Now we analyze seasonality - how data changes across months. From our observations, we know that, for some months, sales tend to be higher, whereas, for other months, sales tend to be lower. We evaluate the differences between the linear trend and the actual sales. Based on the pattern observed in these differences, we produce a model of seasonality to predict sales more accurately for each month:
| Sales for January | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 10.5 | 11.9 | 13.2 | 14.6 | 15.1 | 16.5 | 18.9 | 20 | |
| Sales on the trend line | 13.012 | 14.291 | 15.57 | 16.849 | 18.128 | 19.407 | 20.686 | 21.965 | |
| Difference | -2.512 | -2.391 | -2.37 | -2.249 | -3.028 | -2.907 | -1.786 | -1.965 | -2.401 |
| Sales for February | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 11.9 | 12.6 | 14.4 | 15.4 | 17.4 | 17.9 | 19.5 | 20.8 | |
| Sales on the trend line | 13.1185833333 | 14.3975833333 | 15.6765833333 | 16.9555833333 | 18.2345833333 | 19.5135833333 | 20.7925833333 | 22.0715833333 | |
| Difference | -1.2185833333 | -1.7975833333 | -1.2765833333 | -1.5555833333 | -0.8345833333 | -1.6135833333 | -1.2925833333 | -1.2715833333 | -1.3575833333 |
| Sales for March | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 13.4 | 13.5 | 16.1 | 16.2 | 17.2 | 19.6 | 19.8 | 22.1 | |
| Sales on the trend line | 13.2251666667 | 14.5041666667 | 15.7831666667 | 17.0621666667 | 18.3411666667 | 19.6201666667 | 20.8991666667 | 22.1781666667 | |
| Difference | 0.1748333333 | -1.0041666667 | 0.3168333333 | -0.8621666667 | -1.1411666667 | -0.0201666667 | -1.0991666667 | -0.0781666667 | -0.4641666667 |
| Sales for April | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 12.7 | 13.6 | 14.9 | 17.8 | 17.8 | 20.2 | 19.7 | 20.9 | |
| Sales on the trend line | 13.33175 | 14.61075 | 15.88975 | 17.16875 | 18.44775 | 19.72675 | 21.00575 | 22.28475 | |
| Difference | -0.63175 | -1.01075 | -0.98975 | 0.63125 | -0.64775 | 0.47325 | -1.30575 | -1.38475 | -0.60825 |
| Sales for May | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 13.9 | 14.6 | 15.7 | 17.8 | 18.6 | 19.1 | 20.8 | 21.5 | |
| Sales on the trend line | 13.4383333333 | 14.7173333333 | 15.9963333333 | 17.2753333333 | 18.5543333333 | 19.8333333333 | 21.1123333333 | 22.3913333333 | |
| Difference | 0.4616666667 | -0.1173333333 | -0.2963333333 | 0.5246666667 | 0.0456666667 | -0.7333333333 | -0.3123333333 | -0.8913333333 | -0.1648333333 |
| Sales for June | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 14 | 14.4 | 15.3 | 16.1 | 18.9 | 19.7 | 21.1 | 22.1 | |
| Sales on the trend line | 13.5449166667 | 14.8239166667 | 16.1029166667 | 17.3819166667 | 18.6609166667 | 19.9399166667 | 21.2189166667 | 22.4979166667 | |
| Difference | 0.4550833333 | -0.4239166667 | -0.8029166667 | -1.2819166667 | 0.2390833333 | -0.2399166667 | -0.1189166667 | -0.3979166667 | -0.3214166667 |
| Sales for July | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 13.5 | 15.7 | 16.8 | 17.4 | 18.3 | 19.7 | 21 | 22.6 | |
| Sales on the trend line | 13.6515 | 14.9305 | 16.2095 | 17.4885 | 18.7675 | 20.0465 | 21.3255 | 22.6045 | |
| Difference | -0.1515 | 0.7695 | 0.5905 | -0.0885 | -0.4675 | -0.3465 | -0.3255 | -0.0045 | -0.003 |
| Sales for August | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 14.5 | 14 | 15.7 | 17 | 17.9 | 20.5 | 21 | 22.7 | |
| Sales on the trend line | 13.7580833333 | 15.0370833333 | 16.3160833333 | 17.5950833333 | 18.8740833333 | 20.1530833333 | 21.4320833333 | 22.7110833333 | |
| Difference | 0.7419166667 | -1.0370833333 | -0.6160833333 | -0.5950833333 | -0.9740833333 | 0.3469166667 | -0.4320833333 | -0.0110833333 | -0.3220833333 |
| Sales for September | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 14.3 | 15.5 | 16.8 | 17.2 | 19.2 | 20.3 | 20.6 | 21.9 | |
| Sales on the trend line | 13.8646666667 | 15.1436666667 | 16.4226666667 | 17.7016666667 | 18.9806666667 | 20.2596666667 | 21.5386666667 | 22.8176666667 | |
| Difference | 0.4353333333 | 0.3563333333 | 0.3773333333 | -0.5016666667 | 0.2193333333 | 0.0403333333 | -0.9386666667 | -0.9176666667 | -0.1161666667 |
| Sales for October | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 14.9 | 15.8 | 16.3 | 17.9 | 18.8 | 20.3 | 21.4 | 22.9 | |
| Sales on the trend line | 13.97125 | 15.25025 | 16.52925 | 17.80825 | 19.08725 | 20.36625 | 21.64525 | 22.92425 | |
| Difference | 0.92875 | 0.54975 | -0.22925 | 0.09175 | -0.28725 | -0.06625 | -0.24525 | -0.02425 | 0.08975 |
| Sales for November | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 16.9 | 16.5 | 18.7 | 20.5 | 20.4 | 22.4 | 23.7 | 24 | |
| Sales on the trend line | 14.0778333333 | 15.3568333333 | 16.6358333333 | 17.9148333333 | 19.1938333333 | 20.4728333333 | 21.7518333333 | 23.0308333333 | |
| Difference | 2.8221666667 | 1.1431666667 | 2.0641666667 | 2.5851666667 | 1.2061666667 | 1.9271666667 | 1.9481666667 | 0.9691666667 | 1.8331666667 |
| Sales for December | |||||||||
| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Average |
| Actual sales | 17.4 | 20.1 | 19.7 | 22.5 | 23 | 23.8 | 24.6 | 26.6 | |
| Sales on the trend line | 14.1844166667 | 15.4634166667 | 16.7424166667 | 18.0214166667 | 19.3004166667 | 20.5794166667 | 21.8584166667 | 23.1374166667 | |
| Difference | 3.2155833333 | 4.6365833333 | 2.9575833333 | 4.4785833333 | 3.6995833333 | 3.2205833333 | 2.7415833333 | 3.4625833333 | 3.5515833333 |
We cannot observe any obvious trends in the differences between actual sales and sales on the trend line. Therefore, we just calculate the arithmetic means of these differences for every month.
For example, we notice that sales in December tend to be higher by about 3551.58 USD compared to sales predicted on the trend line. Similarly, sales for January tend to be lower on average by 2401 USD compared to sales predicted on the trend line.
Making the assumption that the month has an impact on the actual sales from our observations of the variation of sales across the months, we take our prediction rule:
sales = 1.279*year -2557.778
We then update it to the new rule:
sales = 1.279*year - 2557.778 + month_difference
Here, sales is the amount of sales for a chosen month and year in the prediction, and month_difference is the average difference in our given data between actual sales and sales on the trend line. More specifically, we get the following 12 equations and predictions for sales for the year 2018 in thousands of USD:
sales_january = 1.279*(year+0/12) - 2557.778 - 2.401
= 1.279*(2018 + 0/12) - 2557.778 - 2.401 = 20.843
sales_february = 1.279*(year+1/12) - 2557.778 - 1.358
= 1.279*(2018+1/12) - 2557.778 - 1.358 = 21.993
sales_march = 1.279*(year+2/12) - 2557.778 - 0.464
= 1.279*(2018+2/12) - 2557.778 - 0.464 = 22.993
sales_april = 1.279*(year+3/12) - 2557.778 - 0.608
= 1.279*(2018+3/12) - 2557.778 - 0.608 = 22.956
sales_may = 1.279*(year+4/12) - 2557.778 - 0.165
= 1.279*(2018+4/12) - 2557.778 - 0.165 = 23.505
sales_june = 1.279*(year+5/12) - 2557.778 - 0.321
= 1.279*(2018+5/12) - 2557.778 - 0.321 = 23.456
sales_july = 1.279*(year+6/12) - 2557.778 - 0.003
= 1.279*(2018+6/12) - 2557.778 - 0.003 = 23.881
sales_august = 1.279*(year+7/12) - 2557.778 - 0.322
= 1.279*(2018+7/12) - 2557.778 - 0.322 = 23.668
sales_september = 1.279*(year+8/12) - 2557.778 - 0.116
= 1.279*(2018+8/12) - 2557.778 - 0.116 = 23.981
sales_october = 1.279*(year+9/12) - 2557.778 + 0.090
= 1.279*(2018+9/12) - 2557.778 + 0.090 = 24.293
sales_november = 1.279*(year+10/12) - 2557.778 + 1.833
= 1.279*(2018+10/12) - 2557.778 + 1.833 = 26.143
sales_december = 1.279*(year+11/12) - 2557.778 + 3.552
= 1.279*(2018+11/12) - 2557.778 + 3.552 = 27.968