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Strategic Cost Management · Business Forecasting Models - Time Series and Regression Analysis

Seasonal Variation and Deseasonalisation of Time Series Data

Updated 11 October 2026 · Fact-checked

Seasonal variation is a regular pattern that repeats within a year, such as higher sales in festival quarters. You measure it with a seasonal index, where 100 means an average period. Compute it by simple average, ratio-to-trend or ratio-to-moving-average. To deseasonalise, divide each actual value by its index and multiply by 100.

Understand Seasonal Variation and Deseasonalisation

A time series has four parts: trend, seasonal variation, cyclical variation and irregular variation. Seasonal variation is the part that repeats within a year (or a shorter period) for the same reasons each time. Examples are higher sales of fans in summer, higher sales of consumer goods in the festival quarter, or heavy demand for fertilisers in the sowing season.

You measure it with a seasonal index. An index of 100 means the period is typical. An index of 120 means that period runs 20% above the average level. An index of 80 means it runs 20% below. Over a full year the indices average 100. So four quarterly indices add up to 400, and twelve monthly indices add up to 1200.

There are three common ways to get the index. The simple average method averages each period across the years and compares it with the overall average. It is quick, but it assumes there is no trend. The ratio-to-trend method divides actual values by trend values, so it removes the trend. The ratio-to-moving-average method divides actual values by a centred moving average, which stands in for trend and cycle. It is the most used method when a trend is present.

Deseasonalisation removes the seasonal effect from actual data. You divide each actual value by its seasonal index (as a fraction of 100). The result shows what the value would have been in an average period. Management uses these figures to see the real direction of sales or costs, and to forecast: you project the trend and then multiply by the seasonal index to get the seasonal forecast.

Key rules to remember

Simple average seasonal index
Seasonal index = (Average of the period across years ÷ Grand average of all periods) × 100
Use when there is no clear trend. The grand average is the average of the period averages.
Ratio-to-trend
Seasonal ratio = (Actual Y ÷ Trend value T) × 100
Fit the trend first (for example by least squares). Then average the ratios for each period across years.
Centred moving average (quarterly data)
Centred MA = (Sum of two consecutive 4-quarter totals) ÷ 8
The first value falls at the third quarter of the first year. You lose two periods at each end. For monthly data use 12-month totals and divide the sum of two consecutive totals by 24.
Ratio-to-moving-average
Seasonal ratio = (Actual Y ÷ Centred MA) × 100
Average the ratios for each quarter (or month) across years. A median is also accepted if the question asks for it.
Adjustment to total 400 (or 1200)
Adjusted index = Average ratio × (400 ÷ Sum of average ratios)
Use 1200 for monthly data. Skip this step if the averages already sum to 400 (or 1200).
Deseasonalised value (multiplicative model)
Deseasonalised value = (Actual Y ÷ Seasonal index) × 100
Model Y = T × S × C × I. An index above 100 lowers the actual value; an index below 100 raises it.
Seasonal forecast
Forecast = Trend forecast × (Seasonal index ÷ 100)
Project the trend first, then apply the index of the target period.
Additive model
Deseasonalised value = Actual Y − Seasonal variation S
Used when the question gives seasonal variations in absolute units. The variations should sum to zero over the year.

How to solve Seasonal Variation and Deseasonalisation questions

Use this order for any seasonal variation question. First read which method the question asks for and whether the model is multiplicative or additive.

  1. 1Arrange the data in a table with years as rows and quarters (or months) as columns.
  2. 2Pick the method as asked. Use the simple average method if no trend is mentioned. Use ratio-to-trend if a trend equation or trend values are given. Use ratio-to-moving-average if the question says moving average.
  3. 3For the simple average method: find the average of each quarter across years, find the grand average, and divide each quarter average by it (×100).
  4. 4For ratio-to-moving-average: compute 4-quarter totals, add consecutive pairs and divide by 8 to get the centred moving average, then compute (Actual ÷ CMA) × 100 for each quarter that has a CMA.
  5. 5Average the ratios quarter-wise across years. Add the four averages. If the total is not 400, multiply each by 400 ÷ total.
  6. 6Check that the final indices sum to 400 (or 1200 for months). Show this check in your answer.
  7. 7To deseasonalise, divide each actual value by its index and multiply by 100. To forecast, multiply the trend value by the index ÷ 100.
  8. 8Write one line of interpretation, for example which quarter is strongest and by how much it is above average.

Quickest way: Total-based shortcut for the simple average method

When to use it: Use when the question asks for a seasonal index by the simple average method and gives several years of quarterly or monthly data. It also works for checking your answer in the other methods.

  1. Add each column (quarter) across all years to get the quarter totals. You do not need to divide by the number of years.
  2. Add the quarter totals and divide by 4 to get the average quarter total. The same ratio results, because the number of years cancels out.
  3. Index = (Quarter total ÷ Average quarter total) × 100.
  4. Check the four indices add up to 400.
  5. For the moving average method, write the 4-quarter totals first and keep the sums of pairs; divide by 8 only when you compute the ratio. Do not round the CMA early.

Common mistakes in Seasonal Variation and Deseasonalisation

  • Taking an ordinary 4-quarter moving average and not centring it

    With an even number of periods, the average falls between two quarters. Students forget that it must be centred to line up with an actual quarter.

    Fix: Add two consecutive 4-quarter totals and divide by 8. Place the result against the third quarter of the first window.

  • Placing the moving average against the wrong quarter

    Students write the first total against the first quarter instead of shifting it down.

    Fix: The first centred moving average belongs to Q3 of Year 1. The ratio table therefore starts at Q3 and ends two quarters before the last observation.

  • Forgetting to adjust the indices so that they total 400 (or 1200)

    Averaging ratios from different years rarely gives exactly 400, and students stop after averaging.

    Fix: Add the averages, then multiply each by 400 ÷ total. Always show the total as a check.

  • Using the simple average method when the data has a strong trend

    It is the easiest method, so students use it without reading the data. Later quarters then look seasonally high only because of growth.

    Fix: If the data rises or falls steadily, say so and prefer ratio-to-trend or ratio-to-moving-average. Use the simple average method when the question names it or states no trend.

  • Multiplying by the index instead of dividing when deseasonalising

    Students confuse deseasonalising with applying seasonality to a forecast.

    Fix: Deseasonalise = divide by index ÷ 100. Forecast = multiply by index ÷ 100. A quick check: a peak quarter's actual figure must come down after deseasonalising.

  • Mixing up additive and multiplicative models

    Both use the word seasonal. Students subtract when the index is a percentage or divide when the variation is in units.

    Fix: If the figures are indices around 100, divide. If they are unit variations (plus and minus) that sum to zero, subtract.

Worked examples

Example 1

A company's quarterly sales (₹ lakh) for three years are: Year 1: 40, 50, 60, 50; Year 2: 44, 54, 66, 56; Year 3: 48, 58, 72, 62. Compute the seasonal indices by the simple average method and deseasonalise the Year 3 sales.

Show the solution
  1. Quarter averages: Q1 = (40 + 44 + 48) ÷ 3 = 132 ÷ 3 = 44. Q2 = (50 + 54 + 58) ÷ 3 = 162 ÷ 3 = 54. Q3 = (60 + 66 + 72) ÷ 3 = 198 ÷ 3 = 66. Q4 = (50 + 56 + 62) ÷ 3 = 168 ÷ 3 = 56.
  2. Grand average = (44 + 54 + 66 + 56) ÷ 4 = 220 ÷ 4 = 55.
  3. Seasonal indices: Q1 = 44 ÷ 55 × 100 = 80. Q2 = 54 ÷ 55 × 100 = 98.18. Q3 = 66 ÷ 55 × 100 = 120. Q4 = 56 ÷ 55 × 100 = 101.82.
  4. Check: 80 + 98.18 + 120 + 101.82 = 400.
  5. Deseasonalised Year 3 sales: Q1 = 48 ÷ 80 × 100 = 60. Q2 = 58 ÷ 98.18 × 100 = 59.07. Q3 = 72 ÷ 120 × 100 = 60. Q4 = 62 ÷ 101.82 × 100 = 60.89.
  6. Interpretation: Q3 is the peak at 20% above average and Q1 is the trough at 20% below. After adjustment, Year 3 sales are almost level at about ₹60 lakh a quarter.

Answer: Seasonal indices: Q1 80, Q2 98.18, Q3 120, Q4 101.82 (total 400). Deseasonalised Year 3 sales (₹ lakh): 60, 59.07, 60 and 60.89.

Example 2

Quarterly production (thousand units) for four years: Year 1: 30, 40, 60, 50; Year 2: 36, 46, 66, 56; Year 3: 42, 52, 72, 62; Year 4: 48, 58, 78, 68. Find the seasonal indices by the ratio-to-moving-average method (to one decimal) and deseasonalise Year 4 Q3 and Q4.

Show the solution
  1. Four-quarter moving totals, starting from Q1 of Year 1: 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252.
  2. Centred MA = (sum of two consecutive totals) ÷ 8. This gives, from Year 1 Q3 to Year 4 Q2: 45.75, 47.25, 48.75, 50.25, 51.75, 53.25, 54.75, 56.25, 57.75, 59.25, 60.75, 62.25.
  3. Ratios (Actual ÷ CMA × 100): Y1 Q3 = 131.15; Y1 Q4 = 105.82; Y2 Q1 = 73.85; Y2 Q2 = 91.54; Y2 Q3 = 127.54; Y2 Q4 = 105.16; Y3 Q1 = 76.71; Y3 Q2 = 92.44; Y3 Q3 = 124.68; Y3 Q4 = 104.64; Y4 Q1 = 79.01; Y4 Q2 = 93.17.
  4. Average ratio by quarter: Q1 = (73.85 + 76.71 + 79.01) ÷ 3 = 76.52. Q2 = (91.54 + 92.44 + 93.17) ÷ 3 = 92.38. Q3 = (131.15 + 127.54 + 124.68) ÷ 3 = 127.79. Q4 = (105.82 + 105.16 + 104.64) ÷ 3 = 105.21.
  5. Total of averages = 76.52 + 92.38 + 127.79 + 105.21 = 401.90, not 400. Adjustment factor = 400 ÷ 401.90 = 0.99527.
  6. Adjusted indices: Q1 = 76.52 × 0.99527 = 76.2. Q2 = 92.38 × 0.99527 = 91.9. Q3 = 127.79 × 0.99527 = 127.2. Q4 = 105.21 × 0.99527 = 104.7. Total = 76.2 + 91.9 + 127.2 + 104.7 = 400.0.
  7. Deseasonalised Year 4: Q3 = 78 ÷ 127.2 × 100 = 61.32. Q4 = 68 ÷ 104.7 × 100 = 64.95.

Answer: Seasonal indices: Q1 76.2, Q2 91.9, Q3 127.2, Q4 104.7 (total 400). Deseasonalised Year 4 production: Q3 about 61.3 thousand units and Q4 about 65.0 thousand units.

Exam tips

  • Read the method name in the question and follow it. Marks are given for the steps, so show the moving totals, the centred values, the ratios and the adjustment.
  • Always show the check that indices total 400 (quarters) or 1200 (months). Examiners look for it.
  • Keep two decimals in ratios and do not round the centred moving average early. Round the final indices as the question asks.
  • In a case-based MCQ, test your answer with logic: a peak season must have an index above 100, and its deseasonalised value must be lower than the actual value.
  • If asked for a recommendation, state the use: deseasonalised data shows the true trend, and seasonal indices help in production, inventory and staffing plans for peak quarters.

Practice questions from Business Forecasting Models - Time Series and Regression Analysis

Seasonal Variation and Deseasonalisation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Seasonal Variation and Deseasonalisation: frequently asked questions

Which method should I use for the seasonal index in the exam?

Use the method named in the question. If none is named, use ratio-to-moving-average when the data shows a trend, and the simple average method when there is no clear trend. State your choice in one line.

Why do the seasonal indices have to total 400 or 1200?

An index of 100 stands for an average period. Over one full year the seasonal effects should cancel out, so four quarterly indices average 100 and total 400, and twelve monthly indices total 1200. If your total differs, adjust each index by the ratio of the required total to your total.

How do I deseasonalise data?

Divide each actual figure by its seasonal index and multiply by 100. This removes the seasonal effect and shows the underlying level. In an additive model you subtract the seasonal variation instead.

Why do we lose values in the moving average method?

A centred 4-quarter average needs two quarters before and after each point, so the first two and last two quarters have no centred moving average. Ratios are computed only for quarters that have one.