Demand forecasting: statistical methods, with worked sums on semi-average, moving average and least squares
Managerial Economics · Unit II · Topic 7 of 15 · 5 min read
On this page
- 1. Statistical methods: overview
- 2. Graphical (free-hand) method
- 3. Semi-average method
- 4. Moving average method
- 5. Method of least squares
- 6. Problem 1 (February 2025): forecast sales for the next five years
- 7. Problem 2 (August 2024): Refresh Co., forecast for the next four years
- 8. Regression and barometric methods, briefly
- 9. Merits and limits of statistical methods
- How this is examined
Cases checked to exist. Read the judgment before quoting a holding.
Every case cited here has been checked to exist — the citation resolves and names the right parties. The propositions attributed to each case have not been verified line by line against the judgment. Case names link through to the full judgment: read it before quoting a holding in an examination answer, and never carry a proposition from notes into court on trust.
1. Statistical methods: overview
Statistical methods use past data to project future demand. They are objective and cheap where records exist, but assume that the past pattern will continue. The main ones are:
- Trend projection — fitting a trend to a time series of sales, by the free-hand (graphical), semi-average, moving average or least squares method;
- Regression and correlation — estimating demand from its determinants (income, price, population);
- Barometric method — using leading indicators that move before demand does (building permits before demand for cement, births before demand for school books).
2. Graphical (free-hand) method
Plot sales against years and draw by eye a smooth line through the points; extend it to read off future values. Simple and quick, but subjective: two people will draw different lines.
3. Semi-average method
Divide the series into two equal halves, find the average of each, and plot each average against the middle of its half. The line joining them is the trend. With an odd number of years, the middle year is usually left out.
Example (February 2025 data). Sales (Rs. crore): 2009 – 100, 2010 – 130, 2011 – 150, 2012 – 180, 2013 – 190. Leave out 2011.
- First half: (100 + 130) ÷ 2 = 115, centred on mid-2009/10.
- Second half: (180 + 190) ÷ 2 = 185, centred on mid-2012/13.
- Rise over 3 years = 70; annual rise = 23.33.
- 2014 = 185 + 1.5 × 23.33 = 220; 2015 = 243.33; 2016 = 266.67; 2017 = 290; 2018 = 313.33.
Merit: objective and simple. Demerit: assumes a straight-line trend and is affected by extreme values in either half.
4. Moving average method
Replace each value by the average of a fixed number of consecutive years (3-year, 5-year), moving one year at a time. The averages smooth out short-term fluctuations and show the trend.
| Year | Sales | 3-year total | 3-year moving average |
|---|---|---|---|
| 2015 | 120 | ||
| 2016 | 140 | 420 | 140.00 |
| 2017 | 160 | 455 | 151.67 |
| 2018 | 155 | 495 | 165.00 |
| 2019 | 180 |
Merit: removes cyclical and irregular movements if the period is chosen well. Demerits: the trend is lost at both ends; it gives no equation, so it is weak for forecasting beyond the data.
5. Method of least squares
Fit the straight line Y = a + bX such that the sum of squared deviations of actual from trend values is the minimum. Taking the middle year as origin (so that ΣX = 0), the normal equations reduce to:
a = ΣY ÷ N b = ΣXY ÷ ΣX²
6. Problem 1 (February 2025): forecast sales for the next five years
| Year | Sales Y | X | XY | X² |
|---|---|---|---|---|
| 2009 | 100 | −2 | −200 | 4 |
| 2010 | 130 | −1 | −130 | 1 |
| 2011 | 150 | 0 | 0 | 0 |
| 2012 | 180 | 1 | 180 | 1 |
| 2013 | 190 | 2 | 380 | 4 |
| Total | 750 | 0 | 230 | 10 |
a = 750 ÷ 5 = 150; b = 230 ÷ 10 = 23. Trend equation: Y = 150 + 23X (origin 2011, X in years).
| Year | X | Forecast (Rs. crore) |
|---|---|---|
| 2014 | 3 | 219 |
| 2015 | 4 | 242 |
| 2016 | 5 | 265 |
| 2017 | 6 | 288 |
| 2018 | 7 | 311 |
Check: the trend values for 2009–2013 are 104, 127, 150, 173, 196, which total 750, the same as actual sales.
7. Problem 2 (August 2024): Refresh Co., forecast for the next four years
| Year | Sales Y | X | XY | X² |
|---|---|---|---|---|
| 2015 | 120 | −2 | −240 | 4 |
| 2016 | 140 | −1 | −140 | 1 |
| 2017 | 160 | 0 | 0 | 0 |
| 2018 | 155 | 1 | 155 | 1 |
| 2019 | 180 | 2 | 360 | 4 |
| Total | 755 | 0 | 135 | 10 |
a = 755 ÷ 5 = 151; b = 135 ÷ 10 = 13.5. Trend: Y = 151 + 13.5X (origin 2017).
| Year | X | Forecast |
|---|---|---|
| 2020 | 3 | 191.5 |
| 2021 | 4 | 205.0 |
| 2022 | 5 | 218.5 |
| 2023 | 6 | 232.0 |
Check: trend values 2015–2019 are 124, 137.5, 151, 164.5, 178 (total 755).
8. Regression and barometric methods, briefly
Regression estimates demand from a causal variable: if sales depend on income, fit Y = a + bX where X is income, then use forecast income to estimate sales. It explains demand rather than merely extending a trend, but needs good data on the determinant. Barometric forecasting uses leading indicators (indices that turn before the series of interest), coincident and lagging indicators; it predicts turning points rather than quantities.
9. Merits and limits of statistical methods
Merits: objective, based on recorded facts, cheap once data exist, and give a definite figure. Limits: assume the past pattern will continue; useless for new products; cannot capture sudden changes in taste, policy or competition; and a straight-line trend may not fit the data.
How this is examined
February 2025: “From the following find out sales for the next five years” (2009–2013 data, 10 marks). August 2024: Refresh Co., sales forecast for the next four years (2015–2019 data, 10 marks). Both are least squares sums. Write the method’s name and formula first, show the full table, give the trend equation with its origin, and present the forecast as a table. A one-line check (trend values total ΣY) shows accuracy.
Now try the real questions: Managerial Economics past question papers.