Question 1200719: For the following sales data, spanning 4 years, perform a trend and seasonal indices analysis, use a multiplicative model. (Note-this example shows only 3 seasons in a year which is not the norm.)
year season sales
1 1 1856
1 2 2012
1 3 985
2 1 1995
2 2 2168
2 3 1072
3 1 2241
3 2 2306
3 3 1105
4 1 2280
4 2 2408
4 3 1120
Answer by GingerAle(43)   (Show Source):
You can put this solution on YOUR website! **1. Calculate the Average Sales for Each Season**
* **Season 1:** (1856 + 1995 + 2241 + 2280) / 4 = 2093
* **Season 2:** (2012 + 2168 + 2306 + 2408) / 4 = 2223.5
* **Season 3:** (985 + 1072 + 1105 + 1120) / 4 = 1070.5
**2. Calculate the Average Sales for Each Year**
* **Year 1:** (1856 + 2012 + 985) / 3 = 1617.67
* **Year 2:** (1995 + 2168 + 1072) / 3 = 1745
* **Year 3:** (2241 + 2306 + 1105) / 3 = 1884
* **Year 4:** (2280 + 2408 + 1120) / 3 = 1936
**3. Calculate the Centered Moving Average (CMA)**
* **Calculate 4-year moving averages:**
* (Year 1 + Year 2) / 2 = (1617.67 + 1745) / 2 = 1681.335
* (Year 2 + Year 3) / 2 = (1745 + 1884) / 2 = 1814.5
* (Year 3 + Year 4) / 2 = (1884 + 1936) / 2 = 1910
* **Calculate Centered Moving Averages:**
* (1681.335 + 1814.5) / 2 = 1747.9175
* (1814.5 + 1910) / 2 = 1862.25
**4. Calculate the Ratio-to-Moving-Average (RMA)**
* Divide the actual sales for each period by the corresponding centered moving average.
| Year | Season | Sales | CMA | RMA |
|---|---|---|---|
| 1 | 1 | 1856 | 1747.9175 | 1.062 |
| 1 | 2 | 2012 | 1747.9175 | 1.151 |
| 1 | 3 | 985 | 1747.9175 | 0.563 |
| 2 | 1 | 1995 | 1747.9175 | 1.141 |
| 2 | 2 | 2168 | 1747.9175 | 1.241 |
| 2 | 3 | 1072 | 1747.9175 | 0.613 |
| 3 | 1 | 2241 | 1862.25 | 1.203 |
| 3 | 2 | 2306 | 1862.25 | 1.238 |
| 3 | 3 | 1105 | 1862.25 | 0.593 |
| 4 | 1 | 2280 | 1862.25 | 1.224 |
| 4 | 2 | 2408 | 1862.25 | 1.292 |
| 4 | 3 | 1120 | 1862.25 | 0.601 |
**5. Calculate the Seasonal Indices**
* **Calculate the average RMA for each season:**
* Season 1: (1.062 + 1.141 + 1.203 + 1.224) / 4 = 1.1575
* Season 2: (1.151 + 1.241 + 1.238 + 1.292) / 4 = 1.2305
* Season 3: (0.563 + 0.613 + 0.593 + 0.601) / 4 = 0.5925
* **Adjust the seasonal indices to sum to 3 (since there are 3 seasons):**
* Season 1: (1.1575 / 1.1575 + 1.2305 + 0.5925) * 3 = 1.101
* Season 2: (1.2305 / 1.1575 + 1.2305 + 0.5925) * 3 = 1.168
* Season 3: (0.5925 / 1.1575 + 1.2305 + 0.5925) * 3 = 0.565
**6. Deseasonalize the Data**
* Divide the actual sales for each period by the corresponding seasonal index.
| Year | Season | Sales | Seasonal Index | Deseasonalized Sales |
|---|---|---|---|---|
| 1 | 1 | 1856 | 1.101 | 1685.68 |
| 1 | 2 | 2012 | 1.168 | 1720.86 |
| 1 | 3 | 985 | 0.565 | 1743.36 |
| 2 | 1 | 1995 | 1.101 | 1811.90 |
| 2 | 2 | 2168 | 1.168 | 1855.17 |
| 2 | 3 | 1072 | 0.565 | 1898.23 |
| 3 | 1 | 2241 | 1.101 | 2035.26 |
| 3 | 2 | 2306 | 1.168 | 1972.22 |
| 3 | 3 | 1105 | 0.565 | 1955.31 |
| 4 | 1 | 2280 | 1.101 | 2070.91 |
| 4 | 2 | 2408 | 1.168 | 2060.87 |
| 4 | 3 | 1120 | 0.565 | 1982.30 |
**7. Trend Analysis**
* **Plot the deseasonalized sales data.**
* **Visually inspect the plot to identify a trend (linear, quadratic, etc.).**
* **Fit a trend line (e.g., linear regression) to the deseasonalized data.**
**8. Forecasting**
* **Use the trend line to forecast future deseasonalized sales.**
* **Multiply the forecasted deseasonalized sales by the appropriate seasonal index to obtain the final forecast.**
**Note:** This analysis provides a basic framework. More sophisticated methods can be used for trend identification and forecasting, such as exponential smoothing or ARIMA models.
This analysis provides a framework for understanding the trend and seasonality in the sales data. You can further refine the analysis by:
* **Visualizing the data:** Create plots of the original data, seasonal indices, and deseasonalized data to better understand the patterns.
* **Considering external factors:** Investigate potential external factors that may influence sales (e.g., economic conditions, marketing campaigns).
* **Using more advanced forecasting methods:** Explore more sophisticated techniques like exponential smoothing or ARIMA models for more accurate forecasts.
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