Summer 7.6 7.7 7.8 7.7 7.9

Autumn 10.9 11.3 11.9 12.6 13.1

Winter 11.7 12.4 12.8 13.5 13.9

Spring 9.9 10.5 10.8 11.4 11.9

Quarter 1 2 3 4

2007 5.8 4.9 3.5 6.7

2008 6.1 5.1 3.2 6.5

2009 5.7 4.5 4.1 7.1



5 It is possible to seasonally adjust time series for other than the usual 4 seasons. Consider an expensive restaurant that wishes to study its customer patterns on a daily basis. In this case a ‘season’ is a single day and there are 7 seasons in a weekly cycle. Data are total revenue each day shown in the table which follows. Modify the spreadsheet solution to allow for these 7 seasons and deseasonalise the following data over a 5-week period. Comment on your result, supporting your statements with mathematical evidence.

Season Week 1 Week 2 Week 3 Week 4 Week 5

Monday 1036 1089 1064 1134 1042

Tuesday 1103 1046 1085 1207 1156

Wednesday 1450 1324 1487 1378 1408

Thursday 1645 1734 1790 1804 1789

Friday 2078 2204 2215 2184 2167

Saturday 2467 2478 2504 2526 2589

Sunday 1895 1786 1824 1784 1755

6 We 11 MC A line-of-best-fit for deseasonalised data was given as:Deseasonalised monthly sales = 1500 × time code + 10 000

where June 2009 represents t = 1.The predicted actual expected sales figure for June 2010, if the June seasonal index is 0.8, would be:A $23 600 B $29 500 C $19 500 D $36 875 E $35 000

7 The following table gives the deseasonalised figures and corresponding seasonal indices for umbrella sales.

Season Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

Number of umbrellas (deseasonalised)

24 24 25 26 25 27 27 28 30 31 33 34

Index 1.15 0.90 0.20 0.20 0.35 0.45 3.0 2.10 2.15 0.95 0.40 0.15

a Find the straight line for the deseasonalised data using the least-squares regression method.

b Predict the umbrella sales for January the following year.

exaM TiP The majority of students are able to predict the deseasonalised value from the trend line, but then fail to seasonalise the value they obtained to determine the actual future value of the data.

[Assessment report 1 2006]

190

8 We 12 Quarterly sales figures for an ice-cream parlour between 2007 and 2009 were used to determine the following seasonal indices.

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

Seasonal index 1.50 1.00 0.25 1.25

Using the seasonal indices provided in the table, calculate the following.a Find the deseasonalised figure, if the actual sales figure for the second quarter in 2008

was $3000.b Find the deseasonalised figure, if the actual sales figure for the third quarter in 2008 was

$800.c Find the predicted value, if the deseasonalised predicted value for the first quarter in 2010

is expected to be $3200.

9 We 13 A newsagency store that is open seven days a week has the following seasonal indices.

Season Mon. Tues. Wed. Thurs. Fri. Sat. Sun.

Index 0.5 0.2 0.6 1.5 2.2 1.1

Find the value of the missing index.

10 Complete the following table of seasonal indices.

Season Summer Autumn Winter Spring

Index 1.23 0.89 1.45

Questions 11 and 12 relate to the following table, which contains the seasonal indices for the daily minimum temperatures of a Victorian town.

Season Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

Index 1.05 1.0 1.0 0.95 0.85 0.8 0.9 0.95 1.05 1.10 1.15

11 MC The seasonal index missing from the table is:A 1.0B 1.05C 1.10D 1.15E 1.20

12 MC If the actual sales figure for June 2009 was $102 000, then the deseasonalised figure would be:A $96 900B $86 700C $107 368.42D $120 000E $102 000



SuMMaRY

Time series

A time series is a set of measurements taken over (usually) equally spaced time intervals, such as hourly, •daily, weekly, monthly or annually.

Trend lines

There are 4 basic types of trend.•Secular: increasing or decreasing steadily1. Seasonal: varying from season to season2. Cyclic: similar to seasonal but not tied to a calendar cycle3. Random: variations caused by external triggers happening at random4.

Fitting trend lines

The trend line is a straight line that can be used to represent the entire time series. Trend lines can be used for •predicting the future values of the time series. The line can be found in several ways.

No smoothing:1. (a) for time series that are clearly linear; that is, slightly random or have secular trends(b) fit the line of best fit by eye, or using the 3-median or least-squares method.With smoothing:2. (a) for time series that are random, secular or have cyclical trends(b) fit the line of best fit using either the 3-median or least-squares method.With deseasonalising:3. (a) for time series that have seasonal trends only(b) fit the line of best fit using either the 3-median or least-squares method(c) re-seasonalise the values determined from the deseasonalised line.

Smoothing time series (primarily for random and secular trends)

Smoothing involves replacing the original time series with another one from which most of the variation has •been removed, in order to see if there is a secular trend. There are three basic smoothing techniques. In all cases, points are ‘lost’ at the start and end of the time series. Refer to the text for detailed descriptions of the techniques involved.

Moving-average smoothing with an odd number of points

Moving-average smoothing works best with an odd number of points. For a •3-point moving average, two points are lost; one point at each end of the time series.

Moving-average smoothing with an even number of points

Moving-average smoothing with an even number of points is a 2-step process. First perform a 4-point •moving average, then centre by averaging pairs of the 4-point averages. For a 4-point centred smoothing, four points are lost; two points at each end of the time series.

Median smoothing

Median smoothing is usually done with an odd number of points. The number of points lost is the same as •for moving-average smoothing.

192

Deseasonalisation (only for seasonal trends)

Deseasonalising a time series involves replacing the original time series with another one where most or all •of the seasonal variation is removed:

Average over all seasons for each year — these are the yearly averages.1. Divide each point in the original time series by its corresponding yearly average.2. Using this new series, average over all years for each season — these are the seasonal indices.3. Divide each point in the original time series by its corresponding seasonal index.4.

To deseasonalise figures:•

Deseasonalised figure or value = actual original figure or valueseasonal index

To seasonalise (predicted) figures:• Seasonalised figure or value = deseasonalised figure or value × seasonal indexThe sum of the seasonal indices is equal to the number of seasons.•



ChaPTeR RevieW

MulTiPle ChoiCe

1 The price of oranges over a 16-month period is recorded in the figure.

t

5040302010

0 2 4 6 8Months

10 12 14

Pric

e of

ora

nges

($)

16

The trend can be described as:A Cyclic B Seasonal C RandomD Secular E There is no trend.

2 A 3-median trend line was fitted to the data from question 1 using the values below. The gradient of this line is:

t 1 2 3 4 5 6 7 8

$ 20 28 10 14 18 24 16 26

t 9 10 11 12 13 14 15 16

$ 16 18 22 20 17 25 20 5

A 4.93 B 0.18 C 0.313D -3.30 E 17.8

3 From another 16-month time series for the price of apples, it was found that the least-squares trend line was: price = 0.415 × month + 8.45. A prediction for the price of apples in month 18 is:

A 8.45 B 0.42 C 6.64 D 15.92E unable to be determined with the above

information 4 A least-squares trend line has been fitted to the time

series in the figure. Its equation is most likely to be:

t

y

706050

1009080

40302010

0 4 82 6 10

A y = 10t B y = -8t + 10 C y = 8tD y = 8t - 10 E y = 8t + 10

5 The following data represent the number of employees in a car manufacturing plant. The data are smoothed using a 3-point moving average.

Year Number

2002 350

2003 320

2004 300

2005 310

2006 270

2007 240

2008 200

2009 160

The first two points in the smoothed trend line are:

A 320 and 300 B 320 and 310 C 323 and 310D 335 and 310 E 323 and 273

6 How many points would be obtained from the smoothed trend in question 5?A 8 B 7 C 6 D 5 E 4

7 Consider the following data.

Time y-value2003 122004 132005 162006 162007 172008 192009 22

The value, after a 4-point smoothing after centring, plotted against the year 2006 is:A 16.25 B 14.25 C 15.5D 17 E 14.875

8 A 3-point median smoothing is performed on the data in the figure below. The last smoothed value is:

t

y3530252015105

0 2 4 6 8 10

A 25 B 21.7 C 20 D 15 E 9

194

9 Seasonal indices and adjustment can be used when:A there are random variations in the dataB there are seasonal variations along with a

secular trendC there are seasonal variations onlyD there are seasonal or cyclic variationsE there are at least 4 seasons’ worth of data

10 The seasonal indices below were obtained from a time series.

Season Spring Summer Autumn Winter

Index 1.12 0.78 0.92

The value of the winter’s seasonal index is:A 1.18 B 0.94 C 1.08 D 1.06E unable to be determined from the given

information 11 Using the data from question 10, a seasonally

adjusted value for the summer of 2000, when the original value was 520, is closest to:

A 406 B 667 C 464 D 614E cannot be determined without additional

information

The following information relates to questions 12–14.The time series plot below shows the revenue from sales (in dollars) each month made by a Queensland souvenir shop over a 3-year period.

0 6

5 000

10 000

15 000

12 18 24 30 36

20 000

Month

Rev

enue

($)

12 This time series plot indicates that, over the 3-year period, revenue from sales each month showed:A no overall trend B no correlationC positive skew D an increasing trend onlyE an increasing trend with seasonal variation

13 A 3-median trend line is fitted to these data. Its slope (in dollars per month) is closest to:

A 125 B 146 C 167D 188 E 255

14 The revenue from sales (in dollars) each month for the first year of the 3-year period is shown below.


Revenue ($) 1236 1567 1240 2178 2308 2512 3510 4234 4597 4478 7034 8978

If this information is used to determine the seasonal index for each month, the seasonal index for September will be closest to:

A 0.80 B 0.82 C 1.16D 1.22 E 1.26

[VCAA 2007]



ShoRT anSWeR

1 The number of uniforms sold in a school uniform shop is reported in the table.

Month Number of uniforms sold

January 118

February 92

March 53

April 20

May 47

June 102

July 90

August 42

September 35

October 26

November 12

December 58

Fit a trend line to these data. What type of trend is best reflected by these data? Can you explain these trends?

2 Fit a least-squares trend line for the following data, which represent the sales at a snack bar during the recent Melbourne show. State the gradient and y-intercept.

Day Sales ($)1 23002 22003 26004 31005 29006 32007 33008 3500

3 Fit a 3-median trend line to the data below.

a State the gradient and y-intercept as exact values.

b Use your line to predict a value when t = 35.

4 A hotel records the number of rooms booked over an 11-day period. Fit a trend line using the least-squares method.

Day 1 2 3 4 5 6 7 8 9 10 11Rooms 12 18 15 20 22 20 25 24 26 28 30

a State the gradient and y-intercept, rounded to 2 decimal places.

b Predict the number of rooms booked for days 12 and 13.

5 Perform a 3-point moving average on the following rainfall data. Plot the original and smoothed data on the same set of axes. Give all answers rounded to 1 decimal place.

Day 1 2 3 4 5 6 7 8

Rain (mm) 2 5 4 6 3 7 6 9

6 Fit a 5-point moving average to the following seasonal data of coat sales.

Season Sales ($)Winter 2008 690Spring 2008 500Summer 2008 400Autumn 2008 720Winter 2009 780Spring 2009 660Summer 2009 550Autumn 2009 440

7 Fit a 4-point centred moving average to the data from question 6. Compare your results. What do you notice about the number of smoothed data points in each case?

8 Perform a 3-point median smoothing on the data shown at right. Plot the smoothed points joined with a straight line.

t

y

20

40

60

8090

10

30

50

70

0 10 20 30 40 50 60 70 80 90

exaM TiP To draw a straight line, students are expected to bring a straight edge (for example, a ruler) into the examination.


t

y

5

10

15

20

25

30

0 2 4 6 8 10

35

196

9 The seasonal indices for the price of shares in CSP fruit canneries are:

Season Index

Winter 1.7

Spring 0.6

Summer 0.5

Autumn 1.2

Deseasonalise the following data:

Season (2009)

Share price

Seasonalised Deseasonalised

Spring 150

Summer 100

Autumn 300

Winter 400

exTended ReSPonSe

Task 1

1 Jazza’s CD store has been opened for the past three weeks. The sales figures for the store were recorded and tabulated as follows.

Jazza’s CD store daily sales figures — number of CDs sold

Mon. Tues. Wed. Thurs. Fri. Sat.Average daily sales

for the week

Week 1 10 8 12 15 24 45

Week 2 12 9 14 18 26 53

Week 3 15 10 16 21 33 58

a Plot the above data as a time series plot and comment on the type of trend that exists. Justify your choice.

b How many seasons are there?c How many cycles are there?d Calculate the average daily sales for each of the weeks.e Complete the table of seasonal indices for each day.

Jazza’s CD Store daily sales figures — number of CDs sold

Mon. Tues. Wed. Thurs. Fri. Sat.

Week 1 1019

= 0.52630.4211 12

19 = 0.7895 1.2632

Week 2 12 = 0.5455 0.4091 14

= 0.63640.8182 2.4091

Week 3 0.5882 0.3922 1.2941 2.2745

f Complete the following table of seasonal indices.

Mon. Tues. Wed. Thurs. Fri. Sat.

Seasonal indices

0.5533 1 22243

. = 0.4075 0.6318 2 4312.

= 0.8104 1.2464

g Interpret what an index of 2.3507 means.



2 Use the results from question 1 to answer question 2.a Complete the table of deseasonalised CD sales figures.

Jazzas’ CD store daily sales figures — number of CDs sold

Mon. Tues. Wed. Thur. Fri. Sat.

Week 1 18.07 19.63 18.99 18.51 19.26 19.14

Week 2 21.69 90 4075.

=

22.16 22.21 20.86

Week 3 27.11 100 4075.

=

25.32 25.91 26.48

b Calculate the equation of the trend line using the least-squares method and interpret the significance of this equation.

c Using the trend line, predict the deseasonalised sales figures for: i Monday week 4 ii Saturday week 4 iii Saturday week 6.d Using the deseasonalised values from part c,

calculate the actual expected future sales for each. Comment on the reliability of the predictions.

Task 2The next 8 questions relate to the following data which represent seasonal rainfall (mm) in an Australian city.

Season 1 2 3 4 5 6 7 8 9 10 11 12

Rainfall (mm) 43 75 41 13 47 78 50 19 51 83 55 25

1 Plot the data points and try to fit a straight trend line by eye. Comment on the ease of making this plot.

2 Now, try to plot a trend line using the 3-median method. Compare the result with that of question 1.

3 Finally, plot a trend line using the least-squares technique. Again, compare your result with the previous ones.

4 To smooth out the seasonal variation, 3-point and 5-point moving average smoothings are tried. Compare the results of these two methods with the results from questions 1 to 3 by plotting the smoothed data.

5 Upon observing the results with the 5-point smoothing, a trend appears. Take the data from the 5-point moving average and fit a straight line using least-squares. Put the first smoothed point at t = 3 and then centre the time data. State the y-intercept and gradient. Compare this trend line with that from question 3.

6 Given the seasonal nature of the data, a 4-point moving average is tried. After calculating the 4-point moving average, fit a straight line using least-squares, following the method of question 5. Compare the results obtained with those from question 5.

7 Finally, try seasonal adjustment. Take t = 1 to be summer and find the seasonal indices. Then, seasonally adjust the data.

8 Take the seasonally adjusted data from question 7 and fit a trend line using least-squares. Comment on this result.

exaM TiP The majority of students are able to predict the deseasonalised value from the trend line, but then fail to seasonalise the value they obtained to determine the actual future value of the data.


exaM TiP Students need to accurately draw a least-squares regression line on a graph by using its formula. Students should determine and connect two points that are widely spaced. Many students draw an inaccurate line by choosing two points that are apparently too close together.


198

Task 3 1 The mean surface temperature (in °C) of Australia for the period 1960 to 2005 is displayed in the time-series

plot below.

[Data Source: ABS 2007]a In what year was the lowest mean surface temperature recorded?The least-squares method is used to fi t a trend line to the time-series plot.b The equation of this trend line is found to be

mean surface temperature = -12.361 + 0.013 × year

i Use the trend line to predict the mean surface temperature (in °C) for 2010. Write your answer correct to 2 decimal places.

The actual mean surface temperature in the year 2000 was 13.55 °C.

ii Determine the residual value (in °C) when the trend line is used to predict the mean surface temperature for this year. Write your answer correct to 2 decimal places.

iii By how many degrees does the trend line predict Australia’s mean surface temperature will rise each year? Write your answer correct to 3 decimal places.

[VCAA 2007]

1960 1965

12.9

13.0

13.1

1970 1975 1980 1985 1990 1995 2000 2005

13.2

13.3

13.4

13.5

13.6

13.7

13.8

13.9

Mea

n su

rfac

e te

mpe

ratu

re (

C)

Year

eBookpluseBookplus

Digital docTest Yourself

Chapter 4

exaM TiP The negative sign was required, as determined by the correct use of the residual formula on the formula sheet.




eBookpluseBookplus aCTiviTieS

Chapter openerDigital doc

10 Quick Questions:• Warm up with a quick quiz on time series. (page 153)

4A Time series and trend linesDigital doc

Spreadsheet 063:• Investigate the least-squares trend line. (page 158)

4B Fitting trend lines and forecastingDigital docs

Skill• SHEET 4.1: Gradient-intercept method for sketching linear graphs (page 163)Spreadsheet 001:• Investigate the 3-median method. (page 163)Work• SHEET 4.1: Plotting time series data and fi tting trend lines using various techniques (page 164)

4C Smoothing time seriesDigital doc

Spreadsheet 077: Investigate the moving average. •(page 169)

Tutorial

We 6 • int-0 : Learn how to display a time series plot.(page 168)

4D Smoothing with an even number of points Tutorial

We 7 • int-0442: Watch a tutorial on performing a centred 4-point moving average on time series data and plotting the result. (page 172)

4E Median smoothingDigital doc

Work• SHEET 4.2: Recognise trends, 3-point moving average, 4-point centred moving average, 6-point centred moving average and 5-point median smoothing. (page 179)

4F Seasonal adjustmentDigital doc

Spreadsheet 115: Make comparisons between •seasonalised and original data. (page 187)

Tutorials

We 10 • int-0443: Watch a tutorial on computing seasonal indices and then using them to deseasonalise data. (page 180) We 11 • int-0 : Learn how to fi nd the equation of the trend line of unemployment fi gures over a fi ve-year period. (page 184) We 12 • int-0445: Watch a tutorial on calculating deseasonalised values, given actual values using seasonal indices. (page 186)

Interactivity int-0185Seasonal adjustment:• Use the interactivity to consolidate your understanding of seasonal adjustment. (page 179)

Chapter reviewDigital doc

Test Yourself: Take the end-of-chapter test to test •your progress. (page 198)

To access eBookPLUS activities, log on to

www.jacplus.com.au

845

846

200200

CoRe — daTa analYSiS

exaM PRaCTiCe 1 ChaPTeRS 1 To 4

MulTiPle ChoiCe 20 minutes

Each question is worth 1 mark.

The following information relates to questions 1 and 2.

The frequency table below shows the number of bedrooms in homes (units and houses) on a street.

1 The mode of the data is:A 1 B 2 C 3D 9 E 28

2 The mean of the data is closest to: A 2 B 2.50 C 2.54D 5.55 E 61.36

x f1 62 93 74 45 2

The following information relates to questions 3 and 4.

The parallel boxplots below represent the scores of two indoor soccer teams over a season containing 12 matches.

1 2 3 4 5 6 7 8 9 10 11 12 13

Scores

Team B

Team A

3 The boxplot for Team B is best described as:A symmetricB negatively skewedC positively skewedD negatively skewed with outliersE positively skewed with outliers

4 From the data, it can be concluded that for the season shown:A Team B won more games than Team AB Team A won more games than Team BC Team A scored the lowest score out of the two

teamsD Team A scored the highest score out of the two

teamsE Team B’s scores were more variable than

Team A’s scores

5 A population has a mean of 82.1 and a standard deviation of 2.3. Approximately 95% of the population will lie in the range:A (5.0, 95.0) B (75.2, 89.0) C (79.8, 84.4)D (77.5, 86.7) E (78.0, 86.2)

6 The height of a student is 182 cm and was taken from a sample with a mean of 175 cm and a standard deviation of 4 cm. The z-score for the student’s height is:A –1.75 B –1.02 C 0.75 D 1.02 E 1.75

7 A set of bivariate data has the following values:r = 0.6754, x = 8.93, sx = 2.38, y = 18.87, sy = 5.09The equation of the least-squares regression line y = a + bx is closest to:A y = 5.97 + 1.44x B y = 1.44 + 5.97xC y = 5.97 – 1.44x D y = 16.1 – 0.32xE y = 16.1 + 0.32x

8 A student constructs a scatterplot from the set of data given.

x 1 2 3 4 5 6 7 8 9

y 12 16 24 25 32 38 56 80 95

To linearise the scatterplot, he applies an x2 transformation. The equation of the least-squares regression line is closest to:A y = -8.08 + 10.02x B y = 9.63 + 1.02x2

C y = 9.63 + 1.02x D y = -8.08 + 10.02x2

E y = -8.44 + 0.95x2

The following information relates to questions 9 and 10.The following table shows the seasonal indices for the quarterly attendances at a swimming pool.

Quarter 1 2 3 4

Seasonal index 1.12 0.95 1.09

9 The value of the missing seasonal index for quarter 3 is:A 0.84 B 0.91 C 0.95D 1.00 E 1.05

10 The actual number of pool visits in the first quarter of a particular year is 146 089. The deseasonalised number is closest to:A 130 437 B 138 785 C 146 000D 153 778 E 163 620

11 A 3-median trend line is fitted to the following time series data.

x 1 2 3 4 5 6 7 8 9 10

y 12 16 15 18 26 22 27 26 29 34


201exam practice 1 Core — data analysis

5

10

15

20

25

30

35

210 43 65 87 9 10 x

y Its slope is closest to:A 2

B 12

C 137

D 2512

E 713

12 For the time series data below, the value of the 3-point moving median centred at t = 5 is:

t 1 2 3 4 5 6 7

C 12 16 13 17 19 18 20

A 5B 17C 18D 19E 20

Total marks = 12

exTended ReSPonSe 10 minutes

A Melbourne-based telephone support service relies on the work of 65 volunteers. A sample of 15 volunteers is selected to look at job satisfaction among volunteers. The 15 volunteers are asked to rate their satisfaction with the volunteer work on a scale of 0 to 10, where 0 is not at all satisfied and 10 is very satisfied. The results are shown below.

Average hours 2 2 2.5 3 3 4 5 5 7 8 10 12 13 15 20

Satisfaction 6 5 3 6 5 7 9 10 8 9 9 8 10 9 10

a Assuming a linear relationship, use the data above to determine the least-squares regression equation that could be used to determine the level of job satisfaction from the number of hours worked. Write your answer in terms of the variables given. [1 mark]

b A residual plot is constructed to test the assumption of linearity for the relationship.

3

1

2

0

2

1

3

2 4 6 8 10 12 14 16 18 20

Residual

Average hours

i Explain the features of this residual plot that suggest the relationship is not linear. The residual plot suggests that a log10 (x) transformation may linearise the data.

ii The table below shows the transformed values. Find the missing value correct to 2 decimal places.

Average hours

2 2 2.5 3 3 4 5 5 7 8 10 12 13 13 20

Log (hours) 0.30 0.30 0.40 0.48 0.48 0.70 0.70 0.85 0.90 1 1.08 1.11 1.18 1.30

Satisfaction 6 5 3 6 5 7 9 10 8 9 9 8 10 9 10

iii Find the equation of the least-squares regression line for the transformed data, in terms of job satisfaction and hours. [1 + 1 + 1 = 3 marks]

Total marks = 4

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