Fundamentals of Business Mathematics and Statistics · Index Numbers and Time Series

Seasonal Variation Measurement: Methods and Seasonal Index

Updated 10 October 2026 · Fact-checked

Seasonal variation is the regular rise and fall in data within a year, such as quarterly or monthly sales. You measure it with a seasonal index, where 100 means an average season. The three methods are simple average, ratio to moving average and link relative. Each ends with indices averaging 100.

Understand Seasonal Variation Measurement

Many businesses have a yearly pattern. Umbrella sales jump in the monsoon. Sweet sales peak around Diwali. Air-conditioner sales rise before summer. This pattern repeats every year and is called seasonal variation. It is one component of a time series, along with trend, cyclical and irregular movements.

We measure it with a seasonal index for each season (each quarter or month). An index of 120 means that season is usually 20% above the average season. An index of 80 means it is usually 20% below. The indices of all seasons average 100. So the quarterly indices add up to 400 and the monthly indices add up to 1200.

The simple average method is the easiest. It assumes there is no strong trend. You average each season over the years, then compare each season's average with the overall average.

The ratio to moving average method is used when a trend is present. A centred moving average estimates trend and cycle. You divide each actual value by it, which removes the trend. The ratios that remain show the seasonal effect plus some random noise. Averaging the ratios season by season reduces the noise.

The link relative method compares each season with the one just before it. You average these link relatives for each season, chain them from a base of 100, correct the chain for the trend left over, and then convert to an index around 100.

Key formulas to remember

Simple average method: Seasonal index = (Average of the season ÷ Grand average) × 100; Grand average is the average of all the seasonal averages. Use it when there is no clear trend.
Sum of indices check: Sum of indices = 100 × number of seasons; 400 for quarters and 1200 for months. Use this to check your answer.
Centred moving average for quarterly data: CMA = (Sum of two consecutive 4-quarter totals) ÷ 8; It falls against a quarter, not between two quarters. For monthly data, the 12-month version divides by 24.
Ratio to moving average: Ratio = (Actual value ÷ CMA) × 100; Average these ratios for each season across the years.
Adjusted index, ratio to moving average: Adjusted index = Average ratio of the season × (100 ÷ Average of all the seasonal average ratios); Do this only when the average of the seasonal ratios is not exactly 100.
Link relative: Link relative = (Current season value ÷ Previous season value) × 100; The first season of the first year has no previous season, so it has no link relative.
Chain relative: Chain relative = (Average link relative of the season × Chain relative of the previous season) ÷ 100; The chain relative of the first season is taken as 100.
Correction for trend in link relative method: L = (Average link relative of first season × Last chain relative) ÷ 100; d = (L − 100) ÷ number of seasons; corrected chain of the k-th season = chain − (k − 1) × d; The first season stays at 100. Then divide each corrected chain by the average of all corrected chains and multiply by 100 to get the index.

How to solve Seasonal Variation Measurement questions

First read which method the question names. If it names none, use the simple average method when the data has no clear trend. Then follow these steps.

1Arrange the data in a table with years in rows and seasons (quarters or months) in columns.
2Simple average method: find the total and average for each season, then the grand average of those averages.
3Simple average method: divide each seasonal average by the grand average and multiply by 100.
4Ratio to moving average method: find 4-quarter (or 12-month) moving totals, then the centred moving average, and place it against the right season.
5Divide each actual value by its CMA and multiply by 100. Average these ratios for each season, then scale them so their average is 100.
6Link relative method: find link relatives, average them by season, build the chain relatives from 100, apply the correction, and divide by the average of the corrected chains.
7Check that the indices add up to 100 × number of seasons (400 or 1200).
8Pick the option that matches your computed index for the season asked.

Quickest way: Fast route for the simple average method

When to use it: Use it when the question gives a small table of quarterly or monthly values and asks for the index of one season, with no mention of trend.

Add each season's column and divide by the number of years to get the seasonal average.
Find the grand average. A shortcut is to add all the data and divide by the total number of values.
Compute only the index the question asks for: seasonal average ÷ grand average × 100.
Use the sum check (400 or 1200) to confirm, or to find a missing index quickly.
Skip the moving-average and link-relative tables unless the question names those methods.

Common mistakes in Seasonal Variation Measurement

Taking the sum of seasonal indices as 100 instead of 100 × number of seasons.
Students remember that the average is 100 and forget how many seasons there are.
Fix: Quarterly indices add to 400 and monthly indices add to 1200. Use this to check or to find a missing index.
Dividing by the wrong number when averaging a season.
Students divide by the number of seasons instead of the number of years.
Fix: Each seasonal average is the column total divided by the number of years. The grand average is the average of the seasonal averages.
Placing the centred moving average against the wrong quarter.
A 4-quarter moving average sits between two quarters, and students forget to centre it.
Fix: Add two consecutive 4-quarter totals and divide by 8. The result sits against the quarter in the middle of those two totals.
Forgetting to adjust the ratios so they average 100.
Students stop once they have averaged the ratios to moving average.
Fix: Find the average of the seasonal average ratios. If it is not 100, multiply each by 100 divided by that average.
Starting the link relative chain from the first link relative instead of 100.
Students mix up the link relative of the first season with its chain relative.
Fix: The first chain relative is always 100. The second is (LR2 × 100) ÷ 100, and each next one uses the previous chain.
Using the simple average method on data with a strong upward trend.
It is the quickest method, so students use it everywhere.
Fix: Under a clear trend, later-season values look high only because of the trend. Use ratio to moving average or link relative, or follow the method the question names.

Worked examples

Example 1

Quarterly sales (₹ lakh) of a Pune garment shop for three years are: Year 1: 30, 36, 46, 40. Year 2: 32, 38, 48, 42. Year 3: 34, 40, 50, 44. Find the seasonal indices by the simple average method.

Show the solution

Total of Q1 = 30 + 32 + 34 = 96, so average = 96 ÷ 3 = 32.
Total of Q2 = 36 + 38 + 40 = 114, so average = 38.
Total of Q3 = 46 + 48 + 50 = 144, so average = 48.
Total of Q4 = 40 + 42 + 44 = 126, so average = 42.
Grand average = (32 + 38 + 48 + 42) ÷ 4 = 160 ÷ 4 = 40.
Q1 index = 32 ÷ 40 × 100 = 80. Q2 index = 38 ÷ 40 × 100 = 95.
Q3 index = 48 ÷ 40 × 100 = 120. Q4 index = 42 ÷ 40 × 100 = 105.
Check: 80 + 95 + 120 + 105 = 400.

Answer: Seasonal indices: Q1 = 80, Q2 = 95, Q3 = 120, Q4 = 105. Q3 is the peak quarter, 20% above the average quarter.

Example 2

A trader's quarterly sales for one year are 20, 30, 45, 25 (Q1 to Q4) and the next year's Q1 is 20. (a) Find the ratio to moving average for Q3 of the first year. (b) Over several years, the average ratios to moving average for Q1 to Q4 come to 100, 120, 150 and 130. Find the adjusted seasonal indices.

Show the solution

(a) The first 4-quarter total is 20 + 30 + 45 + 25 = 120.
The next 4-quarter total (Q2, Q3, Q4 of year 1 and Q1 of year 2) is 30 + 45 + 25 + 20 = 120.
Centred moving average for Q3 = (120 + 120) ÷ 8 = 30.
Ratio for Q3 = 45 ÷ 30 × 100 = 150.
(b) Sum of the average ratios = 100 + 120 + 150 + 130 = 500, so their average = 500 ÷ 4 = 125.
Adjustment factor = 100 ÷ 125 = 0.8.
Adjusted Q1 = 100 × 0.8 = 80. Q2 = 120 × 0.8 = 96. Q3 = 150 × 0.8 = 120. Q4 = 130 × 0.8 = 104.
Check: 80 + 96 + 120 + 104 = 400.

Answer: (a) The ratio to moving average for Q3 is 150. (b) The adjusted seasonal indices are Q1 = 80, Q2 = 96, Q3 = 120 and Q4 = 104.

Exam tips

Since all questions are MCQs, work out only the index asked for. Do not build the full table unless the method needs it.
Use the sum check (400 for quarters, 1200 for months) to test an option or to find a missing index in seconds.
Eliminate options early: indices above 100 mean above-average seasons, so a peak season cannot have an index below 100.
If the question says the data has a trend, expect ratio to moving average or link relative. If it says no trend, use the simple average.
Keep the numbers in a small table on your rough sheet. A wrong column total is the commonest way to lose these 2 marks.

Practice questions from Index Numbers and Time Series

Seasonal Variation Measurement 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 Measurement: frequently asked questions

What does a seasonal index of 120 mean?

It means that season is on average 20% above the average season of the year. An index of 100 is the average level. An index below 100 shows a below-average season.

When should I use the simple average method?

Use it when the data shows no marked upward or downward trend, or when the question asks for it. It is the quickest method, but it can mislead if a trend is present.

Why do we divide by 8 to find the centred moving average for quarterly data?

A 4-quarter total falls between two quarters. You add two consecutive 4-quarter totals so the result sits against one quarter. That sum covers 8 quarter values, so you divide by 8.

Why must the seasonal indices add up to 400 or 1200?

Indices are measured around an average of 100. With 4 seasons the total is 4 × 100 = 400, and with 12 months it is 12 × 100 = 1200. If your total is different, you have made an error or you need to adjust.