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4. Time Series
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VCE 12 General 2023

4.04 Seasonal indices

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Introduction

Instead of calculating a moving mean or a moving median, time series data that contains seasonality can be smoothed by calculating what is known as a seasonal index and using it to deseasonalise the data.

Calculate seasonal indices

The seasonal index is a measure of how a particular season compares with the mean season.

There is a four step process involved to deseasonalise time series data:

  1. Calculate the mean for each cycle

  2. Express each piece of original data as a proportion of the cycle mean

  3. Calculate the seasonal index for each season by finding the mean of the proportions

  4. Deseasonalise the data by dividing by the seasonal index

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 following are the properties of seasonal indices.

  • A seasonal index more than 1 means the season is higher than average.

  • A seasonal index less than 1 means the season is lower than average.

  • The sum of the seasonal indices equals the number of seasons.

  • The mean of the seasonal indices always equals 1.

Deseasonalise the data

To deseasonalise the data each raw value can now be adjusted by dividing it by the relevant seasonal index as follows. This process is also referred to as seasonal adjustment.

\text{Deseasonalised data}=\dfrac{\text{Actual figure}}{\text{Seasonal index}}

If given a deseasonalised figure, then the same formula can also be used to reseasonalise it using the seasonal index.

\text{Reseasonalised data}=\text{Deseasonalised data}\times \text{Seasonal index}

Examples

Example 2

The number of waiters employed by a restaurant chain in each quarter of 1 year, along with some seasonal indices that have been calculated from the previous year's data, are given in the following table.

Quarter 1Quarter 2 Quarter 3Quarter 4
Number of waiters19714483170
Seasonal index1.31-0.571.08
a

What is the seasonal index for the second quarter?

Worked Solution
Create a strategy

Subract the sum of given seasonal indices from the number of seasons.

Apply the idea
\displaystyle \text{Sum of given indices}\displaystyle =\displaystyle 1.31+0.57+1.08Add the given indices
\displaystyle =\displaystyle 2.96Evaluate
\displaystyle \text{Quarter 2 index}\displaystyle =\displaystyle 4-2.96Subtract the sum from the number of seasons
\displaystyle =\displaystyle 1.04Evaluate
b

The seasonal index for quarter 1 is 1.31. What does this mean in terms of the average quarterly numbers of waiters?

A
There are less on average.
B
There are more on average.
C
Each waiter has to do 1.31 times as much as work.
D
There are about the same on average.
Worked Solution
Create a strategy

Use the fact that a seasonal index of 1 means that the season is close to the average.

Apply the idea

Since 1.31>1, there are more on average in terms of quarterly number of waiters.

The answer is option B.

c

Deseasonalise the data. Round to the nearest whole number.

Quarter 1Quarter 2 Quarter 3Quarter 4
Number of waiters19714483170
Seasonal index1.311.040.571.08
Deseasonalised No. of waiters⬚⬚⬚⬚
Worked Solution
Create a strategy

Divide the number of waiters by the seasonal index.

Apply the idea
Quarter 1Quarter 2 Quarter 3Quarter 4
Number of waiters19714483170
Seasonal index1.311.040.571.08
Deseasonalised No. of waiters\dfrac{197}{1.31}=150\dfrac{144}{1.04}=138\dfrac{83}{0.57}=146\dfrac{170}{1.08}=157
Idea summary

Deseasonalised data formula:

\text{Deseasonalised data}=\dfrac{\text{Actual figure}}{\text{Seasonal index}}

Reseasonalised data formula:

\displaystyle \text{Reseasonalised data}\displaystyle =\displaystyle \text{Deseasonalised data}\times \text{Seasonal index}

Outcomes

U3.AoS1.28

identify key qualitative features of a time series plot including trend (using smoothing if necessary), seasonality, irregular fluctuations and outliers, and interpret these in the context of the data

U3.AoS1.29

calculate, interpret and apply seasonal indices

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