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 $106.56%$106.56% which means figures for December are $1.0656$1.0656 times higher than the average (or $6.56%$6.56% above the cycle mean).
We use seasonal indices for two purposes.
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%.
The local police station records the number of speeding fines issued each quarter.
The table alongside has the data for each quarter from 2016 to 2018.
Time Period |
Data |
Percentage of yearly mean |
March 2016 | $105$105 | $106.06%$106.06% |
June 2016 | $91$91 | $x$x |
September 2016 | $101$101 | $102.02%$102.02% |
December 2016 | $99$99 | $100%$100% |
March 2017 | $101$101 | $y$y |
June 2017 | $83$83 | $89.01%$89.01% |
September 2017 | $96$96 | $102.95%$102.95% |
December 2017 | $93$93 | $99.73%$99.73% |
March 2018 | $99$99 | $108.2%$108.2% |
June 2018 | $82$82 | $89.62%$89.62% |
September 2018 | $94$94 | $102.73%$102.73% |
December 2018 | $91$91 | $z$z |
For 2016, 2017 and 2018, calculate the mean number of speeding tickets issued in each time period.
Give your answers to two decimal places.
Year | 2016 | 2017 | 2018 |
Mean | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Use your answers from part (a) to calculate the value of $x$x.
Give your answer to two decimal places.
Use your answers from part (a) to calculate the value of $y$y.
Give your answer to two decimal places.
Use your answers from part (a) to calculate the value of $z$z.
Give your answer to two decimal places.
A local cafe stays open for $5$5 days a week in the city, and records how many cappuccinos they sell on Friday, Saturday and Sunday for four weeks.
The table alongside has the data for the four weeks.
Week |
Day |
Data |
Percentage of weekly mean |
Week $1$1 | Friday | $25$25 | $107.14%$107.14% |
Saturday | $24$24 | $x$x | |
Sunday | $21$21 | $90%$90% | |
Week $2$2 | Friday | $12$12 | $37.89%$37.89% |
Saturday | $36$36 | $y$y | |
Sunday | $47$47 | $148.42%$148.42% | |
Week $3$3 | Friday | $16$16 | $58.54%$58.54% |
Saturday | $12$12 | $43.9%$43.9% | |
Sunday | $54$54 | $197.56%$197.56% | |
Week $4$4 | Friday | $12$12 | $39.56%$39.56% |
Saturday | $23$23 | $75.82%$75.82% | |
Sunday | $56$56 | $z$z |
Calculate the mean number of cappuccinos sold during each week.
Give your answers to two decimal places.
Week | Week $1$1 |
Week $2$2 |
Week $3$3 |
Week $4$4 |
Mean | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Use your answers from part (a) to calculate the value of $x$x.
Give your answer to two decimal places.
Use your answers from part (a) to calculate the value of $y$y.
Give your answer to two decimal places.
Use your answers from part (a) to calculate the value of $z$z.
Give your answer to two decimal places.
Calculate the seasonal index for each day.
Give your answers to four decimal places.
Day | Friday | Saturday | Sunday |
Seasonal index | $\editable{}$ | $\editable{}$ | $\editable{}$ |
The seasonal index for Friday tells us that coffee sales on Friday tend to be
Around $39%$39% greater than the average sales over the three days.
Around $61%$61% less than the average sales over the three days.
Around $39%$39% less than the average sales over the three days.
Around $61%$61% greater than the average sales over the three days.
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.
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.
$\text{Deseasonalised data}=\frac{\text{Raw value}}{\text{Seasonal index}}$Deseasonalised data=Raw valueSeasonal index
Note that the seasonal index should be a decimal when using this formula.
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
The table shows the number of new gaming apps released each quarter, from the beginning of 2016 through to the end of 2018.
Time Period |
Data |
Proportion of yearly mean |
March 2016 | $43$43 | $0.77$0.77 |
June 2016 | $45$45 | $0.81$0.81 |
September 2016 | $54$54 | $0.97$0.97 |
December 2016 | $81$81 | $1.45$1.45 |
March 2017 | $51$51 | $0.82$0.82 |
June 2017 | $50$50 | $0.8$0.8 |
September 2017 | $60$60 | $0.96$0.96 |
December 2017 | $89$89 | $1.42$1.42 |
March 2018 | $57$57 | $0.81$0.81 |
June 2018 | $52$52 | $0.74$0.74 |
September 2018 | $69$69 | $0.98$0.98 |
December 2018 | $103$103 | $1.47$1.47 |
Calculate the seasonal component for the first quarter, correct to two decimal places.
Deseasonalise the data for March 2018. Give your answer to two decimal places.
Calculate the seasonal component for the third quarter. Give your answer to two decimal places.
Calculate the seasonal component for the fourth quarter. Give your answer to two decimal places.
Seasonally adjust the data for December 2017. Give your answer to two decimal places.
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 2016 to the end of 2019.
Time Period |
Growth (in cm) |
Proportion of yearly mean |
April 2016 | $95.6$95.6 | $0.99$0.99 |
August 2016 | $106.7$106.7 | $a$a |
December 2016 | $87.8$87.8 | $0.91$0.91 |
April 2017 | $c$c | $0.99$0.99 |
August 2017 | $101.2$101.2 | $1.1$1.1 |
December 2017 | $84.1$84.1 | $0.91$0.91 |
April 2018 | $86.3$86.3 | $1.01$1.01 |
August 2018 | $93.6$93.6 | $1.09$1.09 |
December 2018 | $77.3$77.3 | $0.9$0.9 |
April 2019 | $76.1$76.1 | $0.99$0.99 |
August 2019 | $83.4$83.4 | $b$b |
December 2019 | $71.8$71.8 | $0.93$0.93 |
Calculate the value of $a$a in the table. Give your answer to two decimal places.
Calculate the value of $b$b in the table. Give your answer to two decimal places.
If the mean for 2017 is $92.2$92.2, calculate the value of $c$c.
Write each line of working as an equation, and give your answer to two decimal places.
Calculate the seasonal component for April. Give your answer to two decimal places.
Seasonally adjust the data for April 2019. Give your answer to two decimal places.
A spreadsheet is a powerful tool for dealing with numbers and formulae. Although many calculators have spreadsheet applications, the screen is very small so it is more practical to use an application on your computer.
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:
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.
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.
Which of the following formulae could be entered into cell $M7$M7 to calculate the cycle mean for 2017?
=AVERAGE(L7:L9)
=AVG(L7:L9)
=($L$6+$L$7+$L$8)/3
(L7+L8+L9)/3
=($L$7+$L$8+$L$9)/3
=3/(L7+L8+L9)
Which of the following formulae could be entered into cell $N13$N13 to calculate the Percentage of Cycle mean for January 2019?
100*(L13/M13)
=L13/M13*100
=L13/$M$13*100
The following formula is entered into cell $K20$K20 to calculate the seasonal index for May
$=(O5+O8+O11+O14)/4$=(O5+O8+O11+O14)/4 . Something is wrong with the formula. Write the correct formula.
$\editable{}$
The following formula is entered into cell $O6$O6 to deseasonalise the data for September
2016: $=L6/K20*100$=L6/K20*100. Something is wrong with the formula. Write the correct formula.
$\editable{}$
One method to check that the calculation of the seasonal indices is correct is to make sure the mean is equal to $100%$100%. What formula could be entered into cell $N18$N18 to check this?
=AVG(J20:L20)
=AVERAGE(J20:L20)
(J20+K20+L20)/3
=(J20+K20+L20)/3
Another method to check that the calculation of the seasonal indices is correct is to check that the sum of the indices is $300$300. What formula could be entered into cell $N18$N18 to check this?
$\editable{}$