Obtain the trend values for the above data using 3-yearly moving averages - Mathematics and Statistics

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Sum

Obtain the trend values for the data, using 3-yearly moving averages

Year 1976 1977 1978 1979 1980 1981
Production 0 4 4 2 6 8
Year 1982 1983 1984 1985 1986  
Production 5 9 4 10 10  
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Solution

Construct the following table for obtaining 3-yearly moving averages for the data.

Year 

t

Production

yt

3-yearly

moving total

 

3-yearly moving
averages trend value

1976 0
1977 4 8 2.6667
198 4 10 3.3333
1979 2 12 4
1980 6 16 5.3333
1981 8 19 6.3333
1982 5 22 7.3333
1983 9 18 6
1984 4 23 7.6667
1985 10 24 8
1986 10
Concept: Measurement of Secular Trend
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Chapter 2.4: Time Series - Q.4

RELATED QUESTIONS

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The method of measuring trend of time series using only averages is _______


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Moving average method of finding trend is very complicated and involves several calculations.


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Least squares method of finding trend is very simple and does not involve any calculations.


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Year 1974 1975 1976 1977 1978 1979 1980 1981 1982
Production 0 4 9 9 8 5 4 8 10

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Solve the following problem :

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Solve the following problem :

Following table shows the number of traffic fatalities (in a state) resulting from drunken driving for years 1975 to 1983.

Year 1975 1976 1977 1978 1979 1980 1981 1982 1983
No. of deaths 0 6 3 8 2 9 4 5 10

Fit a trend line to the above data by graphical method.


Solve the following problem :

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010.

Year 1980 1985 1990 1995 2000 2005 2010
IMR 10 7 5 4 3 1 0

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Solve the following problem :

Following tables shows the wheat yield (‘000 tonnes) in India for years 1959 to 1968.

Year 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968
Yield 0 1 2 3 1 0 4 1 2 10

Fit a trend line to the above data by the method of least squares.


The complicated but efficient method of measuring trend of time series is ______


State whether the following statement is True or False:

The secular trend component of time series represents irregular variations


State whether the following statement is True or False: 

Moving average method of finding trend is very complicated and involves several calculations


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Least squares method of finding trend is very simple and does not involve any calculations


Following table shows the amount of sugar production (in lac tons) for the years 1971 to 1982

Year 1971 1972 1973 1974 1975 1976
Production 1 0 1 2 3 2
Year 1977 1978 1979 1980 1981 1982
Production 4 6 5 1 4 10

Fit a trend line by the method of least squares


The following table gives the production of steel (in millions of tons) for years 1976 to 1986.

Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986
Production 0 4 4 2 6 8 5 9 4 10 10

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The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.

Year 1962 1963 1964 1965 1966 1967 1968 1969
Production
(million barrels)
0 0 1 1 2 3 4 5
Year 1970 1971 1972 1973 1974 1975 1976  
Production
(million barrels)
6 7 8 9 8 9 10  
  1. Obtain trend values for the above data using 5-yearly moving averages.
  2. Plot the original time series and trend values obtained above on the same graph.

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010

Year 1980 1985 1990 1995
IMR 10 7 5 4
Year 2000 2005 2010  
IMR 3 1 0  

Fit a trend line by the method of least squares

Solution: Let us fit equation of trend line for above data.

Let the equation of trend line be y = a + bx   .....(i)

Here n = 7(odd), middle year is `square` and h = 5

Year IMR (y) x x2 x.y
1980 10 – 3 9 – 30
1985 7 – 2 4 – 14
1990 5 – 1 1 – 5
1995 4 0 0 0
2000 3 1 1 3
2005 1 2 4 2
2010 0 3 9 0
Total 30 0 28 – 44

The normal equations are

Σy = na + bΣx

As, Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx2

As, Σx = 0, b =`square`

∴ The equation of trend line is y = `square`


Obtain trend values for data, using 3-yearly moving averages
Solution:

Year IMR 3 yearly
moving total
3-yearly moving
average

(trend value)
1980 10
1985 7 `square` 7.33
1990 5 16 `square`
1995 4 12 4
2000 3 8 `square`
2005 1 `square` 1.33
2010 0

Fit equation of trend line for the data given below.

Year Production (y) x x2 xy
2006 19 – 9 81 – 171
2007 20 – 7 49 – 140
2008 14 – 5 25 – 70
2009 16 – 3 9 – 48
2010 17 – 1 1 – 17
2011 16 1 1 16
2012 18 3 9 54
2013 17 5 25 85
2014 21 7 49 147
2015 19 9 81 171
Total 177 0 330 27

Let the equation of trend line be y = a + bx   .....(i)

Here n = `square` (even), two middle years are `square` and 2011, and h = `square`

The normal equations are Σy = na + bΣx

As Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx2

As Σx = 0, b = `square`

Substitute values of a and b in equation (i) the equation of trend line is `square`

To find trend value for the year 2016, put x = `square` in the above equation.

y = `square`


Complete the table using 4 yearly moving average method.

Year Production 4 yearly
moving
total
4 yearly
centered
total
4 yearly centered
moving average
(trend values)
2006 19  
    `square`    
2007 20   `square`
    72    
2008 17   142 17.75
    70    
2009 16   `square` 17
    `square`    
2010 17   133 `square`
    67    
2011 16   `square` `square`
    `square`    
2012 18   140 17.5
    72    
2013 17   147 18.375
    75    
2014 21  
       
2015 19  

Obtain the trend values for the following data using 5 yearly moving averages:

Year 2000 2001 2002 2003 2004
Production
xi
10 15 20 25 30
Year 2005 2006 2007 2008 2009
Production
xi
35 40 45 50 55

Following table shows the amount of sugar production (in lakh tonnes) for the years 1931 to 1941:

Year Production Year Production
1931 1 1937 8
1932 0 1938 6
1933 1 1939 5
1934 2 1940 1
1935 3 1941 4
1936 2    

Complete the following activity to fit a trend line by method of least squares:


The complicated but efficient method of measuring trend of time series is ______.


The following table shows gross capital information (in Crore ₹) for years 1966 to 1975:

Years 1966 1967 1968 1969 1970
Gross Capital information 20 25 25 30 35
Years 1971 1972 1973 1974 1975
Gross Capital information 30 45 40 55 65

Obtain trend values using 5-yearly moving values.


Fit a trend line to the following data by the method of least square :

Year 1980 1985 1990 1995 2000 2005 2010
IMR 10 7 5 4 3 1 0

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