14.4 Seasonal Indices
LEARNING OBJECTIVES
- Calculate and use seasonal indices to create a forecast for a time series.
Seasonal patterns are repeated patterns that occur over a one-year period due to seasonal influences. Repeated and predictable highs and lows in the time series data that occur at the same time each year indicate seasonality in the data. For example, demand for golf clubs or sunscreen lotion are predictably higher during the summer months. Similarly, retail sales experience predictable highs during the holiday season every year. When a seasonal pattern exists, a seasonal index may be used to create a forecast that accounts for the seasonal component in the time series.
A seasonal index is a numerical measure of the seasonal variation in the time series. There is a seasonal index for each “season” in the time series. For example, if the time series measures monthly data, each month is a season, and there is a seasonal index for each month. If the time series measures quarterly data, each quarter is a season, and there is a seasonal index for each quarter. A seasonal index represents how much a particular season (i.e. month or quarter) deviates from an average season. An average season has a seasonal index of [latex]1[/latex]. An above-average season will have a seasonal index greater than [latex]1[/latex], and a below-average season will have a seasonal index less than [latex]1[/latex].
Calculating Seasonal Indices
The seasonal index for a particular season is found by dividing the average value for that season by the average of all the data.
[latex]\displaystyle{\text{Seasonal Index}=\frac{\text{Average for the Season}}{\text{Average of all Data}}}[/latex]
Follow these steps to calculate the seasonal index for a season:
- Calculate the overall average (mean) of the time series data.
- Calculate the average (mean) of the data for each season. This is easier to do when the time series data is grouped into seasons (i.e. if the time series measures months, group all the January data points together, group all the February data points together, and so on).
- Divide the average for each season by the average of all the data. There will be a seasonal index for each season (i.e. if the time series measures months, there will be [latex]12[/latex] seasonal indices).
NOTES
- The seasonal indices described above are typically used on time series that have a seasonal component but no trend component. The seasonal indices described above only capture the seasonal effect in the time series, and so would not create a good forecast when both seasonality and trend are present in the time series.
- When both seasonality and trend are present in a time series, a change from one season to the next may be due to a trend, a seasonal variation, or just a random fluctuation. In such cases, seasonal indices are calculated using the centred moving average approach. The centred moving average approach to seasonal indices prevents a variation due to a trend from being incorrectly identified as a variation due to seasonality. The seasonal indices based on centred moving averages remove the effect of the season so that the trend effect is easier to identify. The seasonal indices based on centred moving averages are typically used in the decomposition model, which isolates the seasonal and trend components in a time series. Both the seasonal indices based on centred moving averages and the decomposition model are not discussed in this book.
EXAMPLE
Falcon Golf Supply Company sells a wide range of products for golfers. The company recorded two years of monthly sales for its best-selling set of golf clubs, the Peregrine Set.
| Month | Year 1 | Year 2 |
|---|---|---|
| January | 70 | 61 |
| February | 72 | 72 |
| March | 85 | 79 |
| April | 101 | 103 |
| May | 123 | 120 |
| June | 108 | 117 |
| July | 99 | 100 |
| August | 92 | 95 |
| September | 80 | 85 |
| October | 65 | 81 |
| November | 69 | 73 |
| December | 82 | 86 |
- Calculate the seasonal index for each month.
- Identify the average, above-average, and below-average seasons.
- Suppose the company predicts that they will sell a total of [latex]2,400[/latex] sets of Peregrine clubs in year 3. Forecast the sales for each month of year 3.
Solution
- The average of all of the data is [latex]88.25[/latex].
Month Year 1 Year 2 Average per Month Seasonal Index January [latex]70[/latex] [latex]61[/latex] [latex]65.5[/latex] [latex]\frac{65.5}{88.25}=0.7422\ldots[/latex] February [latex]72[/latex] [latex]72[/latex] [latex]72[/latex] [latex]\frac{72}{88.25}=0.8158\ldots[/latex] March [latex]85[/latex] [latex]79[/latex] [latex]82[/latex] [latex]\frac{82}{88.25}=0.9291\ldots[/latex] April [latex]101[/latex] [latex]103[/latex] [latex]102[/latex] [latex]\frac{102}{88.25}=1.1558\ldots[/latex] May [latex]123[/latex] [latex]120[/latex] [latex]121.5[/latex] [latex]\frac{121.5}{88.25}=1.3767\ldots[/latex] June [latex]108[/latex] [latex]117[/latex] [latex]112.5[/latex] [latex]\frac{112.5}{88.25}=1.2747\ldots[/latex] July [latex]99[/latex] [latex]100[/latex] [latex]99.5[/latex] [latex]\frac{99.5}{88.25}=1.1274\ldots[/latex] August [latex]92[/latex] [latex]95[/latex] [latex]93.5[/latex] [latex]\frac{93.5}{88.25}=1.0594\ldots[/latex] September [latex]80[/latex] [latex]85[/latex] [latex]82.5[/latex] [latex]\frac{82.5}{88.25}=0.9384\ldots[/latex] October [latex]65[/latex] [latex]81[/latex] [latex]73[/latex] [latex]\frac{73}{88.25}=0.8271\ldots[/latex] November [latex]69[/latex] [latex]73[/latex] [latex]71[/latex] [latex]\frac{71}{88.25}=0.8045\ldots[/latex] December [latex]82[/latex] [latex]86[/latex] [latex]84[/latex] [latex]\frac{84}{88.25}=0.9518\ldots[/latex] - April, May, June, July, and August are above-average seasons. January, February, March, September, October, November, and December are below-average seasons.
- The forecast for all of year 3 is [latex]2,400[/latex] sets of clubs, which is [latex]\frac{2,400}{12}=200[/latex] sets of clubs per month. The forecast of [latex]200[/latex] per month is not adjusted for the season. To create a forecast adjusted for the season, we multiply the unadjusted forecast of [latex]200[/latex] by the seasonal index for each month.
Month Seasonal Index Forecast for Year 3 January [latex]0.7422\ldots[/latex] [latex]0.7422\ldots\times200=148.44[/latex] February [latex]0.8158\ldots[/latex] [latex]0.8158\ldots\times200=163.17[/latex] March [latex]0.9291\ldots[/latex] [latex]0.9291\ldots\times200=185.84[/latex] April [latex]1.1558\ldots[/latex] [latex]1.1558\ldots\times200=231.16[/latex] May [latex]1.3767\ldots[/latex] [latex]1.3767\ldots\times200=275.35[/latex] June [latex]1.2747\ldots[/latex] [latex]1.2747\ldots\times200=254.96[/latex] July [latex]1.1274\ldots[/latex] [latex]1.1274\ldots\times200=225.50[/latex] August [latex]1.0594\ldots[/latex] [latex]1.0594\ldots\times200=221.90[/latex] September [latex]0.9384\ldots[/latex] [latex]0.9384\ldots\times200=186.97[/latex] October [latex]0.8271\ldots[/latex] [latex]0.8271\ldots\times200=165.44[/latex] November [latex]0.8045\ldots[/latex] [latex]0.8045\ldots\times200=160.91[/latex] December [latex]0.9518\ldots[/latex] [latex]0.9518\ldots\times200=190.37[/latex]
NOTES
- The average of all of the data is found by adding up all of the observations in the time series and dividing by the total number of observations. In this example,[latex]\displaystyle{\text{Average of all Data}=\frac{2118}{24}=88.25}[/latex].
- To calculate the seasonal index for each month, average the observations for the month and divide by the average of all of the data. For example, the average for January is [latex]\displaystyle{\text{Average for January}=\frac{70+61}{2}=65.5}[/latex]. Then the seasonal index for January is [latex]\displaystyle{\text{Seasonal Index for January}=\frac{65.5}{88.25}=0.7422\ldots}[/latex].
- If a season has a seasonal index greater than [latex]1[/latex], the season is above average. If a season has a seasonal index less than [latex]1[/latex], the season is below average. Because April, May, June, July, and August have seasonal indices greater than [latex]1[/latex], they are above-average seasons. Similarly, because January, February, March, September, October, November, and December have seasonal indices less than [latex]1[/latex], they are below-average seasons.
- When calculating the forecast for each season, we need to know the “average”, unadjusted forecast for a season, not the total for the entire year. In this example, the total forecasted sales for year 3 is [latex]2,400[/latex]. Because there are [latex]12[/latex] seasons (months) per year, the “average” per season is [latex]\frac{2,400}{12}=200[/latex]. This [latex]200[/latex] “average” per season is then adjusted by multiplying by the corresponding seasonal index for each season to find the forecast. For example, the forecast for July is [latex]200\times1.1274\ldots=225.50[/latex].
NOTE
The sum of the seasonal indices equals the number of seasons. In the above example, the sum of the seasonal indices equals [latex]12[/latex], the number of seasons.
TRY IT
The table below shows the quarterly sales figures (in [latex]\$1,000,000[/latex]s) for a local business.
| Quarter | Year 1 | Year 2 | Year 3 |
|---|---|---|---|
| 1 | 108 | 114 | 105 |
| 2 | 125 | 116 | 135 |
| 3 | 161 | 148 | 150 |
| 4 | 154 | 163 | 165 |
- Calculate the seasonal index for each quarter.
- Identify the average, below-average, and above-average seasons.
- Suppose the business predicts sales of [latex]\$600,000,000[/latex] for year 4. Calculate the forecast for each quarter of year 4.
Click to see Solution
- The average of all of the data is [latex]137[/latex].
Quarter Year 1 Year 2 Year 3 Average per Quarter Seasonal Index 1 [latex]108[/latex] [latex]114[/latex] [latex]105[/latex] [latex]109[/latex] [latex]0.7956\ldots[/latex] 2 [latex]125[/latex] [latex]116[/latex] [latex]135[/latex] [latex]125.333\ldots[/latex] [latex]0.9148\ldots[/latex] 3 [latex]161[/latex] [latex]148[/latex] [latex]150[/latex] [latex]153[/latex] [latex]1.1167\ldots[/latex] 4 [latex]154[/latex] [latex]163[/latex] [latex]165[/latex] [latex]160.666\ldots[/latex] [latex]1.1727\ldots[/latex] - Quarters 1 and 2 are below-average seasons. Quarters 3 and 4 are above-average seasons.
-
Quarter Forecast for Year 4 1 [latex]0.7956\ldots\times150,000,000=\$119,343,065.69[/latex] 2 [latex]0.9148\ldots\times150,000,000=\$137,226,277.37[/latex] 3 [latex]1.1167\ldots\times150,000,000=\$167,518,248.18[/latex] 4 [latex]1.1727\ldots\times150,000,000=\$175,912,408.76[/latex]
Exercises
- John runs a small business selling ski and snowboard equipment. The quarterly sales (in [latex]\$1000[/latex]s) for the past four years are recorded in the table below.
Quarter Year 1 Year 2 Year 3 Year 4 1 183 208 189 213 2 220 207 191 206 3 139 117 128 119 4 100 132 128 133 - Calculate the seasonal index for each quarter.
- Suppose John forecasts sales totalling [latex]\$900,000[/latex] in year 5. What is the forecast for each quarter of year 5?
- Identify the average, above-average, and below-average seasons.
Click to see Answer
Quarter Seasonal Index Forecast for Year 4 Average, Above, or Below 1 [latex]1.2139\ldots[/latex] [latex]\$273,134.33[/latex] Above 2 [latex]1.2613\ldots[/latex] [latex]\$283,811.71[/latex] Above 3 [latex]0.7699\ldots[/latex] [latex]\$173,249.14[/latex] Below 4 [latex]0.7546\ldots[/latex] [latex]\$169,804.82[/latex] Below - A local garden supply store recorded the number of bags of fertilizer (in [latex]1000[/latex]s) that it sold each month for three years. The data is recorded in the table below.
Month Year 1 Year 2 Year 3 January 1 3 2 February 2 3 4 March 3 2 3 April 15 11 13 May 13 11 12 June 10 12 9 July 11 9 10 August 9 8 8 September 9 8 12 October 4 4 4 November 4 4 5 December 2 2 1 - Calculate the seasonal index for each month.
- Suppose the store predicts it will sell [latex]114,000[/latex] bags of fertilizer in year 4. What is the forecast for each month of year 4?
- Identify the average, above-average, and below-average seasons.
Click to see Answer
Month Seasonal Index Forecast for Year 4 Average, Above, or Below January [latex]0.2962\ldots[/latex] [latex]2,814.82[/latex] Below February [latex]0.4444\ldots[/latex] [latex]4,222.22[/latex] Below March [latex]0.3950\ldots[/latex] [latex]3,753.09[/latex] Below April [latex]1.9259\ldots[/latex] [latex]18,296.30[/latex] Above May [latex]1.7777\ldots[/latex] [latex]16,888.89[/latex] Above June [latex]1.5308\ldots[/latex] [latex]14,543.21[/latex] Above July [latex]1.4814\ldots[/latex] [latex]14,074.07[/latex] Above August [latex]1.2345\ldots[/latex] [latex]11,728.40[/latex] Above September [latex]1.4320\ldots[/latex] [latex]13,604.94[/latex] Above October [latex]0.5925\ldots[/latex] [latex]5,629.63[/latex] Below November [latex]0.6419\ldots[/latex] [latex]6,098.77[/latex] Below December [latex]0.2469\ldots[/latex] [latex]2,345.68[/latex] Below - A local coffee shop recorded the number of coffees (in [latex]100[/latex]s) that they sell each day over a six-week period. The data is recorded in the table below.
Day Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Sunday 9 9 9 11 7 9 Monday 6 5 4 3 5 7 Tuesday 5 5 6 5 7 3 Wednesday 7 6 7 4 7 8 Thursday 7 7 6 3 5 3 Friday 6 4 7 7 3 4 Saturday 8 10 10 10 10 8 - Calculate the seasonal index for each day.
- Suppose the store predicts it will sell [latex]5250[/latex] coffees in week 7. What is the forecast for each day of year 7?
- Identify the average, above-average, and below-average seasons.
Click to see Answer
Day Seasonal Index Forecast for Week 7 Average, Above, or Below Sunday [latex]1.3897\ldots[/latex] [latex]1,042.28[/latex] Above Monday [latex]0.7720\ldots[/latex] [latex]579.04[/latex] Below Tuesday [latex]0.7977\ldots[/latex] [latex]598.35[/latex] Below Wednesday [latex]1.003\ldots[/latex] [latex]752.76[/latex] Above Thursday [latex]0.7977\ldots[/latex] [latex]598.35[/latex] Below Friday [latex]0.7977\ldots[/latex] [latex]598.35[/latex] Below Saturday [latex]1.4417\ldots[/latex] [latex]1,080.88[/latex] Above
