Question 412 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=year−2021u = \text{year} - 2021 (middle year), so u=−3,−2,−1,0,1,2,3u = -3, -2, -1, 0, 1, 2, 3.
  • Σy=630\Sigma y = 630, so mean a=630/7=90a = 630/7 = 90.
  • Σuy=56\Sigma uy = 56, Σu2=28\Sigma u^2 = 28, so slope b=56/28=2b = 56/28 = 2.
  • Trend line: y^=90+2u\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, Σ(y−y^)=0\Sigma(y - \hat{y}) = 0.
  • Trend values are 84,86,88,90,92,94,9684, 86, 88, 90, 92, 94, 96 (sum = 630=Σy630 = \Sigma y), confirming the difference sum is 0.

(b) Expected sales for 2025:


  • For 2025, u=2025−2021=4u = 2025 - 2021 = 4.
  • y^=90+2(4)=98\hat{y} = 90 + 2(4) = 98 (Rs thousands) = Rs 98,000.
Answer keywords:time seriesleast squares trendtrend lineu = year - 2021sum of residuals zeroforecast 2025Rs 98,000slope b = 2

Marking scheme

  1. 1.1 mark: correct set-up of the least-squares trend line y^=90+2u\hat{y} = 90 + 2u using u=year−2021u = \text{year} - 2021 (a=90a = 90, b=2b = 2).
  2. 2.(a) 1 mark: stating/showing Σ(actual−trend)=0\Sigma(\text{actual} - \text{trend}) = 0 (accept the least-squares property with justification).
  3. 3.(b) 1 mark: substituting u=4u = 4 to obtain expected sales = 98 (Rs thousands) i.e. Rs 98,000.
  4. 4.Full 2 marks for whichever alternative (a) or (b) is attempted correctly.

Hint

Fit y^=a+bu\hat{y} = a + bu with u=year−2021u = \text{year} - 2021: a = mean of y, b=Σuy/Σu2b = \Sigma uy / \Sigma u^2. For (a) recall Σ(y−y^)=0\Sigma(y - \hat{y}) = 0; for (b) put u=4u = 4.

Quick oral answer

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

Common mistakes

  1. 1.Using u=5u = 5 or the raw year 2025 instead of u=4u = 4 in the trend equation for part (b).
  2. 2.In part (a), laboriously computing all seven residuals and making an arithmetic slip, instead of citing that Σ(y−y^)=0\Sigma(y - \hat{y}) = 0 for a least-squares fit.
  3. 3.Taking the origin at 2018 (giving Σu≠0\Sigma u \ne 0) yet still using the short-cut formulas a=yˉa = \bar{y} and b=Σuy/Σu2b = \Sigma uy / \Sigma u^2, which are only valid when Σu=0\Sigma u = 0.

Analysis and 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 y^=a+bu\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 Σu=0\Sigma u = 0, which decouples the normal equations into a=Σy/na = \Sigma y / n and b=Σuy/Σu2b = \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 Σ(y−y^)=0\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=5u = 5 or the raw year 2025 in the equation. The origin is 2021, so 2025 corresponds to u=4u = 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.

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 (uˉ,yˉ)(\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=2b = 2 means Mr. Nitin's sales grow on average by Rs 2,000 per year — a steady upward trend despite the dip in 2021.

Question details

Q41 of 44 questions

Board
CBSE
Class
Class 12
Subject
Applied Mathematics
Year
2026
Section
Section E
Marks
2
Chapter
Time Series
Topic
Time Series — Trend by Least Squares
Bloom's level
apply
Syllabus topic
Unit 6: Index Numbers and Time Based Data — Time Series: Fitting a Straight-Line Trend by the Method of Least Squares and Forecasting
Author
CBSE
Reviewed by
Padhantu Academic Team

Spotted a mistake or something unclear?

Tell us and we will fix it.

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 Σ(y−y^)2\Sigma(y - \hat{y})^2. Differentiating with respect to the intercept a and setting it to zero gives the first normal equation Σ(y−y^)=0\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=year−2021u = \text{year} - 2021. For 2025, u=2025−2021=4u = 2025 - 2021 = 4. Substituting into y^=90+2u\hat{y} = 90 + 2u gives 90+8=9890 + 8 = 98, i.e. Rs 98,000.

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

Yes, but then Σu≠0\Sigma u \ne 0 and you must solve the full normal equations Σy=na+bΣu\Sigma y = na + b\Sigma u and Σuy=aΣu+bΣu2\Sigma uy = a\Sigma u + b\Sigma u^2. Choosing the middle year makes Σu=0\Sigma u = 0, which simplifies the work to a=yˉa = \bar{y} and b=Σuy/Σu2b = \Sigma uy / \Sigma u^2, so it is strongly preferred.