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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 y_{t} |
3-yearly moving total |
3-yearly moving |
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 | – | – |
RELATED QUESTIONS
Fill in the blank :
The method of measuring trend of time series using only averages is _______
State whether the following is True or False :
Moving average method of finding trend is very complicated and involves several calculations.
State whether the following is True or False :
Least squares method of finding trend is very simple and does not involve any calculations.
Solve the following problem :
The following table shows the production of pig-iron and ferro- alloys (‘000 metric tonnes)
Year | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 |
Production | 0 | 4 | 9 | 9 | 8 | 5 | 4 | 8 | 10 |
Fit a trend line to the above data by graphical method.
Solve the following problem :
Fit a trend line to the data in Problem 7 by the method of least squares.
Solve the following problem :
Obtain trend values for the data in Problem 7 using 4-yearly moving averages.
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 |
Fit a trend line to the above data by graphical method.
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
State whether the following statement is True or False:
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 |
Obtain the trend value for the year 1990
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 |
- Obtain trend values for the above data using 5-yearly moving averages.
- 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 | x^{2} | 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 + bΣx^{2}
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 | x^{2} | 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 + bΣx^{2}
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 x_{i} |
10 | 15 | 20 | 25 | 30 |
Year | 2005 | 2006 | 2007 | 2008 | 2009 |
Production x_{i} |
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 |