Following are the methods by which we can measure the trend.
(i) Freehand or Graphic Method.
(ii) Method of Semi-Averages.
(iii) Method of Moving Averages.
(iv) Method of Least Squares.

**Measurements
of Trends**

Following are the
methods by which we can measure the trend.

(i) Freehand or Graphic
Method.

(ii) Method of
Semi-Averages.

(iii) Method of Moving
Averages.

(iv) Method of Least
Squares.

It is the simplest and
most flexible method for estimating a trend. We will see the working procedure
of this method.

Procedure:

(a) Plot the time series
data on a graph.

(b) Draw a freehand
smooth curve joining the plotted points.

(c) Examine the
direction of the trend based on the plotted points.

(d) Draw a straight line
which will pass through the maximum number of plotted points.

**Example
9.1**

Fit a trend line by the
method of freehand method for the given data.

*Solution:*

**Note**

The trend drawn by the
freehand method can be extended to predict the future values of the given data.
However, this method is subjective in nature, predictions obtained by this
method depends on the personal bias and judgement of the investigator handling
the data.

In this method, the
semi-averages are calculated to find out the trend values. Now, we will see the
working procedure of this method.

Procedure:

(i) The data is divided
into two equal parts. In case of odd number of data, two equal parts can be
made simply by omitting the middle year.

(ii) The average of each
part is calculated, thus we get two points.

(iii) Each point is
plotted at the mid-point (year) of each half.

(iv) Join the two points
by a straight line.

(v) The straight line
can be extended on either side.

(vi) This line is the
trend line by the methods of semi-averages.

**Example
9.2**

Fit a trend line by the
method of semi-averages for the given data.

*Solution:*

Since the number of
years is odd(seven), we will leave the middle year’s production value and
obtain the averages of first three years and last three years.

**Example
9.3**

Fit a trend line by the
method of semi-averages for the given data.

*Solution:*

Since the number of
years is even(eight), we can equally divide the given data it two equal parts
and obtain the averages of first four years and last four years.

**Note**

(i) The future values
can be predicted.

(ii) The trend values
obtained by this method and the predicted values are not precise.

Moving Averages Method
gives a trend with a fair degree of accuracy. In this method, we take
arithmetic mean of the values for a certain time span. The time span can be
three-years, four -years, five- years and so on depending on the data set and
our interest. We will see the working procedure of this method.

Procedure:

(i) Decide the period of
moving averages (three- years, four -years).

(ii) In case of odd
years, averages can be obtained by calculating,

(iii) If the moving
average is an odd number, there is no problem of centering it, the average
value will be centered besides the second year for every three years.

(iv) In case of even
years, averages can be obtained by calculating,

(v) If the moving
average is an even number, the average of first four values will be
placed between 2 ^{nd} and 3^{rd} year, similarly the average
of the second four values will be placed between 3^{rd} and 4^{th}
year. These two averages will be again averaged and placed in the 3^{rd}
year. This continues for rest of the values in the problem. This process is
called as centering of the averages.

**Example
9.4**

Calculate three-yearly moving averages of number of students studying in a higher secondary school in a particular village from the following data.

*Solution:*

Computation of three- yearly moving averages.

**Example
9.5**

Calculate four-yearly
moving averages of number of students studying in a higher secondary school in
a particular city from the following data.

*Solution:*

Computation of four-
yearly moving averages.

**Note**

The calculated
4-yearly centered moving average belongs to the particular year present in that
row eg; 132.37 belongs to the year 2003.

The line of best fit is
a line from which the sum of the deviations of various points is zero. This is
the best method for obtaining the trend values. It gives a convenient basis for
calculating the line of best fit for the time series. It is a mathematical
method for measuring trend. Further the sum of the squares of these deviations
would be least when compared with other fitting methods. So, this method is
known as the Method of Least Squares and satisfies the following conditions:

(i) The sum of the
deviations of the actual values of *Y* and *Ŷ* (estimated value of *Y*)
is Zero. that is Σ(*Y–Ŷ*) = 0.

(ii) The sum of squares
of the deviations of the actual values of *Y* and *Ŷ* (estimated
value of *Y*) is least. that is Σ(*Y*–*Ŷ*)^{2} is least
;

Procedure:

(i) The straight
line trend is represented by the equation Y = a + bX …(1)

where Y is the
actual value, X is time, a, b are constants

(ii) The constants
‘a’ and ‘b’ are estimated by solving the following two normal

Equations ΣY = n a + b ΣX ...(2)

ΣXY = a ΣX + b ΣX^{2} ...(3)

Where ‘n’ = number of
years given in the data.

(iii) By taking the
mid-point of the time as the origin, we get *ΣX* = 0

(iv) When *ΣX* = 0
, the two normal equations reduces to

The constant ‘*a*’
gives the mean of *Y* and ‘*b*’ gives the rate of change (slope).

(v) By substituting the
values of ‘*a*’ and ‘*b*’ in the trend equation (1), we get the Line of
Best Fit.

**Example
9.6**

Given below are the data
relating to the production of sugarcane in a district.

Fit a straight line
trend by the method of least squares and tabulate the trend values.

*Solution:*

Computation of trend
values by the method of least squares (ODD Years).

Therefore, the required
equation of the straight line trend is given by

Y = a+bX;

Y = 45.143 + 1.036
(x-2003)

The trend values can be
obtained by

When X = 2000 , Yt =
45.143 + 1.036(2000–2003) = 42.035

When X = 2001, Yt =
45.143 + 1.036(2001–2003) = 43.071,

similarly other values
can be obtained.

**Example
9.7**

Given below are the data
relating to the sales of a product in a district.

Fit a straight line
trend by the method of least squares and tabulate the trend values.

*Solution:*

Computation of trend
values by the method of least squares.

In case of EVEN number
of years, let us consider

similarly other values can be
obtained.

**Note**

(i) Future forecasts
made by this method are based only on trend values.

(ii) The predicted
values are more reliable in this method than the other methods.

Seasonal Variations can
be measured by the method of simple average. The data should be available in
season wise likely weeks, months, quarters.

This is the simplest and
easiest method for studying Seasonal Variations. The procedure of simple
average method is outlined below.

**Procedure:**

(i) Arrange the data by
months, quarters or years according to the data given.

(ii) Find the sum of the
each months, quarters or year.

(iii) Find the average
of each months, quarters or year.

(iv) Find the average of
averages, and it is called Grand Average (G)

(v) Compute Seasonal
Index for every season (i.e) months, quarters or year is given by

(vi) If the data is given in months

Similarly we can calculate SI for all other months.

(vii) If the data is given in
quarter

**Example
9.8**

Calculate the seasonal
index for the monthly sales of a product using the method of simple averages.

*Solution:*

Computation of seasonal
Indices by method of simple averages.

Similarly other seasonal index
values can be obtained.

**Example
9.9**

Calculate the seasonal
index for the quarterly production of a product using the method of simple
averages.

*Solution:*

Computation of Seasonal
Index by the method of simple averages.

Tags : Time Series Analysis | Applied Statistics , 12th Business Maths and Statistics : Chapter 9 : Applied Statistics

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12th Business Maths and Statistics : Chapter 9 : Applied Statistics : Measurements of Trends | Time Series Analysis | Applied Statistics

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