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5. Time Series Analysis
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AustraliaAU
Year 12

5.03 Seasonal adjustments and deseasonalising data

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Seasonal indices

The seasonal index (also called seasonal effect or seasonal component) is a measure of how a particular season compares on average to the mean of the cycle. The graph below shows raw seasonal data as well as the data smoothed with a moving average. From the green line we can see that December is always a peak season above the smoothed data line and March is always a low season below the smoothed data line. The seasonal index is a number that can be given as a percentage or as a decimal. The seasonal index for December in this case is 106.564\% which means figures for December are 1.065\,64 times higher than the average (or 6.564\% above the cycle mean).

The image shows a graph of raw seasonal data from 2012 to 2014. Ask your teacher for more information.

We use seasonal indices for two purposes.

  • They can be used to smooth data by a process called deseasonalising.

  • They can be used to help with predicting future scores with time series data. Once an initial predicted value from a smoothed line is calculated, the seasonal index is used to correct that value up or down depending on which season we are predicting for.

Calculating the seasonal index using the average percentage method:

  1. Calculate the mean for each cycle (each year in this case).

  2. Calculate the proportion or average percentage of the cycle mean for each piece of raw data.

  3. Calculate the average proportion for each season (month in this case). It can be written as a decimal or as a percentage.

Examples

Example 1

The local police station records the number of speeding fines issued each quarter. The table below has the data for each quarter from 2012 to 2014.

Time periodDataPercentage of yearly mean
\text{March }2012105106.06\%
\text{June }201291x
\text{September }2012101102.02\%
\text{December }201299100\%
\text{March }2013101y
\text{June }20138389.01\%
\text{September }201396102.95\%
\text{December }20139399.73\%
\text{March }201499108.2\%
\text{June }20148289.62\%
\text{September }201494102.73\%
\text{December }201491z
a

For 2012, 2013 and 2014, calculate the mean number of speeding tickets issued in each time period. Give your answers to two decimal places.

Year201220132014
Mean
Worked Solution
Create a strategy

To find the mean for each year, add up the number of speeding tickets for each month of that year then divide by the number of data values.

Apply the idea
\displaystyle \text{Mean }2012\displaystyle =\displaystyle \dfrac{105+91+101+99}{4}Find the mean of the data for that year
\displaystyle =\displaystyle 99.00Evaluate and round
\displaystyle \text{Mean }2013\displaystyle =\displaystyle \dfrac{101+83+96+93}{4}Find the mean of the data for that year
\displaystyle =\displaystyle 93.25Evaluate and round
\displaystyle \text{Mean }2014\displaystyle =\displaystyle \dfrac{99+82+94+91}{4}Find the mean of the data for that year
\displaystyle =\displaystyle 91.50Evaluate and round
Year201220132014
Mean99.0093.2591.50
b

Divide the time period's data value by the yearly mean then multiply by a hundred percent.

Worked Solution
Create a strategy

To find the percentage of the yearly mean we divide the time period's data value by the yearly mean then multiply by 100\% to get to a percentage figure.

Apply the idea

x is for January 2012 which had 91 fines. From part (a), the mean for 2012 was 99.

\displaystyle x\displaystyle =\displaystyle \dfrac{91}{99}\times 100\%Divide the value by the mean and multiply by 100\%
\displaystyle =\displaystyle 91.92\%Evaluate and round
c

Use your answers from part (a) to calculate the value of y. Give your answer to two decimal places.

Worked Solution
Apply the idea

y is for March 2013 which had 101 fines. From part (a), the mean for 2013 was 93.25.

\displaystyle y\displaystyle =\displaystyle \dfrac{101}{93.25}\times 100\%Divide the value by the mean and multiply by 100\%
\displaystyle =\displaystyle 108.31\%Evaluate and round
d

Use your answers from part (a) to calculate the value of z. Give your answer to two decimal places.

Worked Solution
Apply the idea

z is for December 2014 which had 91 fines. From part (a), the mean for 2014 was 91.5.

\displaystyle z\displaystyle =\displaystyle \dfrac{91}{91.5}\times 100\%Divide the value by the mean and multiply by 100\%
\displaystyle =\displaystyle 99.45\%Evaluate and round
Idea summary

The seasonal index (also called seasonal effect or seasonal component) is a measure of how a particular season compares on average to the mean of the cycle.

Calculating the seasonal index using the average percentage method:

  1. Calculate the mean for each cycle.

  2. Calculate the proportion or average percentage of the cycle mean for each piece of raw data.

  3. Calculate the average proportion for each season. It can be written as a decimal or as a percentage.

Deseasonalised raw data

We use the seasonal index when predicting from time series data. The data is first smoothed either using by a moving average or by deseasonalising (see below). We then calculate a predicted value using the equation of the least-squares regression line from the smoothed data. We then use the seasonal index to adjust the predicted value so that it takes the particular season into consideration. In the above example, a predicted value for December will be adjusted to be higher whereas a predicted value for March will be adjusted lower.

Deseasonalising data is also called making seasonal adjustments. The seasonal indices are used to smooth or deseasonalise our data in a similar way that a moving average is used. Both methods smooth the data as shown in the graphs below.

A graph with raw data, deseasonalised data, and the 7 point moving average. Ask your teacher for more information.

Note that the 7 point moving average line is smoother than the deseasonalised data line, but both methods are used in the real world to assist with predicting from time series data.

Deseasonalised data formula: \text{Deseasonalised data}=\dfrac{\text{Raw value}}{\text{Seasonal index}}

Note that the seasonal index should be a decimal when using this formula.

Examples

Example 2

Every four months Neil records the growth of his bean plant (starting with a new plant every year). The data provided is from the beginning of 2012 to the end of 2015.

Time periodGrowth (in cm)Proportion of yearly mean
\text{April }201295.60.99
\text{August }2012106.7a
\text{December }201287.80.91
\text{April }2013c0.99
\text{August }2013101.21.1
\text{December }201384.10.91
\text{April }201486.31.01
\text{August }201493.61.09
\text{December }201477.30.9
\text{April }201576.10.99
\text{August }201583.4b
\text{December }201571.80.93
a

Calculate the value of a in the table. Give your answer to two decimal places.

Worked Solution
Create a strategy

Divide the corresponding data value by the mean growth for 2012.

Apply the idea

First we will find the mean for 2012.

\displaystyle \text{Mean}\displaystyle =\displaystyle \dfrac{95.6+106.7+87.8}{3}Find the average
\displaystyle =\displaystyle 96.7Evaluate

The growth for August 2012 was 106.7. So we divide this by the mean.

\displaystyle a\displaystyle =\displaystyle 106.7\div 96.7Divide 106.7 by 96.7
\displaystyle =\displaystyle 1.10Evaluate the division
b

Calculate the value of b in the table. Give your answer to two decimal places.

Worked Solution
Create a strategy

Divide the corresponding data value by the mean growth for 2015.

Apply the idea

First we will find the mean for 2015.

\displaystyle \text{Mean}\displaystyle =\displaystyle \dfrac{76.1+83.4+71.8}{3}Find the average
\displaystyle =\displaystyle 77.1Evaluate

The growth for August 2015 was 83.4. So we divide this by the mean.

\displaystyle b\displaystyle =\displaystyle 83.4\div 77.1Divide 83.4 by 77.1
\displaystyle =\displaystyle 1.08Evaluate the division
c

If the mean for 2013 is 92.2, calculate the value of c. Give your answer to two decimal places.

Worked Solution
Create a strategy

Multiply the mean for 2013 by the proportion for April 2013.

Apply the idea

In the previous parts we divided the growth by the mean to get the proportion: \text{Proportion}=\dfrac{\text{Growth}}{\text{Mean}} Now we want to find the growth, so we should do the opposite: multiply the mean for 2013 by the proportion for April 2013: \text{Growth}= \text{Mean} \times \text{Proportion}

The proportion for April 2013 was 0.99 and we are given the mean for 2013 as 92.2.

\displaystyle c\displaystyle =\displaystyle 92.2\times 0.99Multiply the mean by the proportion
\displaystyle c\displaystyle =\displaystyle 91.28Evaluate
d

Calculate the seasonal component for April. Give your answer to two decimal places.

Worked Solution
Create a strategy

Average the proportions for April.

Apply the idea
\displaystyle \text{April}\displaystyle =\displaystyle \dfrac{0.99+0.99+1.01+0.99}{4}Average the proportions for April
\displaystyle =\displaystyle 1.00Evaluate and round
e

Seasonally adjust the data for April 2015. Give your answer to two decimal places.

Worked Solution
Create a strategy

We need to divide the raw data value by the seasonal component from part (d).

Apply the idea

The growth for April 2015 was 76.1.

\displaystyle \text{Deseasonalised data}\displaystyle =\displaystyle 76.1\div 1.00Divide the growth by the mean
\displaystyle =\displaystyle 76.10Evaluate the division
Idea summary

Deseasonalising data is also called making seasonal adjustments.

Deseasonalised data formula: \text{Deseasonalised data}=\dfrac{\text{Raw value}}{\text{Seasonal index}}

The seasonal index should be a decimal when using this formula.

Spreadsheet applications

A spreadsheet is a powerful tool for dealing with numbers and formulae. Although your calculator has a spreadsheet application, the screen is very small so it is more practical to use an application on your computer.

Examples

Example 3

A sports store records the sales of its hockey sticks every 4 months. The finance department create a spreadsheet to record the data and analyse the seasonality of the figures:

The image shows a spreadsheet with a table titled Hockey Stick Sales. Ask your teacher for more information.
The image shows a spreadsheet with a table titled Seasonal indices. Ask your teacher for more information.
a

Which of the following formulae could be entered into cell \text{M7} to calculate the cycle mean for 2017?

A
=\text{AVERAGE}(\text{L7 : L9})
B
=\text{AVG}(\text{L7 : L9})
C
=(\$\text{L}\$6+\$\text{L}\$7+\$\text{L}\$8)/3
D
=(\text{L7+L8+L9})/3
E
=(\$\text{L}\$7+\$\text{L}\$8+\$\text{L}\$9)/3
F
=3/(\text{L7+L8+L9})
Worked Solution
Create a strategy

Choose the options that average the sales for 2017.

Apply the idea

The sales for 2017 are in cells \text{L7, L8,} and \text{L9}. So we need to find the average of the data in these cells. The AVERAGE function will do this, as well as adding the cells and dividing by 3.

The correct options are A: =\text{AVERAGE}(\text{L7 : L9}),\, D: =(\text{L7+L8+L9})/3,\, and E: =(\$\text{L}\$7+\$\text{L}\$8+\$\text{L}\$9)/3.

b

Which of the following formulae could be entered into cell \text{N13} to calculate the Percentage of Cycle mean for January 2019?

A
100 * (\text{L13/M13})
B
=\text{L13/M13} * 100
C
=\text{L13}/\$\text{M}\$13 * 100
Worked Solution
Create a strategy

We need to find a formula that divides the sales for that row, by the cycle mean for 2019 and multiply by 100.

Apply the idea

The sales for January 2019 are in \text{L13}, and the cycle mean for 2019 is in \text{M13}.

The correct options are B: =\text{L13/M13} * 100 and C: =\text{L13}/\$\text{M}\$13 * 100

c

The following formula is entered into cell \text{K20} to calculate the seasonal index for May\text{=(O5+O8+O11+O14)/4}. Something is wrong with the formula. Write the correct formula.

Worked Solution
Create a strategy

Write a formula that averages the percentage cycle means for May.

Apply the idea

Since the Percentage of cycle mean is in the column \text{N}, we should average the values in column \text{N} for the month of May.

The correct formula is:\text{=(N5 + N8 + N11 + N14)/4}

d

The following formula is entered into cell \text{O6} to deseasonalise the data for September 2016: \text{=L6/K20*100}. Something is wrong with the formula. Write the correct formula.

Worked Solution
Create a strategy

Write a formula that divides the September 2016 sales by the seasonal index for September, and multiplies by 100.

Apply the idea

The September 2016 sales are in cell \text{L6} and the seasonal index for September is in cell \text{L20}, so the formula should be:

\text{=L6/L20} * 100

e

One method to check that the calculation of the seasonal indices is correct is to make sure the mean is equal to 100\%. What formula could be entered into cell \text{N18} to check this?

A
=\text{AVG}(\text{J20 : L20})
B
=\text{AVERAGE}(\text{J20 : L20})
C
(\text{J20+K20+L20})/3
D
=(\text{J20+K20+L20})/3
Worked Solution
Create a strategy

Choose the options that averages the seasonal indices.

Apply the idea

We can average the seasonal indices in cells \text{J20, K20}, and \text{L20} either by using the AVERAGE function or by adding the values and dividing by 3.

The correct options are B: =\text{AVERAGE (J20:L20)} and D: \text{=(J20+K20+L20)/3}.

f

Another method to check that the calculation of the seasonal indices is correct is to check that the sum of the indices is 300. What formula could be entered into cell \text{N18} to check this?

Worked Solution
Create a strategy

Write a formula that adds the seasonal indices.

Apply the idea

We need to add the seasonal indices in cells \text{J20, K20}, and \text{L20}:=\text{J20+K20+L20}

Idea summary

A spreadsheet can be used to find seasonal indices and deseasonalise data using tables and formulas.

All formulas need to start with an equals (=) sign.

The AVERAGE function is useful for finding the means.

Outcomes

ACMGM090

calculate seasonal indices by using the average percentage method

ACMGM091

deseasonalise a time series by using a seasonal index, including the use of spreadsheets to implement this process

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