What is a moving average and why is it used?

Moving average: average of consecutive data points over a fixed period (m). It smooths out short-term fluctuations to reveal the long-term trend. For m-period moving average: add m consecutive values, divide by m, then move forward one period and repeat. Odd period (3, 5): averages fall at middle period. Even period (4): requires centring.

Samacheer Kalvi 12th Statistics Chapter 4 Exercise Solutions

Chapter 4 — Time Series Analysis

Key Concepts Quick Recap

Component	Symbol	Nature	Example
Secular Trend	T	Long-term, smooth movement	Population growth, GDP growth
Seasonal Variation	S	Regular yearly pattern	Sales during Diwali, monsoon rainfall
Cyclical Variation	C	Business cycles (2-10 years)	Economic booms and recessions
Irregular Variation	I	Unpredictable	Earthquake, pandemic, war

Models:

Additive: Y = T + S + C + I (components additive)
Multiplicative: Y = T × S × C × I (components multiplicative — more common)

I. Choose the Best Answer (MCQs)

1. Which component of time series is responsible for the regular repetition of pattern within a year? (a) Trend (b) Seasonal variation (c) Cyclical variation (d) Irregular variation

Answer(b) Seasonal variation Seasonal variation occurs within a year — tied to seasons, festivals, weather, or calendar effects. Examples: ice cream sales peak in summer, blanket sales in winter, greeting card sales before Christmas.

2. Moving averages are used primarily to: (a) Measure seasonal variation (b) Eliminate irregular fluctuations and reveal the trend (c) Predict future values exactly (d) Calculate the cyclical component

Answer(b) Eliminate irregular fluctuations and reveal the trend Moving averages smooth out short-term fluctuations (seasonal and irregular), revealing the underlying trend. Cannot be used for the first and last few data points.

3. For a 4-year moving average, to make the averages correspond to actual time periods, we need: (a) No adjustment (b) Centring (c) Seasonal adjustment (d) Logarithmic transformation

Answer(b) Centring Even-period (4-year) moving averages fall between time periods. To align them with actual periods, we take a 2-period moving average of the moving averages (centring).

4. The secular trend is the: (a) Short-term fluctuation (b) Long-term smooth movement of data over many years (c) Seasonal variation (d) Irregular variation

Answer(b) Long-term smooth movement of data over many years Trend indicates the general direction (upward, downward, or stationary) of data over a long period. Measured by least squares method, moving averages, or semi-average method.

5. Business cycles (booms and recessions) are classified as: (a) Trend (b) Seasonal variation (c) Cyclical variation (d) Irregular variation

Answer(c) Cyclical variation Cyclical variation: oscillations around the trend; period usually 2-10+ years. Related to business cycles (expansion → peak → contraction → trough → expansion). Unlike seasonal variation, cycles don't repeat at fixed intervals.

6. If the trend equation is Ŷ = 50 + 3t (t in years from base year), the trend value for t = 5 is: (a) 55 (b) 65 (c) 50 (d) 80

Answer(b) 65 Ŷ = 50 + 3(5) = 50 + 15 = 65.

7. The method of least squares for trend fitting minimises: (a) Sum of deviations (b) Sum of absolute deviations (c) Sum of squared deviations (d) Product of deviations

Answer(c) Sum of squared deviations Least squares: finds the line Ŷ = a + bt that minimises Σ(Y − Ŷ)². Normal equations: ΣY = na + bΣt and ΣtY = aΣt + bΣt².

II. Short Answer Questions

Q1. Calculate 3-year moving averages for the following data:

Answer

Year	Sales (₹ lakh)
2018	40
2019	44
2020	42
2021	46
2022	50

Solution

Year	Sales	3-Year Moving Total	3-Year Moving Average
2018	40	—	—
2019	44	40+44+42 = 126	126/3 = 42.0
2020	42	44+42+46 = 132	132/3 = 44.0
2021	46	42+46+50 = 138	138/3 = 46.0
2022	50	—	—

Q2. What are the uses of time series analysis?

Answer

1. Understanding past behaviour: Identifying trend, seasonal and cyclical patterns. 2. Forecasting: Predicting future values based on historical patterns. 3. Planning: Businesses use seasonal patterns for inventory, production, staffing. 4. Policy making: Governments analyse economic time series for budget and fiscal policy. 5. Comparison: Comparing performance across periods after removing seasonal effects.

Common Mistakes

Confusing seasonal (within year, regular) and cyclical (multi-year, irregular timing) variation.
Not centring when computing even-period moving averages.
Using Y-on-X regression equation instead of time-series trend equation.

Exam Checklist

4 components: T (trend), S (seasonal), C (cyclical), I (irregular).
Models: additive (Y = T+S+C+I) and multiplicative (Y = T×S×C×I).
Moving average: odd period → no centring needed; even period → centring required.
Least squares trend: Ŷ = a + bt; normal equations ΣY = na + bΣt; ΣtY = aΣt + bΣt².
If Σt = 0 (origin at middle): a = ΣY/n, b = ΣtY/Σt².

Frequently Asked Questions

1. Secular Trend (T): long-term movement (upward or downward) over many years. 2. Seasonal Variation (S): regular periodic fluctuations within a year (festivals, climate). 3. Cyclical Variation (C): wave-like ups and downs over several years (business cycles). 4. Irregular/Random Variation (I): unpredictable, erratic fluctuations.

Samacheer Kalvi 12th Statistics Chapter 4 Exercise Solutions

Chapter 4 — Time Series Analysis

Key Concepts Quick Recap

I. Choose the Best Answer (MCQs)

II. Short Answer Questions

Common Mistakes

Exam Checklist

Frequently Asked Questions

Other question types — same chapter

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