5.03 Seasonal adjustments and deseasonalising data

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%$106.564% which means figures for December are $1.06564$1.06564 times higher than the average (or $6.564%$6.564% above the cycle mean).

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.

 

Exploration

Step 1: Calculating the mean for each cycle

We'll use the data from the graph above to illustrate each of the steps involved. There are $4$4 time periods in each year. From the graph and the table below we can determine that there are $4$4 seasons in a cycle.

First we need to calculate the mean for the 2012, 2013 and 2014 cycles.

The first and last are already done for us, we need to calculate the mean for 2013 cycle.

$\text{2013 mean}=\frac{427+463+484+494}{4}=467$2013 mean=427+463+484+4944​=467

 

Step 2: Expressing the raw data as a proportion of the cycle mean

To calculate $X$X, $Y$Y and $Z$Z in the table, we need to calculate each of their corresponding raw data values as a proportion of the mean for the cycle (or year) that they belong to.

$X=\frac{480}{490.75}=0.9781$X=480490.75​=0.9781 We could also write $97.81%$97.81%. This is less than $100%$100% and indicates that this raw data value is $2.19%$2.19% below the cycle mean.

$Y=\frac{427}{467}=0.9143$Y=427467​=0.9143 We could also write $91.43%$91.43%. This is less than $100%$100% and indicates that this raw data value is $8.57%$8.57% below the cycle mean.

$Z=\frac{499}{462.25}=1.0795$Z=499462.25​=1.0795 We could also write $107.95%$107.95%. This is more than $100%$100% and indicates that this raw data value is $7.95%$7.95% above the cycle mean.

Step 3: Calculating the average proportion for each season

We will now calculate the average proportion for each season our data. This is known as the seasonal index or seasonal component.

The seasonal component for June is $98.97%$98.97% and for December is $106.56%$106.56%.

To determine the seasonal index for March we calculate the average seasonal change for each March quarter. The seasonal index for March is then:

$\frac{0.9598+0.9143+0.9194}{3}=0.9312=93.12%$0.9598+0.9143+0.91943​=0.9312=93.12%

We can similarly calculate the seasonal index for September, which is:

$\frac{1.0025+1.0364+1.0016}{3}=1.0135=101.35%$1.0025+1.0364+1.00163​=1.0135=101.35%

Summary of seasonal indices

Season	Seasonal Index	Meaning
March	$93.12%$93.12%	This season is on average $6.88%$6.88% below the year mean.
June	$98.97%$98.97%	This season is on average $1.03%$1.03% below the year mean.
September	$101.35%$101.35%	This season is on average $1.35%$1.35% above the year mean.
December	$106.56%$106.56%	This season is on average $6.56%$6.56% above the year mean.

 

Note that the average for the four seasons will be $100%$100% or the sum of the four seasons will be $400$400. This is because seasonal indices are calculated as a proportion of the average, so the average of these proportions will always return the true average, the mean, which is represented as $100%$100%. When working with decimal values for seasonal indices, we have the same relationship, except instead of $100%$100% we convert to the decimal $1.00$1.00.

This also means that if we know all the seasonal indices of a season except for one, we can calculate the missing score using an average of $100%$100%.

 

Practice question

Question 1

Question 2

 

How we use the seasonal index

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 the raw data

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.

Note that the $7$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.

 

Exploration

Returning to the exploration above, this means all March raw data should be divided by $0.9312$0.9312. March is the lowest season so dividing by a number less than one will increase the scores.

All June raw data should be divided by $0.9897$0.9897. June is also a below average cycle season so again this will increase the scores.

All September raw data should be divided by $1.0135$1.0135. September is above the cycle average season so dividing by a number greater than $1$1 will lower the scores.

All December raw data should be divided by $1.0656$1.0656. December is the highest season so again this will lower the scores.

So dividing each raw score by the seasonal index, we get the following deseasonalised data.

$X=\frac{492}{1.0135}=485.4465$X=4921.0135​=485.4465

$Y=\frac{463}{0.9897}=467.8185$Y=4630.9897​=467.8185

$Z=\frac{425}{0.9312}=456.4003$Z=4250.9312​=456.4003

 

Practice questions

Question 3

Question 4

 

Using a spreadsheet application to calculate seasonal indices and deseasonalise data

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.

 

Worked example

The Beach Cafe records the following customer numbers quarterly. The owners wish to gain a clear picture of seasonal variations over time. Use a spreadsheet application to calculate the seasonal index for each month and hence deseasonalise the data. Display the original data and the deseasonalised data on the same graph.

	Jan	Apr	July	Oct
2016	1670	1218	815	1384
2017	1865	1327	903	1449
2018	2325	1731	1112	1835
2019	2520	1890	1232	2016

 

Step 1: Set up your table structure in your spreadsheet application

Note that it is a good idea to write the data vertically in order from 2016 Jan to 2019 October. It also helps to give each time period a number as the numbers will be used for predictions later. In order to graph the data with time numbers on the horizontal axis we need to enter the time numbers as text.

Step 2: Create the formulae

Below is a screen shot of the finished spreadsheet. Do NOT type these numbers in. We will use formulae to calculate them.

Here are the formula used

Note:

• Using horizontal borders to separate the $4$4 cycle groups makes the spreadsheet clearer to read.
• We know that the $4$4 values for 2016 have the same cycle mean as do the values for the other years so therefore merging the cells for cycle mean for each year makes sense and allows you to copy the formula for 2016 data down the column.
• In this example the formula in cell E3 is =AVERAGE(D3 : D6) has been used. We could also use the formula $=\frac{(D3+D4+D5+D6)}{4}$=(D3+D4+D5+D6)4​
• In column F we are calculating the percentage of cycle mean for each data point. The four 2016 numbers all need to be divided by cell C3. If we use an absolute reference ($C$3) in the formula in cell F3 then we will be able to copy the formula down to cell F6. The $ signs ensure the C3 is 'locked' in place.
• The four seasonal indices have been calculated in a separate table to the right. It's a good idea to check that the average of these four numbers is $100%$100% (or the total is $400$400). That way you know that your formulae are working correctly.

Step 4: Create the graph

Select the cells you wish to display on the graph. In this case, it is cells C3 to D18 and also F3 to F18.

Select Insert ►Chart from the menu and choose the 2D line with markers type.

The finished chart should look something like the one below. We can see the original data has been smoothed by being deseasonalised. Note that the title can be changed by double clicking in the title box and typing an appropriate name.

 

Practice question

Question 5