SECTION 2 : STATISTICS
UNIT 4
TIME SERIES & INDEX NUMBER (20 MARKS) 
TIME SERIES ANALYSIS
A: DEFINATION : A TIME SERIES IS A SEQUENCE OF VALUES OF A PHENOMENON ARRANGED IN ORDER OF THEIR OCCURANCE
B: COMPONENTS OF TIME SERIES (V.IMP): FOUR COMPONENT 
1)  SECULAR TREND (T) : TREND OBSERVED OVER A LONG PERIOD. TREND CAN BE INCREASING OR DECREASING TREND.
EG : RISE IN SALES OF VEHICLE, RISE IN USE OF MOBILES , RISE IN POPULATION, FALL IN PROFIT OF A COMPANY
2)  SEASONAL VARIATION (S) :  REGULAR CHANGES WITH OCCUR IN THE DATA DUE TO SEASONS, CUSTOMS OR TRADITIONS. PERIOD IS MOSTLY ONE YEAR  
EG :SALES OF  ICE-CREAM IN SUMMER, SALES OF RAINCOATS IN RAINY SEASON, SALES OF SWEETS IN DIWALI, SALES OF CLOTHES DURING  MARRIAGE/FESTIVALS
3 )  CYCLICAL VARIATIONS (C) : THESE FLUCTUATIONS ARE DUE TO CHANGES IN BUSINESS CYCLE. PERIOD IS MORE THAN A YEAR.
FOUR PHASES OF ANY BUSINESS ACTIVITY : PROSPERITY, RECESSION, DEPRESSION AND RECOVERY
THEY ARE RECURRING AND PERIODIC IN NATURE
MOSTLY EVERY BUSINESS HAS A BUSINESS CYCLE
4)  IRREGULAR VARIATIONS (I) : VARIATIONS WHICH CANNOT BE PREDICTED AND ARE ERACTIC IN NATURE,NO  FORECAST IS POSSIBLE
EG : FLOOD, WAR, STRIKES, LOCKDOWN, EARTHQUAKE
C :  ANALYSIS OF TIME SERIES 
LET O- ORIGINAL TIME SERIES
T - SECULAR TREND
S- SEASONAL VARIATION
C- CYCLICAL VARIATION
I - IRREGULAR VARIATION
ADDITIVE MODEL 
0= T + S + C+ I
MUTLIPLICATIVE MODEL 
0 = T X S X C X I
D :  ESTIMATION OF TREND 
1- FREEHAND CURVE METHOD 
2 - METHOD OF MOVING AVERAGES (3, 5 & 4) (VIMP)
3 - LEAST SQUARES METHOD ( N EVEN OR ODD) (VIMP)
DIV A
1 FREEHAND CURVE METHOD (SIMPLE SELF STUDY)
Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Profit 13 15 23 17 25 30 28 35 40 45
To plot graph 
X Axis will be year
Y Axis will be time series value ( Profit values)
Points are plotted on graph paper and they are joined with help of free hand curve
2) METHOD OF MOVING AVERAGES (VIMP)
i) 3 YEARLY MOVING AVERAGES
Eg 1 : Estimate the trend using 3 yearly Moving Average Method. Plot the original time series and the moving averages on a graph paper.
Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Profit in 000(Rs) 13 15 23 17 25 30 28 35 40 45
Soln
Y Axis 
Year Profit 3 yearly Moving Total  3 yearly Moving Average Year (X Axis)  Profit 3 yearly Moving Average (Trend Values)
2005 13 -- -- 2005 13 --
2006 15 51 17 2006 15 17
2007 23 55 18.33 2007 23 18.33
2008 17 65 21.67 2008 17 21.67
2009 25 72 24 2009 25 24
2010 30 83 27.67 2010 30 27.67
2011 28 93 31 2011 28 31
2012 35 103 34.33 2012 35 34.33
2013 40 120 40 2013 40 40
2014 45 -- -- 2014 45 --  
To plot graph 
X Axis will be year
Y Axis will be time series value ( data is profit values)
Points are plotted and joined with help of straight line
In this example 
Profit is original data or time series
3 yearly moving averages in the trend value
(TWO LINES ARE DRAWN) 
Graph 
Eg 2 : Estimate the trend using 3 yearly Moving Average Method
Year 2001 2002 2003 2004 2005 2006 2007 2008 2009
Production 110 108 90 120 130 100 140 145 150
Soln
Year  Production 3 yearly Moving Total  3 yearly Moving Average (DIVIDED BY 3)
2001 110 - -
2002 108 308 102.67
2003 90 318 106
2004 120 340 113.33
2005 130 350 116.67
2006 100 370 123.33
2007 140 385 128.33
2008 145 435 145
2009 150 - -
Eg 3 : Estimate the trend using 3 yearly Moving Average Method. 
Year 2000 2001 2002 2003 2004 2005 2006 2007
Income in '000 Rs  13 15 23 17 25 30 28 35
Soln 
Year Income in '000 Rs  3 yearly Moving Total  3 yearly Moving Average
2000 13 - -
2001 15 51 17
2002 23 55 18.33
2003 17 65 21.67
2004 25 72 24
2005 30 83 27.67
2006 28 93 31
2007 35 - -
ii) 5 YEARLY MOVING AVERAGES
Eg 1 : Estimate the trend using 5 yearly Moving Average Method. Plot the Moving Averages and the Trend Line on the Graph Paper.
Year 2001 2002 2003 2004 2005 2006 2007 2008 2009
Sales 30 35 30 40 50 45 60 70 80
Soln
Y Axis
Year Sales 5 yearly Moving Total  5 yearly Moving Average Year (X Axis) Sales 5 yearly Moving Averages
2001 30 - - 2001 30  
2002 35 - - 2002 35  
2003 30 185 37 2003 30 37
2004 40 200 40 2004 40 40
2005 50 225 45 2005 50 45
2006 45 265 53 2006 45 53
2007 60 305 61 2007 60 61
2008 70 - - 2008 70  
2009 80 - - 2009 80  
Eg 2 : Estimate the trend using 5 yearly Moving Average Method
Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Profit 30 45 50 60 50 65 70 80 80 90
Soln
Year Profit 5 yearly Moving Total  5 yearly Moving Average (DIVIDED BY 5)
2005 30 - -
2006 45 - -
2007 50 235 47
2008 60 270 54
2009 50 295 59
2010 65 325 65
2011 70 345 69
2012 80 385 77
2013 80 - -
2014 90 - -
iii) Four yearly Centered Moving Average 
Eg 1 : Estimate the trend using 4 yearly Centred Moving Average Method. Plot the original time series and the moving averages on a graph paper.
Year 2001 2002 2003 2004 2005 2006 2007 2008 2009
Sales 30 35 30 40 50 45 60 70 80
Soln
Year Sales (Original Data)  4 yearly moving total Centered Total (2 yearly moving total) Moving Average (Divide by 8) (Trend values)      
2001 30 - - -      
               
2002 35 - - -      
    135          
2003 30   290 36.25      
    155          
2004 40   320 40      
    165          
2005 50   360 45      
    195          
2006 45   420 52.5      
    225          
2007 60   480 60      
    255          
2008 70 - - -      
               
2009 80 - - -      
     
     
Year Sales (Original Data)  Moving Averages
     
2001 30        
2002 35  
2003 30 36.25
2004 40 40
2005 50 45
2006 45 52.5
2007 60 60
2008 70  
2009 80  
Eg 2 : Estimate the trend using 4 yearly  Moving Average Method
Year 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
Production 20 15 25 30 45 40 50 60 75 85
Soln
Year Production 4 yearly moving total Centered Total (2 yearly moving total) Moving Average (Divide by 8)
2011 20 - - -
         
2012 15 - - -
    90    
2013 25   205 25.63
    115    
2014 30   275 34.38
    140    
2015 45   305 38.13
    165    
2016 40   360 45.00
    195    
2017 50   420 52.50
    225    
2018 60   495 61.88
    270    
2019 75 - - -
         
2020 85 - - -
3 ) FITTING A STRAIGHT LINE TREND BY LEAST SQUARE METHOD 
∑
Y= a + bx
X should be chosen in such a way that ∑x =0
a = ∑y/n
 
b= ∑xy/ ∑x²
TWO TYPES ARE THERE
i) n : no of years is odd
ii) n : no of years is even
i) n : no of years is odd
Eg 1 Fit a straight line trend using a least square method for the following data. Also estimate the trend value for the year 2018      
                   
  Year 2011 2012 2013 2014 2015 2016 2017  
  Production 20 15 25 30 45 40 50  
                   
Soln
ORIGINAL DATA TREND VALUE 
Year Production (y)  X= Year -mid year (2014) X² XY Y= a +bX = 32.14+ 5.71X Short Cut
2011 20 -3 9 -60 15.01 -115
2012 15 -2 4 -30 20.72 +b 275
2013 25 -1 1 -25 26.43 +b 160
2014 30 0 0 0 32.14 +b
2015 45 1 1 45 37.85 +b
2016 40 2 4 80 43.56 +b
2017 50 3 9 150 49.27 +b 54.98
∑Y=225 ∑X =0 ∑X² = 28 ∑XY =160
n=7
X should be chosen in such a way that ∑X =0 5.71
a = ∑y/n = 32.14
   
b= ∑xy/ ∑x² = 5.71 (b will be positive if it is increasing data and negative if decreasing data)
Equation of Straight line 
Y= a + bx
 
Y= 32.14+ 5.71X  
 
The Trend Value for 2018 is 54.98  
 
FOR 2018 X = 4  Y = 54.98  
 
Eg 2 Fit a straight line trend using a least square method for the following data. Also estimate the trend value for the year 2016
             
  Year 2010 2011 2012 2013 2014
  Sales 12 15 20 18 25
             
Soln
ORIGINAL DATA TREND VALUE 
Year Sales (Y) X= Year -mid year (2012) X² XY Y= a +bX =18+2.9X Short Cut
2010 12 -2 4 -24 12.2 -39
2011 15 -1 1 -15 15.1 +b 68
2012 20 0 0 0 18 +b 29
2013 18 1 1 18 20.9 +b
2014 25 2 4 50 23.8 +b
90 ∑X =0 ∑X² = 10 ∑XY =29
2.9
X should be chosen in such a way that ∑X =0 18
a = ∑y/n = 18
b= ∑xy/ ∑x² = 2.9
Equation of Straight line 
Y= a + bx = 18+2.9X
FOR YEAR 2016 X = 4 Y= 29.6 29.6
ii) n : no of years is even
Eg 1 Fit a straight line trend using a least square method for the following data. Also estimate the trend value for the year 2007    
                 
  Year 2001 2002 2003 2004 2005 2006  
  Profit 30 40 55 60 50 70  
                 
Soln
ORIGINAL DATA TREND VALUE
Year Profit (Y) X= 2(Year -mid year (2003.5)) X² XY Y= a +bX =50.83+3.36X Short Cut  
2001 30 -5 25 -150 34.03    
2002 40 -3 9 -120 40.75 +2b -325
2003 55 -1 1 -55 47.47 +2b 560
2004 60 1 1 60 54.19 +2b 235
2005 50 3 9 150 60.91 +2b  
2006 70 5 25 350 67.63 +2b
305 ∑X =0 ∑X² =70 ∑XY= 235     6.72
235 6.72
X should be chosen in such a way that ∑X =0
a = ∑y/n = 50.83
b= ∑xy/ ∑x² = 3.36
Equation of Straight line 
Y= a +bX =50.83+3.36X X= 7
For the year 2007 X=7  Y= 74.35
Eg 2 Fit a straight line trend using a least square method for the following data. Also estimate the trend value for the year 2019
Year 2010 2011 2012 2013 2014 2015 2016 2017
Sales 12 15 20 25 20 30 35 40
Soln
Year Sales (Y) X= 2(Year -mid year (2013.5)) X² XY Y= a +bX = 24.63+1.91X    
2010 12 -7 49 -84 11.26    
2011 15 -5 25 -75 15.08 +2b  
2012 20 -3 9 -60 18.9 +2b  
2013 25 -1 1 -25 22.72 +2b  
2014 20 1 1 20 26.54 +2b  
2015 30 3 9 90 30.36 +2b  
2016 35 5 25 175 34.18 +2b  
2017 40 7 49 280 38 +2b  
197 ∑X =0 ∑X² =168 ∑XY =321    
3.82
X should be chosen in such a way that ∑X =0
a = ∑y/n = 24.63
b= ∑xy/ ∑x² = 1.91
Equation of Straight line  5.5
Y= a +bX = 24.63+1.91X 
FOR 2019  X= 11
Y= 24.63 +1.91*11 = 45.64
DIV A, C & D
4)  METHOD TO ESTIMATE SEASONAL INDICES
G= Grand Total
GRAND AVERAGE =  G / TOTAL NO OF VALUES
SEASONAL INDEX = AVERAGE FOR SEASON * 100 / GRAND AVERAGE 
Eg 1 :  Find the seasonal component of the time series using method of seasonal indices
  SALES FOR DIFFERENT QUARTERS
Year  Jan-March April-June July- Sept Oct-Dec
2003 107 120 114 113
2004 109 123 115 112
2005 110 122 113 114
2006 108 125 117 113
Soln   SEASONS  
  Jan-March April-June July- Sept Oct-Dec GRAND
Total 434 490 459 452 1835 (G)
Average 108.5 122.5 114.75 113 114.69 (G/16)
Seasonal Index 94.60 106.81 100.04 98.53  
Eg 2 :  Find the seasonal component of the time series using method of seasonal indices
  SEASONS
Year  I II III IV
2002 54 55 56 50
2003 58 60 65 60
2004 55 66 68 65
2005 54 68 70 60
2006 58 65 72 68
Soln   SEASONS  
  I II III IV GRAND
Total 279 314 331 303 1227 (G)
Average 55.8 62.8 66.2 60.6 61.35 (G/16)
Seasonal Index 90.95 102.36 107.91 98.78