Obtain the trend line to the given data.

Time Series Analysis

Trend Line by Least Squares

Official Answer

With 7 years, shift the origin to the middle year 2021 and let $X = \text{year} - 2021$ , so $X = -3, -2, -1, 0, 1, 2, 3$ .

Compute the sums:

$\Sigma Y = 80 + 90 + 92 + 83 + 94 + 99 + 92 = 630$ , so $a = \bar{y} = 630/7 = 90$ .
$\Sigma XY = (-3)(80) + (-2)(90) + (-1)(92) + 0(83) + 1(94) + 2(99) + 3(92) = -512 + 568 = 56$ .
$\Sigma X^2 = 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28$ .
$b = \Sigma XY / \Sigma X^2 = 56 / 28 = 2$ .

Trend line: $Y = 90 + 2X$ , where $X = \text{year} - 2021$ (Y in Rs thousands).

trend lineleast squaresmethod of least squaresorigin shiftslope b=2Y=90+2Xtime seriesline of best fit

Marking Scheme

10.5 mark: correct origin shift $X = \text{year} - 2021$ and the values of $a = 90$ .
20.5 mark: correct slope $b = \Sigma XY / \Sigma X^2 = 56/28 = 2$ and final trend line $Y = 90 + 2X$ .
3Accept an equivalent equation using a different consistent coding of X.

Hint

There are 7 years (odd), so take the middle year 2021 as origin and code $X = \text{year} - 2021$ (so $X = -3 \dots 3$ ). Then the trend line $Y = a + bX$ has $a = \bar{y} = \Sigma Y / 7$ and $b = \Sigma XY / \Sigma X^2$ .

Quick Oral Answer

Taking 2021 as the origin, the sales average 90 thousand and rise by 2 thousand each year, so the least-squares trend line is $Y = 90 + 2X$ where $X = \text{year} - 2021$ .

Analysis & Explanation

This part applies the least-squares method to fit a straight-line trend to time-series data.

Concept

Because $\Sigma X = 0$ when the origin is the middle year, the normal equations decouple, giving the simple formulae $a = \Sigma Y / n$ and $b = \Sigma XY / \Sigma X^2$ . This origin-shift trick is the standard exam technique for an odd number of years.
The positive slope $b = 2$ means sales rise on average by Rs 2,000 per year.

Exam trap

Forgetting to shift the origin (or shifting to the wrong year) makes $\Sigma X \ne 0$ and forces the full two-equation solution — more work and more error.
Mislabelling units: sales are in Rs thousands, so $Y = 90$ means Rs 90,000.

Real-world

Trend lines are the foundation of business forecasting; here Mr. Nitin can project future soap sales, a direct commercial use of the least-squares line.

Common Mistakes

1Not shifting the origin to the middle year, leaving $\Sigma X \ne 0$ and complicating the normal equations.
2Arithmetic errors in $\Sigma XY$ (the mixed positive and negative products are easy to slip on).
3Writing the trend line in terms of the actual year without stating $X = \text{year} - 2021$ , causing confusion in later parts.

Interesting Facts

The method of least squares was published by Adrien-Marie Legendre in 1805 and used by Gauss to predict the orbit of the asteroid Ceres.

When the number of time points is odd, coding the middle point as zero always makes $\Sigma X = 0$ , which is why exam problems favour odd-length series.

A slope of 2 (thousand rupees per year) means the trend predicts steady annual growth regardless of the year-to-year zig-zag in the actual data.

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Frequently Asked Questions

Why take 2021 as the origin instead of 2018?

2021 is the middle of the seven years. Coding $X = \text{year} - 2021$ makes the X-values symmetric (−3 to 3) so that $\Sigma X = 0$ . This simplifies the least-squares formulae to $a = \Sigma Y / n$ and $b = \Sigma XY / \Sigma X^2$ , reducing both effort and error.

What does the slope of 2 tell us?

The slope $b = 2$ means that, according to the fitted trend, soap sales increase on average by Rs 2,000 (2 thousand) per year over the period, smoothing out the year-to-year fluctuations.