Mathematics and Statistics · Ch 12 — Time Series

Meaning and Uses of a Time Series

Many quantities in business and economics are recorded at regular intervals of time — the annual sales of a firm, the monthly output of a factory, the quarterly profit of a company, the daily closing price of a share. A time series is a set of observations of a variable arranged in the chronological order in which they occur. Formally, if a variable $y$ is observed at successive time points $t_1, t_2, \ldots, t_n$ , the ordered pairs $(t_1, y_1), (t_2, y_2), \ldots, (t_n, y_n)$ form a time series.

The distinguishing feature is that time is the independent variable: the value of $y$ is studied as it changes over time. This chapter of the Maharashtra Std XII (SYJC) commerce Mathematics and Statistics course explains what a time series is made of and then develops three ways of measuring its long-term movement (the trend). These are the same time-series principles set out in the wider national mathematics-and-statistics curriculum on which this syllabus draws.

Why study a time series?

It describes the past behaviour of the variable compactly.
It helps us understand the present by separating a genuine long-term movement from short, temporary ups and downs.
Above all it lets us forecast the future — a firm that knows the trend of its sales can plan production, staffing and stock.
It allows comparison of one series with another (e.g. two firms' sales) on a common time footing.

Note

Time series vs. an ordinary frequency distribution

In a frequency distribution the order of the classes carries no special meaning. In a time series the order is everything: the observations must stay in their exact chronological sequence, because it is the movement through time that we analyse.

Definition 1Time series

A set of observations of a variable recorded at successive, equally spaced points of time and arranged in chronological order, i.e. the pairs $(t_i, y_i)$ with time as the independent variable.

Definition 2Trend (secular trend)

The smooth, long-term movement of a time series — its general direction over a long period — which is the component the methods of this chapter aim to measure.

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