Measurement of Trend: The Method of Least Squares

Meaning and Principle

The method of least squares fits a straight-line trend

where $Y_c$ is the estimated (trend) value, $a$ is the value of $Y_c$ at $X=0$, and $b$ is the trend's slope — the average change in $Y$ per unit of $X$. The line is chosen so that the sum of the squares of the deviations of the actual values from the trend values, $\Sigma (Y - Y_c)^2$, is the minimum possible — hence "least squares".

Coding the time variable ($X$)

To keep the arithmetic simple, years are coded as deviations from a convenient origin so that $\Sigma X = 0$:

• Odd number of years: take the middle year as the origin ($X=0$) and code the years on either side as $-2,-1,0,1,2,\ldots$ — each unit of $X$ equals one year.
• Even number of years: there is no single middle year, so the two central years are coded $-1$ and $+1$, and the rest as $-3,-1,1,3,\ldots$ (the "$2X$" method) — here each unit of $X$ equals half a year.

Normal equations

With $\Sigma X = 0$, the two normal equations of least squares reduce to:

where $n$ is the number of years in the series.

A built-in check

A genuinely useful property of the least squares line is that the trend values it produces always satisfy

Recomputing $\Sigma Y_c$ after fitting the line and confirming it equals the original $\Sigma Y$ is therefore a free, mechanical cross-check on the whole calculation — exactly the kind of independent second check a Gujarat board Std-12 statistics answer should show, not just the final equation.

Why it is preferred over moving averages …