Q41

2 marksSection E

(a) Find the sum of the differences between the actual sales and the trend values (for 2018 - 2024).

OR

(b) What are the expected sales for the year 2025 ?

Time Series

Time Series — Trend by Least Squares

Official Answer

Both alternatives use the least-squares trend line fitted to the seven years of sales.

Setting up the trend line (needed for both parts):

Take deviations $u = \text{year} - 2021$ (middle year), so $u = -3, -2, -1, 0, 1, 2, 3$ .
$\Sigma y = 630$ , so mean $a = 630/7 = 90$ .
$\Sigma uy = 56$ , $\Sigma u^2 = 28$ , so slope $b = 56/28 = 2$ .
Trend line: $\hat{y} = 90 + 2u$ (in Rs thousands).

(a) Sum of differences between actual and trend values:

By the property of the method of least squares, $\Sigma(y - \hat{y}) = 0$ .
Trend values are $84, 86, 88, 90, 92, 94, 96$ (sum = $630 = \Sigma y$ ), confirming the difference sum is 0.

(b) Expected sales for 2025:

For 2025, $u = 2025 - 2021 = 4$ .
$\hat{y} = 90 + 2(4) = 98$ (Rs thousands) = Rs 98,000.

time seriesleast squares trendtrend lineu = year - 2021sum of residuals zeroforecast 2025Rs 98,000slope b = 2

Marking Scheme

11 mark: correct set-up of the least-squares trend line $\hat{y} = 90 + 2u$ using $u = \text{year} - 2021$ ( $a = 90$ , $b = 2$ ).
2(a) 1 mark: stating/showing $\Sigma(\text{actual} - \text{trend}) = 0$ (accept the least-squares property with justification).
3(b) 1 mark: substituting $u = 4$ to obtain expected sales = 98 (Rs thousands) i.e. Rs 98,000.
4Full 2 marks for whichever alternative (a) or (b) is attempted correctly.

Hint

Fit $\hat{y} = a + bu$ with $u = \text{year} - 2021$ : a = mean of y, $b = \Sigma uy / \Sigma u^2$ . For (a) recall $\Sigma(y - \hat{y}) = 0$ ; for (b) put $u = 4$ .

Quick Oral Answer

With the origin at the middle year 2021, the trend line is $\hat{y} = 90 + 2u$ ; the residuals always add up to zero, and putting $u = 4$ forecasts 2025 sales of 98 thousand, i.e. Rs 98,000.

Analysis & Explanation

This part completes the least-squares trend analysis begun earlier in the case study, and CBSE deliberately tests two contrasting ideas within the same OR.

Concept:

The straight-line trend $\hat{y} = a + bu$ is fitted so that the sum of squared residuals is minimised; a direct consequence is that the residuals themselves sum to zero.
Using the middle year as origin makes $\Sigma u = 0$ , which decouples the normal equations into $a = \Sigma y / n$ and $b = \Sigma uy / \Sigma u^2$ .

Exam trap:

In part (a) many students waste time computing each residual individually. The examiner is rewarding the insight that $\Sigma(y - \hat{y}) = 0$ for any correctly fitted least-squares line — you can state this and back it with one supporting line.
In part (b) the common slip is using $u = 5$ or the raw year 2025 in the equation. The origin is 2021, so 2025 corresponds to $u = 4$ .

Real-world:

Trend forecasting like this is exactly how an FMCG factory such as Mr. Nitin's projects next year's demand to plan raw-material purchase and production capacity.

Common Mistakes

1Using $u = 5$ or the raw year 2025 instead of $u = 4$ in the trend equation for part (b).
2In part (a), laboriously computing all seven residuals and making an arithmetic slip, instead of citing that $\Sigma(y - \hat{y}) = 0$ for a least-squares fit.
3Taking the origin at 2018 (giving $\Sigma u \ne 0$ ) yet still using the short-cut formulas $a = \bar{y}$ and $b = \Sigma uy / \Sigma u^2$ , which are only valid when $\Sigma u = 0$ .

Interesting Facts

The zero-sum-of-residuals result is one of the 'normal equation' identities that Carl Friedrich Gauss and Adrien-Marie Legendre established around 1805-1809 when the method of least squares was first published.

Because the fitted line always passes through the point $(\bar{u}, \bar{y})$ , the trend value for the central year 2021 equals the mean sales of Rs 90 thousand exactly.

A positive slope $b = 2$ means Mr. Nitin's sales grow on average by Rs 2,000 per year — a steady upward trend despite the dip in 2021.

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Frequently Asked Questions

Why is the sum of the differences between actual and trend values always zero?

The least-squares line is obtained by minimising $\Sigma(y - \hat{y})^2$ . Differentiating with respect to the intercept a and setting it to zero gives the first normal equation $\Sigma(y - \hat{y}) = 0$ . So for any correctly fitted straight-line trend, the positive and negative residuals cancel exactly and their sum is zero.

How do I choose the value of u for the forecast year 2025?

The origin here is the middle year 2021, so $u = \text{year} - 2021$ . For 2025, $u = 2025 - 2021 = 4$ . Substituting into $\hat{y} = 90 + 2u$ gives $90 + 8 = 98$ , i.e. Rs 98,000.

Can I take the origin at 2018 instead of the middle year?

Yes, but then $\Sigma u \ne 0$ and you must solve the full normal equations $\Sigma y = na + b\Sigma u$ and $\Sigma uy = a\Sigma u + b\Sigma u^2$ . Choosing the middle year makes $\Sigma u = 0$ , which simplifies the work to $a = \bar{y}$ and $b = \Sigma uy / \Sigma u^2$ , so it is strongly preferred.