State whether the raw data is higher or lower than average for the following seasonal indexes:
The seasonal index is greater than 1.0.
The seasonal index is less than 1.0.
The seasonal index is greater than 100\%
The seasonal index is less than 100\%.
Explain how to deseasonalise raw data, using the seasonal index.
Explain how to convert a deseasonalised figure to the raw data figure, using the seasonal index.
On a time series graph there are 'peaks' and 'troughs'. Explain the meaning of these terms in relation to the seasonal index.
Explain the meaning of the following seasonal indices in context:
The seasonal index for heater sales in winter is 1.30.
The seasonal index for Wednesday's earnings is 0.81.
The seasonal index for the sales of cold drinks in a shop in January is 1.6. To correct the January sales of cold drinks for seasonality, what calculation should be applied?
A petrol store owner records the number of cars visiting the station each day for a number of weeks. The seasonal indices for each day are displayed in the table below:
Which day of the week is the most popular day to buy petrol?
Which day of the week is the least popular day to buy petrol?
In a certain week, the number of cars on Friday is 152. Deseasonalise this number. Round your answer to nearest whole number.
| Day | Seasonal indices |
|---|---|
| \text{Monday} | 0.66 |
| \text{Tuesday} | 0.67 |
| \text{Wednesday} | 0.71 |
| \text{Thursday} | 0.63 |
| \text{Friday} | 2.46 |
| \text{Saturday} | 0.87 |
In a certain week, the number of cars on Monday is 65. Deseasonalise this number. Round your answer to nearest whole number.
Describe the purpose of deaseasonalising the data.
The number of customers for an online food delivery service is recorded for periods of 4 months. The seasonal indices calculated as a percentage are displayed in the given table:
| Months | \text{Jan - Apr} | \text{May - Aug} | \text{Sept - Dec} |
|---|---|---|---|
| Seasonal index | 38.36 | 173.11 | 88.54 |
Which is the least popular period to order food online?
Which is the most popular period to order food online?
In a certain year the number of customers from September to December was 1024. Deseasonalise this number. Round your answer to nearest whole number.
The deseasonalised figure from May to August in a certain year was 1432. What was the raw data for this period in that year? Round your answer to nearest whole number.
From the beginning of 2016, the number of new houses built in the suburb of Woodford was recorded and figures are released every four months. The following table contains the data from the beginning of 2016 to the end of 2019:
For 2016, 2017 and 2018, calculate the mean number of houses built in each time period to two decimal places:
2016
2017
2018
Hence, calculate the value of the following to two decimal places:
x
y
z
| Time period | Houses built | Percentage of yearly mean |
|---|---|---|
| \text{April }2016 | 6 | 60\% |
| \text{August }2016 | 21 | x |
| \text{December }2016 | 3 | 30\% |
| \text{April }2017 | 11 | y |
| \text{August }2017 | 24 | 180\% |
| \text{December }2017 | 5 | 37.5\% |
| \text{April }2018 | 15 | 84.91\% |
| \text{August }2018 | 25 | 141.51\% |
| \text{December }2018 | 13 | z |
| \text{April }2019 | 15 | 77.59\% |
| \text{August }2019 | 28 | 144.83\% |
| \text{December }2019 | 15 | 77.59\% |
The local police station records the number of speeding fines issued each quarter. The following table has the data for each quarter from 2016 to 2018:
For 2016, 2017 and 2018, calculate the mean number of speeding tickets issued in each time period to two decimal places:
2016
2017
2018
Hence, calculate the value of the following to two decimal places:
x
y
z
| Time period | Data | Percentage of yearly mean |
|---|---|---|
| \text{March }2016 | 105 | 106.06\% |
| \text{June }2016 | 91 | x |
| \text{September }2016 | 101 | 102.02\% |
| \text{December }2016 | 99 | 100\% |
| \text{March }2017 | 101 | y |
| \text{June }2017 | 83 | 89.01\% |
| \text{September }2017 | 96 | 102.95\% |
| \text{December }2017 | 93 | 99.73\% |
| \text{March }2018 | 99 | 108.2\% |
| \text{June }2018 | 82 | 89.62\% |
| \text{September }2018 | 94 | 102.73\% |
| \text{December }2018 | 91 | z |
A local grocery store records the number of lemons sold in the first quarters of 2016, 2017, 2018 and 2019. The table shows the data for each first quarter up to 2019:
Calculate the mean number of lemons sold in each year's first quarter to two decimal places:
2016
2017
2018
2019
Hence, calculate the value of the following to two decimal places:
x
y
z
Calculate the seasonal index of each month to four decimal places:
January
February
March
| Time period | Data | Percentage of quarterly mean |
|---|---|---|
| \text{January 2016} | 20 | 71.43\% |
| \text{February 2016} | 28 | x |
| \text{March 2016} | 36 | 128.57\% |
| \text{January 2017} | 12 | 50.7\% |
| \text{February 2017} | 24 | y |
| \text{March 2017} | 35 | 147.89\% |
| \text{January 2018} | 42 | 138.46\% |
| \text{February 2018} | 17 | 56.04\% |
| \text{March 2018} | 32 | 105.49\% |
| \text{January 2019} | 39 | 105.41\% |
| \text{February 2019} | 51 | 137.84\% |
| \text{March 2019} | 21 | z |
A local cafe stays open for 5 days a week, and records how many cappuccinos they sell on Friday, Saturday and Sunday for four weeks, as shown in the table:
Calculate the mean number of cappuccinos sold during each week to two decimal places:
Week 1
Week 2
Week 3
Week 4
Hence, calculate the values of the following to two decimal places:
x
y
z
Calculate the seasonal index for each day to four decimal places:
Friday
Saturday
Sunday
What does the seasonal index for Friday tell us about coffee sales on Friday?
| Week | Day | Data | Percentage of weekly mean |
|---|---|---|---|
| \text{Week 1} | \text{Friday} | 25 | 107.14\% |
| \text{Saturday} | 24 | x | |
| \text{Sunday} | 21 | 90\% | |
| \text{Week 2} | \text{Friday} | 12 | 37.89\% |
| \text{Saturday} | 36 | y | |
| \text{Sunday} | 47 | 148.42\% | |
| \text{Week 3} | \text{Friday} | 16 | 58.54\% |
| \text{Saturday} | 12 | 43.9\% | |
| \text{Sunday} | 54 | 197.56\% | |
| \text{Week 4} | \text{Friday} | 12 | 39.56\% |
| \text{Saturday} | 23 | 75.82\% | |
| \text{Sunday} | 56 | z |
The table shows the number of new gaming apps released each quarter, from the beginning of 2016 through to the end of 2018:
Calculate the seasonal component for the following quarters to two decimal places:
First quarter
Third quarter
Fourth quarter
Deseasonalise the data for March 2018.
Seasonally adjust the data for December 2017.
| Time period | Data | Proportion of yearly mean |
|---|---|---|
| \text{March }2016 | 43 | 0.77 |
| \text{June }2016 | 45 | 0.81 |
| \text{September }2016 | 54 | 0.97 |
| \text{December }2016 | 81 | 1.45 |
| \text{March }2017 | 51 | 0.82 |
| \text{June }2017 | 50 | 0.8 |
| \text{September }2017 | 60 | 0.96 |
| \text{December }2017 | 89 | 1.42 |
| \text{March }2018 | 57 | 0.81 |
| \text{June }2018 | 52 | 0.74 |
| \text{September }2018 | 69 | 0.98 |
| \text{December }2018 | 103 | 1.47 |
Consider the data provided for each of the following time periods from 2016 to 2019:
Calculate the seasonal index for the April season to two decimal places.
Hence, deseasonalise the data for April 2018.
Calculate the seasonal index for the August season.
Seasonally adjust the data for August 2016.
Calculate the seasonal index for the December season.
Seasonally adjust the data for December 2017.
| Time period | Data | Percentage of yearly mean |
|---|---|---|
| \text{April }2016 | 6 | 60\% |
| \text{August }2016 | 21 | 210\% |
| \text{December }2016 | 3 | 30\% |
| \text{April }2017 | 11 | 83\% |
| \text{August }2017 | 24 | 180\% |
| \text{December }2017 | 5 | 38\% |
| \text{April }2018 | 15 | 85\% |
| \text{August }2018 | 25 | 142\% |
| \text{December }2018 | 13 | 74\% |
| \text{April }2019 | 15 | 78\% |
| \text{August }2019 | 28 | 145\% |
| \text{December }2019 | 15 | 78\% |
Every four months Neil records the growth of his bean plant (starting with a new plant every year). The data provided in the table is from the beginning of 2016 to the end of 2019:
Round all answers to two decimal places.
Calculate the value of:
a
b
If the mean for 2017 is 92.2, calculate the value of c.
Calculate the seasonal component for April.
Seasonally adjust the data for April 2019.
| Time period | Growth (in cm) | Proportion of yearly mean |
|---|---|---|
| \text{April }2016 | 95.6 | 0.99 |
| \text{August }2016 | 106.7 | a |
| \text{December }2016 | 87.8 | 0.91 |
| \text{April }2017 | c | 0.99 |
| \text{August }2017 | 101.2 | 1.1 |
| \text{December }2017 | 84.1 | 0.91 |
| \text{April }2018 | 86.3 | 1.01 |
| \text{August }2018 | 93.6 | 1.09 |
| \text{December }2018 | 77.3 | 0.9 |
| \text{April }2019 | 76.1 | 0.99 |
| \text{August }2019 | 83.4 | b |
| \text{December }2019 | 71.8 | 0.93 |
Consider the data provided for each of the following time periods from 2016 to 2018:
Round all answers to two decimal places.
Calculate the value of:
a
b
If the mean for 2018 is 86.075, calculate the value of c.
Calculate the seasonal component for the third quarter.
Deseasonalise the data for September 2017.
| Time period | Data | Proportion of yearly mean |
|---|---|---|
| \text{March }2016 | 108.2 | 1.12 |
| \text{June }2016 | 80.9 | 0.84 |
| \text{September }2016 | 102.4 | a |
| \text{December }2016 | 94.6 | 0.98 |
| \text{March }2017 | 95.3 | 1.1 |
| \text{June }2017 | 68.7 | b |
| \text{September }2017 | 93.5 | 1.08 |
| \text{December }2017 | 88.1 | 1.02 |
| \text{March }2018 | 91.3 | 1.06 |
| \text{June }2018 | c | 0.74 |
| \text{September }2018 | 87.5 | 1.02 |
| \text{December }2018 | 81.5 | 0.95 |
The table below shows the number of complaints for neighbourhood disturbance received by the Springfield Council during weekdays for 3 weeks:
Round all answers to two decimal places.
Calculate the value of:
a
b
If the mean for Week 2 is 1302.20, calculate the value of c.
Calculate the seasonal index for Wednesday.
Seasonally adjust the data for Thursday in Week 2.
| Day | Complaints | Proportion of weekly mean |
|---|---|---|
| \text{Week One} | ||
| \text{Monday} | 1656 | 1.47 |
| \text{Tuesday} | 254 | a |
| \text{Wednesday} | 1705 | 1.51 |
| \text{Thursday} | 878 | 0.78 |
| \text{Friday} | 1155 | 1.02 |
| \text{Week Two} | ||
| \text{Monday} | 1979 | 1.52 |
| \text{Tuesday} | c | 0.54 |
| \text{Wednesday} | 1774 | 1.36 |
| \text{Thursday} | 794 | 0.61 |
| \text{Friday} | 1259 | 0.97 |
| \text{Week Three} | ||
| \text{Monday} | 1433 | 0.9 |
| \text{Tuesday} | 917 | 0.57 |
| \text{Wednesday} | 1609 | 1.01 |
| \text{Thursday} | 1413 | b |
| \text{Friday} | 2609 | 1.63 |
To provide information to traffic engineers, the number of vehicles using an intersection each day between 6 am and 9:30 am was recorded over three weeks. The data is tabulated below:
| Week | Day | Vehicles | Percentage of weekly mean |
|---|---|---|---|
| 1 | \text{Monday} | 836 | 77\% |
| \text{Tuesday} | 1898 | 175\% | |
| \text{Wednesday} | 1648 | a | |
| \text{Thursday} | 1339 | 124\% | |
| \text{Friday} | 640 | 59\% | |
| \text{Saturday} | 980 | 90\% | |
| \text{Sunday} | 241 | 22\% | |
| 2 | \text{Monday} | 1321 | 94\% |
| \text{Tuesday} | 1989 | 142\% | |
| \text{Wednesday} | b | 129\% | |
| \text{Thursday} | 2064 | 147\% | |
| \text{Friday} | 648 | 46\% | |
| \text{Saturday} | 1301 | 93\% | |
| \text{Sunday} | 692 | 49\% | |
| 3 | \text{Monday} | 1503 | 85\% |
| \text{Tuesday} | 2635 | 148\% | |
| \text{Wednesday} | 2532 | 143\% | |
| \text{Thursday} | 2292 | c | |
| \text{Friday} | 1136 | 64\% | |
| \text{Saturday} | 1738 | 98\% | |
| \text{Sunday} | 586 | 33\% |
Calculate the value of the following to the nearest percentage:
a
c
If the mean for Week 2 is 1402.57, calculate the value of b. Round your answer to two decimal places.
Calculate the seasonal index for Monday. Round your answer to two decimal places.
The following table displays the quarterly newspaper sales (in thousands of dollars) of a corner store. Also shown are the seasonal indices for the four quarters:
| Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | |
|---|---|---|---|---|
| Sales | 1058 | 1852 | 1650 | |
| Seasonal Index | 0.9 | 0.8 | 1.4 |
What quantity is the sum of the seasonal indices is equal to?
Hence, find the seasonal index for Quarter 4.
Which quarter is peak season for newspaper sales? Explain your answer.
Calculate the deseasonalised sales figure for Quarter 2.
Calculate the deseasonalised sales figure for Quarter 3.
The deseasonalised sales figure for Quarter 1 is 1238. Calculate the actual sales figure for Quarter 1.
Calculate the seasonal index for Winter for the following data:
| Summer | Autumn | Winter | Spring | |
|---|---|---|---|---|
| Seasonal Index | 1.20 | 0.92 | 1.48 |
A pizza shop that is open seven days a week has the following seasonal indices:
| Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday | |
|---|---|---|---|---|---|---|---|
| Seasonal Index | 0.4 | 0.3 | 0.5 | 0.8 | 1.9 | 1.2 |
The index for Friday has not been recorded. Calculate the missing index.
The number of waiters employed by a restaurant chain in each quarter of one year, along with the quarterly seasonal indices are displayed in the following table:
| Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | |
|---|---|---|---|---|
| Number of waiters | 197 | 144 | 83 | 170 |
| Seasonal Index | 1.31 | 0.57 | 1.08 |
Calculate the seasonal index for the second quarter.
Explain the meaning of seasonal index for Quarter 1 this context.
Deseasonalise the data for each quarter. Round your answers to the nearest whole number.
A company sells central heating systems. The following table shows the quarterly seasonal indices for sales in the last three quarters of a year. One year represents one cycle.
| Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | |
|---|---|---|---|---|
| Seasonal Index | 1.30 | 1.45 | 0.58 |
Calculate the seasonal index for Quarter 1.
The deseasonalised sales for Quarter 1 in 2018 were \$2\,500\,000 million. Calculate the actual sales in dollars.
The following table contains the seasonal indices for the monthly sales of spring water in a particular supermarket. The full year represents one cycle:
Calculate the seasonal index for February.
If the actual sales figure for June was \$108\,000, calculate the deseasonalised figure.
If the deseasonalised figure for December is \$110\,000, calculate the actual sales for December.
| Month | Seasonal index |
|---|---|
| \text{January} | 1.00 |
| \text{February} | |
| \text{March} | 1.00 |
| \text{April} | 1.00 |
| \text{May} | 0.95 |
| \text{June} | 0.80 |
| \text{July} | 0.75 |
| \text{August} | 0.95 |
| \text{September} | 0.95 |
| \text{October} | 1.10 |
| \text{November} | 1.10 |
| \text{December} | 1.15 |
The seasonal indices for the first 11 months of the year for sales in a sporting equipment store are shown in the table. The full year represents one cycle:
Calculate the seasonal index for December.
In May, the store sold \$210\,000 worth of sporting equipment. Calculate the deseasonalised value of the May sales.
In February, the deseasonalised value of sales was \$200\,000. Calculate the actual sales figures for February.
| Month | Seasonal Index |
|---|---|
| \text{Jan} | 1.23 |
| \text{Feb} | 0.99 |
| \text{Mar} | 1.12 |
| \text{Apr} | 1.11 |
| \text{May} | 0.89 |
| \text{Jun} | 0.96 |
| \text{Jul} | 0.86 |
| \text{Aug} | 0.73 |
| \text{Sep} | 0.77 |
| \text{Oct} | 0.96 |
| \text{Nov} | 1.10 |
| \text{Dec} |
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:
Write the formula, that when entered into cell \text{M7} calculates the cycle mean for 2017.
Write the spreadsheet formula, that when entered into cell \text{N13} calculates the Percentage of Cycle mean for January 2019.
The following formula is entered into cell \text{K20} to calculate the seasonal index for May:
\text{=(O5+O8+O11+O14)/4} This formula is incorrect. Write the correct formula.
An incorrect formula: \text{=L6/K20*100} is entered into cell \text{O6} to deseasonalise the data for September 2016. Write the correct formula.
One method to check that the calculation of the seasonal indices is correct is to make sure the mean is equal to 100\%. Write the formula that could be entered into cell \text{N18} to check this.
Another method to check that the calculation of the seasonal indices is correct is to check that the sum of the indices is 300. Write the formula that could be entered into cell \text{N18} to check this.
A bus company records the number of passengers using public transport on a particular route each day over a three week period. The spreadsheet with the recorded data is shown below. Note the table in columns I and J has been displayed on top of column F and G in the image:
Which cell should contain the following information:
The cycle mean for Week 1.
The seasonal index for Wednesday.
The deseasonalised score for Monday of Week 3.
John enters the following incorrect formula into cell \text{E11} to calculate the cycle mean for Week 2: \text{=AVERAGE(D11:D18)}. Write the correct formula.
What formula must be entered into cell \text{F18} to calculate the percentage of cycle mean for Monday of Week 3?
What formula must be entered into cell \text{J2} to calculate the seasonal index for Monday?
What formula must be entered into cell \text{G18} to deseasonalise the data for Monday Week 3?
One method to check that the calculation of the seasonal indices is correct is to check that the sum of the indices is 700. What formula could be used to do this?
Another 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 used to do this?
A taxi company records the weekly number of taxis ordered to go to the airport over a four month period. The data is displayed in the spreadsheet below:
Week 2 in April corresponds to which time value (t)?
Which week does the time value t = 7 correspond to?
What formula must be entered into cell \text{G11} to calculate the cycle mean for March?
What formula must be entered to calculate the percentage of cycle mean for time period t = 10?
A mistake has been made and the raw data figure for t = 5 should be 822. Will the cycle mean for this cycle now be higher or lower?
The seasonal indices are calculated and shown in the table below:
Explain how to check that a mistake has not been made.
Which week represents a 'peak' season in each cycle?
Are the taxi orders in Week 1 of each cycle higher or lower than average?
The raw data taxi orders for t = 18 are recorded as 1610 orders. Deseasonalise this figure, rounding your answer to two decimal places.
The deseasonalised figure for t = 20 is 1321 . Calculate the number of taxis ordered in this time period.