Calculate 3-yearly moving averages for the following data :
Years (t): 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
Variables (x): 3, 5, 7, 10, 12, 14, 15, 16
Calculate 3-yearly moving averages for the following data :
Years (t): 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
Variables (x): 3, 5, 7, 10, 12, 14, 15, 16
Official answer
The 3-yearly moving averages are 5, 7.33, 9.67, 12, 13.67, 15 (centred on 2014–2019).
Method: For each set of 3 consecutive years, take the 3-year total and divide by 3; the average is placed against the middle year.
| Year | x | 3-year total | Moving average |
|---|---|---|---|
| 2013 | 3 | — | — |
| 2014 | 5 | 15 | 5.00 |
| 2015 | 7 | 22 | 7.33 |
| 2016 | 10 | 29 | 9.67 |
| 2017 | 12 | 36 | 12.00 |
| 2018 | 14 | 41 | 13.67 |
| 2019 | 15 | 45 | 15.00 |
| 2020 | 16 | — | — |
Marking scheme
- 1.1 mark: correct 3-year moving totals (15, 22, 29, 36, 41, 45) placed against the correct central years.
- 2.1 mark: correct moving averages 5, 7.33, 9.67, 12, 13.67, 15 with the first and last years left blank.
- 3.Accept averages rounded to one or two decimals.
Hint
Add each block of 3 consecutive values, divide by 3, and place the result against the middle year.
Quick oral answer
For a 3-yearly moving average I add each group of three consecutive values, divide by 3, and place the result against the middle year, giving 5, 7.33, 9.67, 12, 13.67 and 15 for the years 2014 to 2019.
Common mistakes
- 1.Placing the moving average against the first year of each block instead of the central year.
- 2.Computing moving averages for 2013 and 2020, which cannot have a 3-year centred average.
- 3.Adding the wrong three values, e.g. skipping a year or overlapping the windows incorrectly.
Analysis and explanation
This question tests the moving-average method of smoothing a time series to reveal its underlying trend.
Concept: A 3-yearly moving average replaces each middle value by the average of itself and its two neighbours. This smooths out short-term fluctuations and highlights the general direction of the data.
Placement: Because the period (3) is odd, each moving average is written directly against the central year. The first year (2013) and last year (2020) have no moving average, as they lack a full window on both sides.
Exam trap: Always centre an odd-order moving average on the middle year — do not shift it. Keep the 3-year totals aligned with the correct central year, and round consistently (here to 2 decimals).
Application: Businesses use moving averages to smooth monthly sales, and analysts use them on stock prices to identify trends free of daily noise.
Interesting facts
Odd-order moving averages (3, 5, 7 years) need no 'centring' step, but even-order ones (like 4-yearly) require an extra two-item moving average to align with a year.
Moving averages are the backbone of technical stock-market analysis, where the '50-day' and '200-day' moving averages are watched closely by traders.
Question details
Q24 of 44 questions
- Board
- CBSE
- Class
- Class 12
- Subject
- Applied Mathematics
- Year
- 2026
- Section
- Section B
- Marks
- 2
- Type
- Very Short Answer
- Chapter
- Time Series and Forecasting
- Topic
- Time Series — 3-Yearly Moving Averages
- Bloom's level
- apply
- Syllabus topic
- Unit: Index Numbers and Time-Based Data — Moving Averages in Time Series
- Author
- CBSE
- Reviewed by
- Padhantu Academic Team
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Frequently asked questions
Why do the first and last years have no moving average?
A 3-yearly moving average is centred on the middle of a block of three consecutive years. The first year (2013) has no year before it and the last year (2020) has no year after it, so neither can sit at the centre of a complete 3-year window — their moving averages are undefined.
Why do we use moving averages instead of the raw data?
Raw time-series data contains short-term ups and downs (seasonal or random noise). Averaging over a moving window smooths these fluctuations, making the long-term trend easier to see and to use for forecasting future values.