Question 391 markSection E

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=year−2021X = \text{year} - 2021, so X=−3,−2,−1,0,1,2,3X = -3, -2, -1, 0, 1, 2, 3.


Compute the sums:


  • ΣY=80+90+92+83+94+99+92=630\Sigma Y = 80 + 90 + 92 + 83 + 94 + 99 + 92 = 630, so a=yˉ=630/7=90a = \bar{y} = 630/7 = 90.
  • ΣXY=(−3)(80)+(−2)(90)+(−1)(92)+0(83)+1(94)+2(99)+3(92)=−512+568=56\Sigma XY = (-3)(80) + (-2)(90) + (-1)(92) + 0(83) + 1(94) + 2(99) + 3(92) = -512 + 568 = 56.
  • ΣX2=9+4+1+0+1+4+9=28\Sigma X^2 = 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28.
  • b=ΣXY/ΣX2=56/28=2b = \Sigma XY / \Sigma X^2 = 56 / 28 = 2.

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

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

Marking scheme

  1. 1.0.5 mark: correct origin shift X=year−2021X = \text{year} - 2021 and the values of a=90a = 90.
  2. 2.0.5 mark: correct slope b=ΣXY/ΣX2=56/28=2b = \Sigma XY / \Sigma X^2 = 56/28 = 2 and final trend line Y=90+2XY = 90 + 2X.
  3. 3.Accept 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=year−2021X = \text{year} - 2021 (so X=−3…3X = -3 \dots 3). Then the trend line Y=a+bXY = a + bX has a=yˉ=ΣY/7a = \bar{y} = \Sigma Y / 7 and b=ΣXY/ΣX2b = \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+2XY = 90 + 2X where X=year−2021X = \text{year} - 2021.

Common mistakes

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

Analysis and explanation

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


Concept


  • Because ΣX=0\Sigma X = 0 when the origin is the middle year, the normal equations decouple, giving the simple formulae a=ΣY/na = \Sigma Y / n and b=ΣXY/ΣX2b = \Sigma XY / \Sigma X^2. This origin-shift trick is the standard exam technique for an odd number of years.
  • The positive slope b=2b = 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 ΣX≠0\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=90Y = 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.

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 ΣX=0\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.

Question details

Q39 of 44 questions

Board
CBSE
Class
Class 12
Subject
Applied Mathematics
Year
2026
Section
Section E
Marks
1
Chapter
Time Series Analysis
Topic
Trend Line by Least Squares
Bloom's level
apply
Syllabus topic
Unit 6: Index Numbers and Time-based Data — Time Series: Trend by Method of Least Squares
Author
CBSE
Reviewed by
Padhantu Academic Team

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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=year−2021X = \text{year} - 2021 makes the X-values symmetric (−3 to 3) so that ΣX=0\Sigma X = 0. This simplifies the least-squares formulae to a=ΣY/na = \Sigma Y / n and b=ΣXY/ΣX2b = \Sigma XY / \Sigma X^2, reducing both effort and error.

What does the slope of 2 tell us?

The slope b=2b = 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.