Seasonal Indices
When considering a set of data over time it is sometimes visible that there is a seasonal fluctuation in
the data. An example of this would be average monthly temperatures over several years.
A seasonal adjustment can be made to the data in order that a better picture of the trends can be
gained so that more reliable predictions are possible. This is done using Seasonal Indices.
Seasonal Indices are calculated so that their average is 1. This means the
SUM OF SEASONAL INDICES = NUMBER OF SEASONS.
For example, if seasons are months, the SUM SEASONAL INDICES = 12.
Example1: Find the missing Seasonal Index.
Quarter
Q1
Q2
Q3
Seasonal Indices
1.3
1.4
0.4
Q4
A seasonal index of:



1.3 tells us data in Q1 are typically 30% above average.
1.4 tells us data in Q2 are typically 40% above average.
0.4 tells us data in Q3 are typically 60% below average.
A season index is defined by the formula:
Seasonal index =
𝒗𝒂𝒍𝒖𝒆 𝒇𝒐𝒓 𝒔𝒆𝒂𝒔𝒐𝒏
𝒔𝒆𝒂𝒔𝒐𝒏𝒂𝒍 𝒂𝒗𝒆𝒓𝒂𝒈𝒆
Question 1
A time series plot is to be constructed using the data
contained in the table below:
Month no.
1
2
3
4
5
6
7
8
9
10
11
12
Cost ($)
58
92
123
136
151
162
159
139
172
180
185
179
The data in the table is used to calculate the seasonal
index for each month. The seasonal index for month 4 is
closest to:
A. 0.81
B. 0.85
C. 0.94
D. 1.04
E. 1.06
where the season is a month, quarter or the like.
Question 2
The following (incomplete) table shows seasonal indices
at Crockett’s dress shop for 2012.
Summer
1.20
Autumn
Winter
Spring
1.05
Which one of the following statements regarding sales in
2012 is true?
A. The highest sales season could not be Autumn or
Winter
B. Both Autumn and Winter must have sales below
the season average
C.
Sales figures for Autumn and Winter could be
identical
D. Sales in Spring were generally greater than sales
in Summer
E.
The seasonal indices for Autumn and Winter will
both be greater than 1
A seasonal index is used to deseasonalise (smooth) the data, using the rule:
Deseasonalised figure =
𝒂𝒄𝒕𝒖𝒂𝒍 𝒇𝒊𝒈𝒖𝒓𝒆
𝒔𝒆𝒂𝒔𝒐𝒏𝒂𝒍 𝒊𝒏𝒅𝒆𝒙
Example 2: using example 1 above,
deseasonalise the following data.
Quarter
Actual data
Deseasonalised
Data
2000
Q1
Q2 Q3 Q4
1500 1650 300 1100
1500
Autumn
1.10
Winter
data
0
1
Question 3
a. On Tommy’s farm, tomatoes can be grown in a
hothouse all year round but production varies from one
season to another.
The seasonal indices for some of the seasons are given in
the table below.
Summer
1.25
Series2
Deseasonalised
500
What does this look like graphically?
Season
Seasonal Index
Actual data
Series1
1000
Spring
0.99
2
3
4
b. Tommy calculated the deseasonalised summer
production figure to be 1500 tonnes. The actual summer
production figure would have been:
A.
B.
C.
D.
E.
1200
1650
1875
1364
1500
a. The seasonal index for winter is:
A.
B.
C.
D.
E.
0.66
1.66
1.99
0.99
1.00
Steps to Deseasonalise Time Series Data
Step 1: Yearly (cycle) Average – Find the average value for the entire seasonal cycle.
Step 2: Seasonal Average: Divide each seasonal value by the yearly average. This shows how each
season compares to the yearly average.
Step 3: Seasonal Index: Average the seasonal values in step 2.
Step 4: Deseasonalised figures (seasonally adjusted figures) – Divide each original value by each
corresponding seasonal Index.
Step 5: You can now plot the seasonally adjusted data to show the trend more clearly than the original
data would have. Any marked change in figures will stand out more on the graph of deseasonalised
data.
Example 3
Step 4: Deseasonalise figures
Summer Autumn
Actual
Sales
figures
920
Winter
Spring
1241
446
Desea
Sales
Figures
=
Step 5: You can now graph both seasonal data
and deseasonalised data. Sketch below.
1085
Step 1: Yearly average=
920+1085+1241+446
4
Summer Autumn
Step 2: Seasonal Averages.
Summer Autumn Winter Spring
Seasonal
averages
Step 3: Seasonal Indices: as this is only over 1
cycle, the seasonal averages are the seasonal
Indices.
Example 4: The average daily sales (in dozens) of soft drinks in a milk bar are as follows.
1985
1986
1987
1988
Summer
32
34
37
38
Autumn
17
16
19
24
Winter
12
13
17
16
Spring
22
25
26
29
Step 1: Yearly averages
Step 2: Seasonal Averages: divide each season by its yearly average.
Summer
Autumn
Winter
Spring
1985
1986
1987
1988
Seasonal
Indices
Step 3: Seasonal Indices: Average the seasonal values.
Step 4: Deseasonalise Data by dividing actual values by seasonal indices.
Summer
1985
1986
1987
1988
Step 5: Sketch the seasonal
and deseasonalised data.
Autumn
Winter
Spring
Winter
Spring
Question 4
The sales results for Rashid’s car yard over eight
consecutive quarters are:
2008
2009
Q1
Q2
Q3
Q4
42 000
52 000
21 000
32 000
15 000
26 000
22 000
50 000
The seasonal index for Q2 is 0.82.
Yearly
average
25 000
40 000
a. The seasonal index for the first quarter is:
A.
B.
C.
D.
E.
b. The deseasonalised figure to the nearest 10 of Q2 in
2009 is:
A.
B.
C.
D.
E.
17 220
25 610
26 240
39 020
64 630
1.3
1.49
1.68
47000
94000
7D
Fitting a Trend line and Forecasting
A trend line would then be fitted to this Deseasonalised data. Independent variable is time (t) and
dependent variable is number of cans (y). When you predict from a trend line, the values is
deseasonalised (smoothed). Hence you must re-adjust it to be seasonal if needed.
Since 𝑑𝑒𝑠𝑒𝑎𝑠𝑜𝑛𝑎𝑙𝑖𝑠𝑒𝑑 𝑓𝑖𝑔𝑢𝑟𝑒 =
then
𝐴𝑐𝑡𝑢𝑎𝑙 𝑓𝑖𝑔𝑢𝑟𝑒
𝑆𝑒𝑎𝑠𝑜𝑛𝑎𝑙 𝑖𝑛𝑑𝑒𝑥
Seasonalised figure = deseasonalised figure x seasonal index
i.e. raw figure = smoothed prediction x seasonal index
From previous example, the Least Squares regression line of deseasonalised values is: y =
So a prediction for summer 1989 would be t =
y=
Now seasonalise it:
yactual =
Question 5
Hunter owns a bike shop and has noted that there is
seasonal variation apparent when considering the number
of bikes sold each season. Over a period of three years
from 2010 to 2012 inclusive the seasonal indices for bike
sales are calculated and given in the following
(incomplete) table.
Season
Seasonal Index
Summer
1.3
Autumn
Winter
0.72
Using Summer 2010 as time period 1, Autumn 2010 as
time period 2 etc, the following least squares regression
equation was calculated:
Deseasonalised sales = 540.1 + 4.25 × time period
d. Calculate the residual value of deseasonalised sales for
Spring 2012.
Spring
1.09
a. Complete the table with the seasonal index for
Autumn.
b. Interpret the seasonal index for Winter in the context
of the problem.
e. Use the regression equation and other available
information to forecast the actual number of bikes Hunter
will sell during Winter 2014. Give your answer to the
nearest whole number.
c. During Spring 2012, Hunter sold 654 bikes. Determine
the deseasonalised sales figure for this particular season.
7E