Business Mathematics and Statistics · Ch 9 — Applied Statistics (Time Series, Index Numbers, Statistical Quality Control)

Measuring Trend: Method of Least Squares

The method of least squares fits a mathematical trend line to a time series so that trend can be measured (and, importantly, extended to forecast future years) without losing values at either end the way the moving average method does. For a straight-line trend,

Y = a + bX

where $Y$ is the trend value, $X$ represents time (the year), and $a$ , $b$ are constants found by solving the two normal equations:

\sum Y = na + b\sum X \qquad \qquad \sum XY = a\sum X + b\sum X^2

The arithmetic is made simple by a standard trick: code the years so that the middle year becomes $X=0$ and the years on either side become $-1, -2, \ldots$ and $1, 2, \ldots$ . This makes $\sum X = 0$ , which collapses the normal equations to

a = \dfrac{\sum Y}{n} \qquad \qquad b = \dfrac{\sum XY}{\sum X^2}

Worked Example. A shop's sales (₹ lakh) for five years were:

Year	2017	2018	2019	2020	2021
Sales ( $Y$ )	18	20	22	28	32

Coding the middle year (2019) as $X=0$ :

Year	$X$	$Y$	$XY$	$X^2$
2017	$-2$	18	$-36$	4
2018	$-1$	20	$-20$	1
2019	$0$	22	$0$	0
2020	$1$	28	$28$	1
2021	$2$	32	$64$	4
Total	$\sum X = 0$	$\sum Y = 120$	$\sum XY = 36$	$\sum X^2 = 10$

a = \dfrac{120}{5} = 24 \qquad \qquad b = \dfrac{36}{10} = 3.6

So the fitted trend line is $Y = 24 + 3.6X$ . Substituting each $X$ back gives the trend values:

| Year | 2017 | 2018 | 2019 | 2020 | 2021 |

|---|---|---|---|---|---| …

Definition 8Method of Least Squares

A method of fitting a trend line to a time series such that the sum of the squares of the deviations of the actual values from the trend value …

Definition 9Normal Equations

The pair of simultaneous equations, derived from the least-squares condition, that are solved to find the constants $a$ and $b$ of a fitted str …

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