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Statistics Online Tutor Help Illustrations and Solutions
Illustration 1.
Convert the following annual trend equation into a quarterly one:
Yc = 32 + 4.8X
Solution:
By the formula of conversion of an annual trend into a quarterly, one we have,
Yc =
𝑎
4
+
𝑏𝑋
4 ×12
Substituting the respective values in the above we get,
Yc =
32
4
+
4.8𝑋
48
= 8 + 0.1 X
Where, X unit = 1 quarter, and Y unit = quarterly value of a variable.

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Illustration 2.
Given the trend equation, Yc = 240 – 1.8X, where X unit = 6 months, and Y unit = annual
production. Convert the equation into a monthly trend.
Solution:
Here, Y unit = annual production, and X unit is 6 months, So to convert the given equation
into a monthly trend, we have to divide a by 12, but bX by 12 × 6. Thus the required
equation will be :
Yc =
𝑎
12
-
𝑏𝑋
12 ×6
= 20 – 0.025X
Where, X unit = 1 month, and Y unit = monthly production.

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Illustration 3.
Convert the following trend equation into an annual basis:
Yc = 50 + 0.5X
Where, the point of origin is Jan 2004, X unit = one month and Y unit = monthly sales (in
’000$)
Solution
Here, the origin of the equation is 15th ’04 (i.e. the midpoint of January) which needs to be
shifted first to the 1st July ’04 (i.e. the midpoint of the yr. ’04) by 5.5 months forward.
Hence, K = 5.5
By the formula of shifting a trend we have,
Yc = a + b (X + K)
= 50 + 0.5 (X + 5.5)
= 50 + 0.5X + 2.75
= 52.75 + 0.5X
Now, by the formula of conversion of a monthly equation into an annual trend we have,
Yc = a × 12 × bX (12 × 12)
Substituting the respective values in the above we get,
Yc = 52.75 × 12 + 0.5 × (144)

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= 633 + 72X
Where, the point of origin = 2004, X unit = 1 year and Y unit = annual sales (in ’000 Rs).
Note. The above problems required both shifting of the trend origin, and conversion of the
trend equation. Since, the conversion was to be made from a lower time basis (i.e. monthly)
to a higher time basis (i.e. annual) shifting, was made first and the conversion at the last.
Illustration 4.
The equation for yearly sales (in ’000$) for a commodit6y with the year, 2003 as origin is
Yc =91.6 + 28.8X. Determine the trend equation to give the monthly trend values with Jan
’04. Since, the conversion is required to be made from a higher periodical base (annual) to
a lower periodical base (monthly), it is necessary to attempt on the conversion first, and
then on shifting of the trend origin as follows :
(ii) Conversion of the trend
By the formula of conversion of an annual trend into a monthly one we have,
Yc =
𝑎
12
+
𝑏𝑋
12 ×12
Substituting the respective values, in the above we get,
Yc =
91 .6
12
+
28.8𝑋
12 ×12
= 7.63 + 0.2X
Thus, Yc = 7.63 + 0.2 Y

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When X unit = 1 month, and Y unit = monthly sales
(ii) Shifting of the Trend Origin
Here, it is required to shift forward the trend origin by 6.5 months i.e. from 1st July, 03 (the
midpoint of the year, 03) to the 15th Jan, 04 (the midpoint of the month).
Thus, K = 5.5
By the formula of shifting a trend we have,
Yc = a + b (X + K)
Substitution the respective values in the above we get,
Yc = 7.63 + 0.2 (X + 6.5)
= 7.63 + 0.2X + 1.3
= 8.93 + 0.2 Y
Yc = 8.93 + 0.2 X
Where, the point of origin is Jan ’03, X unit = 1 month, and Y unit monthly sales.
(iii) Calculation of the trend value for March ’04
The time deviation of March ’04 from the trend origin Jan ’04, or X = 2
Thus, when X = 2, Yc = 8.93 + 0.2 (2)
= 8.93 + 0.4 = 9.33
Hence, the trend value for March ’04 will be 9.33.

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Illustration 5.
Convert the annual trend equation Yc = 48 + 1.2 X into monthly trend equation, and shift
the origin from 2003 to Jan 1, 2004.
Solution
Here, conversion is to be made first as it is required to be made from a higher period
(annual) to a lower period (monthly) basis.
(i)Conversion of the trend equation
By the formula of conversion of an annual trend to a monthly one we have,
Yc =
𝑎
12
+
𝑏𝑋
12×12

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