The projected quarterly sale figures for a manufacturer of solar panels for three years in millions of Australian dollars are shown in the table below:
What is the seasonal index for Q1?
What is the seasonal index for Q4?
What would be the most appropriate moving average to smooth this data?
Describe the underlying trend in this data.
| Q1 | Q2 | Q3 | Q4 | |
|---|---|---|---|---|
| 2019 | 0.73 | 0.85 | 1.07 | 1.36 |
| 2020 | 0.68 | 0.84 | 1.25 | 1.28 |
| 2021 | 0.84 | 0.92 | 1.41 | 1.53 |
Data on dam levels is collected tri-annually in January, May and September over a period of 4 years. The seasonal indices are calculated in order to deseasonalise the data.
Find the seasonal index for January.
The dam level in September of the first year was 72.3\% full. Deseasonalise this score.
The deseasonalised score for May of the second year was 70.1\%. Calculate the actual dam level for May.
| Seasonal index | |
|---|---|
| January | |
| May | 96.23\% |
| September | 117.51\% |
Data on the number of traffic accidents at a set of traffic lights is collected weekly over a period of 6 months. The seasonal indices are calculated in order to deseasonalise the data. Assume there are 4 seasons in a cycle.
Find the missing value in the table of seasonal indices below:
| Week 1 | Week 2 | Week 3 | Week 4 | |
|---|---|---|---|---|
| Seasonal index | 1.34 | 0.87 | 1.05 |
The number of traffic accidents in Week 4 of the first month was 34. Deseasonalise this score, rounding your answer to the nearest whole number.
The deseasonalised score for Week 1 of the second month was 32. Calculate the actual number of traffic accidents in this week. Round your answer to the nearest whole number.
Data on the number of cartons of chocolate milk sold at the school canteen is collected every day over a 6 week time period. The seasonal indices for each day are calculated in order to deseasonalise the data.
Find the missing value in the table of seasonal indices below:
| Monday | Tuesday | Wednesday | Thursday | Friday | |
|---|---|---|---|---|---|
| Seasonal index | 102.21\% | 97.54\% | 114.65\% | 88.98\% |
Which day is the most popular day for buying chocolate milk?
Will deseasonalised data for Wednesday be higher or lower than the original raw data for Wednesday?
The number of chocolate milk cartons sold on Thursday of Week 2 was 57. Calculate the deseasonalised score for Thursday of Week 2, rounding your answer to the nearest whole number.
Data on the number of cloudy days is collected quarterly over a number of years. The seasonal indices are calculated in order to deseasonalise the data.
Find the missing value in the table of seasonal indices below:
| Spring | Summer | Autumn | Winter | |
|---|---|---|---|---|
| Seasonal index | 0.8 | 1.3 | 1.4 |
Will deseasonalised data for Spring be higher or lower than the original raw data for Spring?
Will deseasonalised data for Autumn be higher or lower than the original raw data for Autumn?
The number of cloudy days in Winter 2018 was 65. Calculate the deseasonalised score for Winter 2018. Round your answer to the nearest whole number.
Data on the number of customers at a local cafe was collected daily over a number of weeks. The seasonal indices are calculated in order to deseasonalise the data.
Find the missing value in the table of seasonal indices below:
| Mon | Tues | Wed | Thurs | Fri | |
|---|---|---|---|---|---|
| Seasonal index | 67\% | 78\% | 82\% | 139\% |
Which days of the week have raw scores that are higher than the average in a cycle?
Which days of the week have raw scores that are lower than the average in a cycle?
If the number of customers for Wednesday of Week 3 was 84, calculate the deseasonalised score. Round your answer to the nearest whole number.
Data on the number of people cycling to work is collected monthly over a number of years. The seasonal indices are calculated for each month.
Find the missing value in the given table of seasonal indices.
Which months have raw scores that are higher than the average in a cycle?
Which months have raw scores that are lower than the average in a cycle?
The number of people cycling to work in March of the second year is 125. Calculate the deseasonalised score, rounding your answer to the nearest whole number.
Which month would have the strongest 'peak' in a time series graph of the data?
| Seasonal index | |
|---|---|
| Jan | 1.1 |
| Feb | 1.0 |
| Mar | 0.9 |
| Apr | 0.8 |
| May | 0.6 |
| Jun | 0.8 |
| Jul | 0.9 |
| Aug | |
| Sep | 1.1 |
| Oct | 1.2 |
| Nov | 1.5 |
| Dec | 1.2 |
The table below shows some time series data where t represents time:
| t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| x | 25.4 | 27.2 | 24.1 | 25.6 | 28.3 | 25.1 | y | 29.1 |
Calculate the 3 point moving average at t = 3. Round your answer to one decimal place.
If the 4 point centred moving average at t = 5 is 26.0, find the value of y.
The table below shows some time series data where t represents time:
| t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| x | 15.2 | 17.3 | 14.6 | 12.6 | 18.3 | 15.8 | y | 18.2 |
Calculate the 4 point centred moving average at t = 3.
If the 5 point moving average at t = 5 is 25.44, find the value of y.
The table below shows some time series data where t represents time:
| t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| x | y | 98 | 90 | 85 | 107 | 100 | 91 | 86 |
Calculate the 6 point centred moving average at t = 5, correct to one decimal place.
If the 3 point moving average at t = 2 is 96.67, find the value of y, correct to the nearest whole number.
A bank collects data on the number of customers being served at the branch on a weekly basis and uses a spreadsheet to calculate a number of moving averages:
| Time | Data | 3MA | 4CMA | 5MA | 8CMA |
|---|---|---|---|---|---|
| 1 | 110 | ||||
| 2 | 135 | 117.33 | |||
| 3 | 107 | 102.67 | 103.875 | 104.6 | |
| 4 | 66 | 92.67 | 103.125 | 109.4 | |
| 5 | 105 | 101.67 | 102.125 | 102.4 | 101.38 |
| 6 | 134 | 113 | 100.5 | 93 | D |
| 7 | 100 | 98 | 98.875 | 99.4 | 98.19 |
| 8 | 60 | 86 | B | 102.8 | 96.69 |
| 9 | 98 | 93.33 | 94.25 | 94.8 | 95.19 |
| 10 | 122 | 104.67 | 92.875 | C | 92.19 |
| 11 | 94 | 90.33 | 91.5 | 92.2 | 89.31 |
| 12 | 55 | 80.33 | 87.875 | 92.4 | 88 |
| 13 | 92 | A | 84.375 | 85.8 | 86.06 |
| 14 | 99 | 93.33 | 83.125 | 77 | 82.44 |
| 15 | 89 | 79.33 | 80.625 | 81.4 | 76.44 |
| 16 | 50 | 72 | 77 | 80 | |
| 17 | 77 | 70.67 | 68.5 | 67.2 | |
| 18 | 85 | 65.67 | |||
| 19 | 35 |
Calculate the values of the following to one decimal place:
State the moving average(s) that show a strictly decreasing trend.
Explain why 4CMA is the most suitable for this data.
A gym owner records attendance numbers in the table below:
| Week | Attendance numbers | Cycle mean | Percentage of cycle mean | Deseasonalised data | |
|---|---|---|---|---|---|
| Jan | 1 | 760 | 1064.5 | 71.40\% | 1093.68 |
| 2 | 1123 | 105.50\% | 1012.73 | ||
| 3 | A | 75.56\% | 1052.22 | ||
| 4 | 1560 | 146.55\% | 1097.31 | ||
| Feb | 1 | 830 | 1172.5 | 70.79\% | 1194.41 |
| 2 | 1267 | 108.06\% | D | ||
| 3 | 903 | 77.01\% | 1165.83 | ||
| 4 | 1690 | 144.14\% | 1188.75 | ||
| Mar | 1 | 835 | 1222 | C | 1201.61 |
| 2 | 1387 | 113.50\% | 1250.80 | ||
| 3 | 954 | 78.07\% | 1231.68 | ||
| 4 | 1712 | 140.10\% | 1204.23 | ||
| Apr | 1 | 880 | B | 67.45\% | 1266.37 |
| 2 | 1520 | 116.50\% | 1370.74 | ||
| 3 | 1020 | 78.18\% | 1316.89 | ||
| 4 | 1799 | 137.88\% | 1265.42 |
The seasonal indices are as follows:
| Week 1 | Week 2 | Week 3 | Week 4 | |
|---|---|---|---|---|
| Seasonal index | 64.49\% | 110.89\% | E | 142.168\% |
Calculate the values of the following, giving answers to two decimal places if necessary:
A
B
C
D
Calculate the value of E as a percentage, correct to three decimal places.
Soup kitchen volunteers cook meals for the homeless on Friday, Saturday and Sunday nights. Data is recorded for four weeks and entered into the table below:
| Day | Meals cooked | Weekly mean | Perentage of weekly mean | Deseasonalised data | |
|---|---|---|---|---|---|
| Week 1 | \text{Fri} | 225 | 192.33 | 116.98\% | 192 |
| \text{Sat} | 202 | 105.03\% | 193 | ||
| \text{Sun} | 150 | C | 191 | ||
| Week 2 | \text{Fri} | 243 | B | 116.45\% | 208 |
| \text{Sat} | 218 | 104.47\% | 209 | ||
| \text{Sun} | 165 | 79.07 | 210 | ||
| Week 3 | \text{Fri} | 251 | 214.67 | 116.93\% | 214 |
| \text{Sat} | 223 | 103.88\% | 213 | ||
| \text{Sun} | 170 | 79.19\% | 217 | ||
| Week 4 | \text{Fri} | 272 | 230.67 | 117.92\% | D |
| \text{Sat} | A | 108.48\% | 231 | ||
| \text{Sun} | 179 | 77.60\% | 228 |
Seasonal indices are as follows:
| Fri | Sat | Sun | |
|---|---|---|---|
| Seasonal index | 1.171 | E | 0.785 |
Calculate the values of the following, giving answers to two decimal places if necessary:
A
B
C
D
Calculate the value of E to three decimal places.
Describe the underlying trend identified by the deseasonalised data.
The quarterly power bills of a suburban household are recorded in the table below:
| \text{Month} | \text{Time }\left(t\right) | \text{Bill } \left(\$\right) | \text{Yearly} \\ \text{mean} | \text{Percentage} \\ \text{of yearly} \\ \text{mean} | \text{Deseasonalised} \\ \text{data} | |
|---|---|---|---|---|---|---|
| 2016 | \text{Jan} | 1 | 456.24 | 348.37 | 130.97\% | 341.37 |
| \text{Apr} | 2 | A | 95.40\% | 347.23 | ||
| \text{Jul} | 3 | 300.43 | 86.24\% | 354.12 | ||
| \text{Oct} | 4 | 304.45 | 87.39\% | 354.85 | ||
| 2017 | \text{Jan} | 5 | 477.05 | B | 132.85\% | 356.94 |
| \text{Apr} | 6 | 343.77 | 95.73\% | 359.16 | ||
| \text{Jul} | 7 | 305.98 | 85.21\% | 360.66 | ||
| \text{Oct} | 8 | 309.54 | 86.20\% | 360.78 | ||
| 2018 | \text{Jan} | 9 | 494.22 | 367.47 | 134.49\% | 369.79 |
| \text{Apr} | 10 | 352.56 | C | 368.34 | ||
| \text{Jul} | 11 | 310.65 | 84.54\% | 366.17 | ||
| \text{Oct} | 12 | 312.43 | 85.02\% | 364.15 | ||
| 2019 | \text{Jan} | 13 | 510.45 | 374.56 | 136.28\% | 381.94 |
| \text{Apr} | 14 | 358.76 | 95.78\% | D | ||
| \text{Jul} | 15 | 312.25 | 83.37\% | 368.05 | ||
| \text{Oct} | 16 | 316.76 | 84.57\% | 369.20 |
Seasonal indices are displayed in the table below:
| Jan | Apr | Jul | Oct | |
|---|---|---|---|---|
| Seasonal index | E | 95.716\% | 84.839\% | 85.797\% |
Calculate the values of the following to two decimal places:
A
B
C
D
Calculate the value of E as a percentage, correct to three decimal places.
Is the underlying trend identified by the deseasonalised data increasing, decreasing or remaining stable?
A cafe owner in a suburban supermarket recorded the number of customers they had for each quarter over four years in the table below:
| Month | No. of Customers | Yearly mean | Percentage of yearly mean | Deseasonalised number | |
|---|---|---|---|---|---|
| 2016 | \text{Jan} | 1687 | 1293.75 | 130.396 | 1284 |
| \text{Apr} | 1218 | 94.145 | 1256 | ||
| \text{July} | 886 | 68.483 | D | ||
| \text{Oct} | 1384 | 106.976 | 1318 | ||
| 2017 | \text{Jan} | 1789 | A | 130.825 | 1361 |
| \text{Apr} | 1327 | 97.038 | 1369 | ||
| \text{July} | 905 | 66.179 | 1358 | ||
| \text{Oct} | 1449 | 105.960 | 1380 | ||
| 2018 | \text{Jan} | 2325 | 1748.5 | 132.971 | 1769 |
| \text{Apr} | 1745 | 99.800 | 1800 | ||
| \text{July} | B | 63.597 | 1669 | ||
| \text{Oct} | 1812 | 103.632 | 1726 | ||
| 2019 | \text{Jan} | 2565 | 1951 | 131.471 | 1952 |
| \text{Apr} | 1890 | C | 1949 | ||
| \text{July} | 1333 | 68.324 | 2000 | ||
| \text{Oct} | 2016 | 103.332 | 1920 |
The following table also shows the seasonal index for the quarters:
| Jan | Apr | July | Oct | |
|---|---|---|---|---|
| Seasonal index | 131.42 | 96.964 | E | 104.975 |
A
B
C
D
E
Using deseasonalised number column, state whether the trend is increasing, decreasing, or remaining stable.
The accountant recommends that the cafe owner employ more staff in the January quarter. Do you agree? Explain your answer.
A car yard records the sales of a particular brand of car and records the data in the table below:
| Month | Sales | Yearly mean | Percentage of yearly mean | Deseasonalised data | |
|---|---|---|---|---|---|
| 2016 | \text{Jan} | 10 | 30.33 | 32.97 | 26 |
| \text{May} | 55 | 181.32 | 32 | ||
| \text{Sept} | X | 85.71 | 29 | ||
| 2017 | \text{Jan} | 14 | Y | 38.89 | 36 |
| \text{May} | 63 | 175.00 | 36 | ||
| \text{Sept} | 31 | 86.11 | 35 | ||
| 2018 | \text{Jan} | 16 | 39.34 | 32.97 | 42 |
| \text{May} | 68 | Z | 39 | ||
| \text{Sept} | 38 | 93.44 | 43 | ||
| 2019 | \text{Jan} | 19 | 45.00 | 42.22 | P |
| \text{May} | 76 | 168.89 | 44 | ||
| \text{Sept} | 40 | 88.89 | 45 |
The following table also shows the seasonal index for the months:
| Jan | May | Sept | |
|---|---|---|---|
| Seasonal index | 38.36 | 173.11 | K |
Calculate the value of:
X
Y
Z
P
K
Does the value of Z imply that the raw data for May 2018 is above or below the average for that year?
Without using your calculator, state the average of the percentage of yearly means for January.
The accountant recommends that a good time for staff to take holidays is January. Do you agree? Explain your answer.
A university records the number of students cycling to the library each day and records the data in the table below:
| Day | Number | Weekly mean | Percentage of weekly mean | Deseasonalised data | |
|---|---|---|---|---|---|
| Week 1 | \text{Mon} | 66 | 103.57 | 63.72 | 100 |
| \text{Tue} | 68 | 65.66 | P | ||
| \text{Wed} | 71 | 68.55 | 100 | ||
| \text{Thur} | 65 | 62.76 | 103 | ||
| \text{Fri} | L | 246.21 | 104 | ||
| \text{Sat} | 111 | 107.17 | 107 | ||
| \text{Sun} | 89 | 85.93 | 107 | ||
| Week 2 | \text{Mon} | 68 | M | 66.57 | 103 |
| \text{Tue} | 67 | 65.59 | 100 | ||
| \text{Wed} | 72 | 70.49 | 101 | ||
| \text{Thur} | 66 | 64.62 | 105 | ||
| \text{Fri} | 250 | 244.76 | 102 | ||
| \text{Sat} | 107 | 104.76 | 103 | ||
| \text{Sun} | 85 | 83.22 | 102 | ||
| Week 3 | \text{Mon} | 66 | 98.57 | 66.96 | 100 |
| \text{Tue} | 68 | N | 101 | ||
| \text{Wed} | 72 | 73.04 | 101 | ||
| \text{Thur} | 62 | 62.09 | 98 | ||
| \text{Fri} | 244 | 247.54 | 99 | ||
| \text{Sat} | 99 | 100.43 | 95 | ||
| \text{Sun} | 79 | 80.14 | 95 |
The following table also shows the seasonal index for the days:
| Mon | Tue | Wed | Thur | Fri | Sat | Sun | |
|---|---|---|---|---|---|---|---|
| Seasonal index | 0.66 | 0.67 | 0.71 | Q | 2.46 | 1.04 | 0.83 |
Calculate the value of:
L
M
N
P
Q
Does the value of the percentage of weekly mean for Wednesday in Week 1 imply that the number of students cycling that day is above or below the average for that week?
Without using your calculator, state the average of the percentage of weekly means for Sunday.
Using the deseasonalised number column, determine whether the trend decreasing, increasing or remaining stable.
A pizza delivery company records the number of daily deliveries over a period of 3 weeks and records the data in the table below:
| \text{Day} | \text{Time }\left(t\right) | \text{No. Orders} | \text{Weekly} \\ \text{mean} | \text{Percentage} \\ \text{of weekly} \\ \text{mean} | \text{Deseasonalised} \\ \text{data} | |
|---|---|---|---|---|---|---|
| Week 1 | \text{Mon} | 1 | 35 | 69.1 | 50.62\% | 44.25 |
| \text{Tues} | 2 | 37 | 53.51\% | 69.22 | ||
| \text{Wed} | 3 | 42 | 60.74\% | 71.41 | ||
| \text{Thur} | 4 | A | 75.21\% | 71.72 | ||
| \text{Fri} | 5 | 104 | 150.41\% | 72.46 | ||
| \text{Sat} | 6 | 114 | 164.88\% | 72.51 | ||
| \text{Sun} | 7 | 100 | 144.63\% | 73.86 | ||
| Week 2 | ||||||
| \text{Mon} | 8 | 40 | 75.6 | 52.93\% | 50.57 | |
| \text{Tues} | 9 | 42 | B | 78.57 | ||
| \text{Wed} | 10 | 47 | 62.19 | 79.91 | ||
| \text{Thur} | 11 | 57 | 75.43\% | 78.62 | ||
| \text{Fri} | 12 | 114 | 150.85\% | 79.43 | ||
| \text{Sat} | 13 | 124 | 164.08\% | 78.87 | ||
| \text{Sun} | 14 | 105 | 138.94\% | 77.55 | ||
| Week 3 | ||||||
| \text{Mon} | 15 | 120 | 89.7 | 133.76\% | 151.70 | |
| \text{Tues} | 16 | 46 | 51.27\% | 86.05 | ||
| \text{Wed} | 17 | 48 | 53.50\% | 81.61 | ||
| \text{Thur} | 18 | 60 | 66.88\% | 82.75 | ||
| \text{Fri} | 19 | 116 | 129.80\% | 80.82 | ||
| \text{Sat} | 20 | 128 | 142.68\% | 81.42 | ||
| \text{Sun} | 21 | 110 | 122.61\% | C |
The seasonal indices are as follows:
| Mon | Tues | Wed | Thurs | Fri | Sat | Sun |
|---|---|---|---|---|---|---|
| 79.10\% | D | 58.81\% | 72.50\% | 143.52\% | 157.21\% | 135.39\% |
Calculate the values of the following:
A
B
C
D
Construct a time series graph for the number of orders against t.
There appears to be an outlier in the seasonal trend. Which value of t contains the outlier?
On what day of the week does the outlier occur?
A shop owner collects data on the number of thefts per quarter and uses a spreadsheet to calculate a number of moving averages:
| Year | Quarter | Time | Data | 3 pt moving avg | 4 pt moving avg | 5 pt moving avg | 6 pt CMA | 8 pt CMA |
|---|---|---|---|---|---|---|---|---|
| 1 | \text{Jan} | 1 | 15 | |||||
| \text{Apr} | 2 | 18 | 19.33 | |||||
| \text{Jul} | 3 | 25 | 21.00 | 19.75 | 19.00 | |||
| \text{Oct} | 4 | 40 | A | 20.25 | 20.00 | 20.17 | ||
| 2 | \text{Jan} | 5 | 17 | 19.00 | 20.75 | 21.80 | 21.50 | 20.75 |
| \text{Apr} | 6 | 20 | 21.33 | 21.25 | 21.20 | 21.33 | 21.25 | |
| \text{Jul} | 7 | 27 | 23.00 | 21.75 | 21.00 | 21.00 | 21.75 | |
| \text{Oct} | 8 | 22 | 22.67 | B | 22.00 | 22.17 | 22.25 | |
| 3 | \text{Jan} | 9 | 19 | 21.00 | 22.75 | 23.80 | 23.50 | 22.688 |
| \text{Apr} | 10 | 22 | 23.33 | 23.25 | 23.20 | 23.25 | 23.188 | |
| \text{Jul} | 11 | 29 | 25.00 | 23.625 | 22.80 | 22.92 | 23.688 | |
| \text{Oct} | 12 | 24 | 24.33 | 24.125 | 24.00 | C | 24.063 | |
| 4 | \text{Jan} | 13 | 20 | 23.00 | 24.625 | 25.60 | 25.25 | 24.438 |
| \text{Apr} | 14 | 25 | 25.00 | 24.875 | 24.80 | 24.92 | 24.938 | |
| \text{Jul} | 15 | 30 | 26.67 | 25.25 | 24.40 | 24.58 | ||
| \text{Oct} | 16 | 25 | 25.67 | 25.75 | 25.80 | 26.08 | ||
| 5 | \text{Jan} | 17 | 22 | 24.67 | 26.625 | 27.80 | ||
| \text{Apr} | 18 | 27 | 28.00 | |||||
| \text{Jul} | 19 | D |
Calculate the value of:
A
B
C
D
State the two purposes of a moving average.
Which centred moving average clearly shows an increasing trend using the most data points?
Give two reasons as to why this centred moving average is the most suitable for this data.
Which other centred moving average also shows an increasing trend?
Graph the original data and the 4 pt moving average data.