A time series is a set of observations arranged:

Chronologically. According to time.

Changes within a year that repeat every year are:

Seasonal variations. E.g. festival or monsoon effects.

Variations caused by floods, strikes or wars are:

Irregular. They are unpredictable.

Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 2 Chapter 4: Time Series

Short notes, key concepts, quick revision and practice MCQs with answers

A time series records values of a variable at regular intervals of time. The chapter explains the uses and components of a time series, the additive and multiplicative models and three methods of measuring secular trend: the graphical method, moving averages and least squares.

Key concepts: Time Series

Time series: A set of observations made at regular intervals of time and arranged chronologically, e.g. yearly sales or monthly rainfall. It helps to understand past behaviour, forecast and compare.
Secular trend: The smooth, regular long-term movement of a time series, rising, falling or constant.
Seasonal variation: Patterns of change within a year that repeat every year, due to climate, festivals, customs and habits (e.g. umbrella sales in the monsoon).
Cyclical variation: Long-term oscillatory movement over more than a year; a business cycle has four phases: prosperity, decline, depression and recovery.
Irregular variation: Random changes caused by unforeseen events such as floods, famines, earthquakes, strikes or wars.
Models: Additive: Xₜ = Tₜ + Sₜ + Cₜ + Iₜ. Multiplicative: Xₜ = Tₜ × Sₜ × Cₜ × Iₜ.
Moving averages: A k-period moving average is the series of means of k successive values, placed against the middle period. For an even k the averages are centred by averaging pairs of them. Trend values are lost at the beginning and end.
Least squares: Fit Xₜ = a + bu using the normal equations ΣX = na + bΣu and ΣuX = aΣu + bΣu². Code time as u = t − middle t (n odd) or u = 2(t − mean of the two middle t) (n even) so that Σu = 0; then a = ΣX/n and b = ΣuX/Σu².

Quick revision: important points

Four components: trend, seasonal, cyclical, irregular.
Three methods of trend: graphical, moving averages, least squares.
With Σu = 0: a = ΣX/n, b = ΣuX/Σu².
Moving average of 3 periods of a, b, c is (a + b + c)/3, placed against b.
Least squares gives trend values for every period and can be used for forecasting.

Practice MCQs with answers

A time series is a set of observations arranged:
1. Alphabetically
2. In order of size
3. Chronologically
4. Randomly
Show answer
Answer: C. Chronologically
According to time.
The smooth long-term movement of a time series is called:
1. Seasonal variation
2. Secular trend
3. Cyclical variation
4. Irregular variation
Show answer
Answer: B. Secular trend
Definition.
Changes within a year that repeat every year are:
1. Seasonal variations
2. Secular trend
3. Cyclical variations
4. Irregular variations
Show answer
Answer: A. Seasonal variations
E.g. festival or monsoon effects.
Oscillatory movements lasting more than a year are:
1. Seasonal
2. Cyclical
3. Irregular
4. Trend
Show answer
Answer: B. Cyclical
Business cycles.
Variations caused by floods, strikes or wars are:
1. Seasonal
2. Cyclical
3. Irregular
4. Secular
Show answer
Answer: C. Irregular
They are unpredictable.

All Class 12 Maths and Statistics chapters

Class 12 Maths and Statistics notes: all chapters