A trend is a long-term increase or decrease in dependent variable (positive secular trend or
negative secular trend)
Seasonal variation/pattern are systematic, calendar-related movements, where similar
patterns occur in the same place each year at regular intervals, e.g. ice-cream sales
Cyclical variation is when significant peaks and troughs occur at irregular intervals, e.g.
earthquakes
Random variations or fluctuations are unsystematic, short-term fluctuations seen in all data
(no peaks and troughs but random fluctuation around a relatively stable mean)
Outliers are points extreme from the mean
Smoothing levels fluctuations and displays trends
YR. NO. THREE-POINT AVG.
clearer; smoothing using moving averages involves
1 25
taking a fixed number of data points and averaging
2 18 25 + 18 + = 22
them to give new points to produce a smoothed
time series, reducing the effects of variation 3 23 18 + 23 + = 20.67
A three-point moving average takes the average 4 21 23 + 21 + = 21
of the data point before and after, and with an odd 5 19 21 + 19 + = 20
number of points, average is on one of time values 6 20
Deseasonalising is the removal of the seasonal trend to better understand it
If given seasonalised average and seasonal index, actual sales can be found, ex. SI of 0.72 = 72% of expected sales
If all seasonal index values are added, it equals to the number of quarters
ex. deseasonalised sales are $100000; predict actual quarterly sales if SI is: summer 74%, autumn 110%, winter 120%,
spring 96%. → 100000 x 0.74 = $74000, 100000 x 1.1 = $110000, 100000 x 1.2 = $12000, 100000 x 0.96 = $96000
ex. comment on the effect the seasonal index had on the value found. → inc/dec value to underlying trend.
1. Find average for each year Yearly average
(yearly average) Q1 Q2 Q3 Q4
2. Divide each month by the 2010 5 7 9 3 Σ/4 = 6
yearly average. 2011 4 8 9 4 Σ/4 = 6.25
3. Find the average of each
2012 5 9 10 5 Σ/4 = 7.25
month (seasonal index)
4. Divide original data by its Quarterly proportion and quarterly average (seasonal index)
seasonal index. Q1 Q2 Q3 Q4
2010 5/6 = 0.83 7/6 = 1.16 9/6 = 1.5 3/6 = 0.5
2011 4/6.25 = 0.64 8/6.25 = 1.28 9/6.25 = 1.44 4/6.25 = 0.64
2012 5/7.25 = 0.69 9/7.25 = 1.24 10/7.25 = 1.38 5/7.25 = 0.69
Σ/3 = 0.72 Σ/3 = 1.23 Σ/3 = 1.44 Σ/3 = 0.61
Deseasonalised data: deseasonalised value = actual value/SI
Q1 Q2 Q3 Q4
2010 5/0.72 = 6.95 7/1.23 = 5.70 9/1.44 = 6.25 3/0.61 = 4.92
2011 4/0.72 = 5.56 8/1.23 = 6.52 9/1.44 = 6.25 4/0.61= 6.56
2012 5/0.72 = 6.98 9/1.23 = 7.33 10/1.44 = 6.94 5/0.61 = 8.20
Regression lines are used to model long term trends
ex. determine the equation of the least-squares regression line for the deseasonalised
time series data. → y = ax + b → y = 1.242t + 275.204.
Plot the least-squares regression line on the time series plot. → (choose any point, e.g. t
= 5) y = 1.242(5) + 275.204 = 281.414 ∴ 5, 281.414. → (select a value of t, not too close
to the first, and calculate y, e.g. t = 30) y= 1.242(30) + 275.204 = 312.464 ∴ 30, 312.464.
ex. the least-squares line using deseasonalised data is R = -12.071n + 1681.25. Use this
line to predict the total number of rooms occupied during Spring 2020/21. → n = 21,
R = (-12.071 x 21 + 1681.25) = 1427.8 → R x SI ∴ 1427.8 x 1.0432 = 1489.4
negative secular trend)
Seasonal variation/pattern are systematic, calendar-related movements, where similar
patterns occur in the same place each year at regular intervals, e.g. ice-cream sales
Cyclical variation is when significant peaks and troughs occur at irregular intervals, e.g.
earthquakes
Random variations or fluctuations are unsystematic, short-term fluctuations seen in all data
(no peaks and troughs but random fluctuation around a relatively stable mean)
Outliers are points extreme from the mean
Smoothing levels fluctuations and displays trends
YR. NO. THREE-POINT AVG.
clearer; smoothing using moving averages involves
1 25
taking a fixed number of data points and averaging
2 18 25 + 18 + = 22
them to give new points to produce a smoothed
time series, reducing the effects of variation 3 23 18 + 23 + = 20.67
A three-point moving average takes the average 4 21 23 + 21 + = 21
of the data point before and after, and with an odd 5 19 21 + 19 + = 20
number of points, average is on one of time values 6 20
Deseasonalising is the removal of the seasonal trend to better understand it
If given seasonalised average and seasonal index, actual sales can be found, ex. SI of 0.72 = 72% of expected sales
If all seasonal index values are added, it equals to the number of quarters
ex. deseasonalised sales are $100000; predict actual quarterly sales if SI is: summer 74%, autumn 110%, winter 120%,
spring 96%. → 100000 x 0.74 = $74000, 100000 x 1.1 = $110000, 100000 x 1.2 = $12000, 100000 x 0.96 = $96000
ex. comment on the effect the seasonal index had on the value found. → inc/dec value to underlying trend.
1. Find average for each year Yearly average
(yearly average) Q1 Q2 Q3 Q4
2. Divide each month by the 2010 5 7 9 3 Σ/4 = 6
yearly average. 2011 4 8 9 4 Σ/4 = 6.25
3. Find the average of each
2012 5 9 10 5 Σ/4 = 7.25
month (seasonal index)
4. Divide original data by its Quarterly proportion and quarterly average (seasonal index)
seasonal index. Q1 Q2 Q3 Q4
2010 5/6 = 0.83 7/6 = 1.16 9/6 = 1.5 3/6 = 0.5
2011 4/6.25 = 0.64 8/6.25 = 1.28 9/6.25 = 1.44 4/6.25 = 0.64
2012 5/7.25 = 0.69 9/7.25 = 1.24 10/7.25 = 1.38 5/7.25 = 0.69
Σ/3 = 0.72 Σ/3 = 1.23 Σ/3 = 1.44 Σ/3 = 0.61
Deseasonalised data: deseasonalised value = actual value/SI
Q1 Q2 Q3 Q4
2010 5/0.72 = 6.95 7/1.23 = 5.70 9/1.44 = 6.25 3/0.61 = 4.92
2011 4/0.72 = 5.56 8/1.23 = 6.52 9/1.44 = 6.25 4/0.61= 6.56
2012 5/0.72 = 6.98 9/1.23 = 7.33 10/1.44 = 6.94 5/0.61 = 8.20
Regression lines are used to model long term trends
ex. determine the equation of the least-squares regression line for the deseasonalised
time series data. → y = ax + b → y = 1.242t + 275.204.
Plot the least-squares regression line on the time series plot. → (choose any point, e.g. t
= 5) y = 1.242(5) + 275.204 = 281.414 ∴ 5, 281.414. → (select a value of t, not too close
to the first, and calculate y, e.g. t = 30) y= 1.242(30) + 275.204 = 312.464 ∴ 30, 312.464.
ex. the least-squares line using deseasonalised data is R = -12.071n + 1681.25. Use this
line to predict the total number of rooms occupied during Spring 2020/21. → n = 21,
R = (-12.071 x 21 + 1681.25) = 1427.8 → R x SI ∴ 1427.8 x 1.0432 = 1489.4
