Q.Distinguish between seasonal variation and cyclical variation in a time series.

The random (irregular) component cannot be predicted.

A time series has four components: trend, seasonal, cyclical and random (irregular). Trend, seasonal and cyclical movements follow patterns and can be estimated, but the random component is caused by chance, unforeseen events (natural calamities, accidents, strikes) and follows no regular pattern, so it is impossible to predict.

The cyclical component of a time series is shown by the notation $C$.

The four components of a time series are usually written as: Trend $=T$, Seasonal $=S$, Cyclical $=C$, and Random/Irregular $=R$ (or $I$). Thus the cyclical component is denoted by $C$ (as in the additive model $Y=T+S+C+R$ or multiplicative model $Y=T\times S\times C\times R$).

An increase in umbrella sales caused by the monsoon is a seasonal variation, because it recurs at the same period every year — option (c).

The GSEB Class-12 Statistics Time Series chapter identifies four components: trend, seasonal, cyclical and irregular variation.

• Seasonal variation repeats regularly within a period of one year or less, driven by seasons, weather, festivals, customs, etc.

• The monsoon is a season that returns every year, and umbrella demand rises with it — a textbook seasonal effect.

In a time series, the independent variable is time, denoted $t$ — option (c).

A time series (GSEB Class-12 Statistics) is a set of observations of a variable arranged in order of time. The observed value $y_t$ changes with time, so:

• $t$ (time) is the independent variable,
• $y_t$ is the dependent variable.

Hence the independent variable is $t$.

A fall in agricultural production due to floods is an unforeseen, non-repeating event — a random (irregular) variation. Option (A).

The components of a time series are trend, seasonal, cyclical and random (irregular). Trend, seasonal and cyclical variations are regular and recurring, whereas variations caused by unpredictable, one-off events such as floods, earthquakes, wars or strikes are random (irregular) variations. A flood damaging crops is exactly such an unforeseen event.

Short-term variation is produced by the seasonal and irregular (random) components.

A time series has four components: trend, seasonal, cyclical and irregular. Trend and cyclical variations operate over long periods (several years), so they are long-term. Variations that occur within a short period (a year or less) are the:

• Seasonal variations — regular, repeating changes within a year (seasons, festivals, weather); and

• Irregular (random) variations — short, unpredictable changes due to one-off events.

A drop in production caused by a strike is a random (irregular) variation — a sudden, unpredictable, non-recurring effect.

Reasoning. The four components of a time series are trend, seasonal, cyclical and random (irregular). A strike is an unforeseeable, one-off event that does not repeat in a regular pattern, so its effect is classified as random/irregular variation.

A time series is a sequence of numerical observations of a variable arranged in order of time.

Definition. A time series is a collection of observations of a variable taken at successive, usually equally-spaced, points or periods of time and arranged in chronological order — for example, monthly sales, yearly production, or daily temperature. Time is the independent variable, and analysing the series lets us study its trend, seasonal, cyclical and random components.

The seasonal component produces regular (short-term, periodic) variations — option (c).

A time series has four components: trend (long-term), seasonal, cyclical, and irregular. The seasonal component arises from causes that operate in a fixed, repeating rhythm within a year (weather, festivals, customs), so the fluctuations it causes appear at regular intervals and in a predictable pattern.

• Long-term change is the trend, not the seasonal component.

• Irregular/zero variations come from random causes, not seasonal ones.

It is the decomposition of an observed time series into its components (trend, seasonal, cyclical, irregular) so that each can be measured and interpreted.

A time series is a set of values of a variable recorded over successive time periods. The observed values are the combined effect of several forces. Analysis of a time series means identifying and isolating these forces — its components:

1. Secular trend — the long-term general movement.
2. Seasonal variation — regular short-term (within-a-year) fluctuations.
3. Cyclical variation — medium-term ups and downs (business cycles).
4. Irregular / random variation — unpredictable, accidental fluctuations.

By separating and measuring these components, we can understand past behaviour, compare periods after removing seasonal effects, and forecast future values — the practical purpose of time-series analysis in business.