A quantitative method of forecasting or smoothing a time series by averaging each successive group (no. of observations) of data values. Term MOVING is used because it is obtained by summing and averaging the values from a given no of periods, each time deleting the oldest value and adding a new value.

1. MOVING AVERAGE METHOD
&
METHOD OF LEAST SQUARES
Department of Management Sciences
University of Gujrat
Lahore Sub-Campus
8-H Main Canal Bank Near Tajbagh Bridge Harbanspura Lahore
Submitted by: Hassan Jalil
Roll No.: F15-10
Deg. Program.: MBA 3.5
Course: Business Mathematics & Statistics
Submitted to: Miss Madiha Maqsood

2. MOVING AVERAGE METHOD
• A quantitative method of forecasting or smoothing a time series
by averaging each successive group (no. of observations) of data
values.
• Term MOVING is used because it is obtained by summing and
averaging the values from a given no of periods, each time
deleting the oldest value and adding a new value.

3. • For applying the method of moving averages the
period of moving averages has to be selected
• This period can be 3- yearly moving averages 5yr
moving averages 4yr moving averages etc.
• For ex:- 3-yearly moving averages can be
calculated from the data : a, b, c, d, e, f can be
computed as :
CONT.

4. • If the moving average is an odd no of values e.g., 3
years, there is no problem of centring it. Because the
moving total for 3 years average will be centred besides
the 2nd year and for 5 years average be centred
besides 3rd year.
• But if the moving average is an even no, e.g., 4 years
moving average, then the average of 1st 4 figures will
be placed between 2nd and 3rd year.
• This process is called centring of the averages. In case
of even period of moving averages, the trend values
are obtained after centring the averages a second
time.
CONT.

5. GOALS
• Smooth out the short-term fluctuations.
• Identify the long-term trend.
2
5
2 2
7
6
3 3
3.67
5
0
1
2
3
4
5
6
7
8
1994 1995 1996 1997 1998 1999
MOVING AVG.
Sales M.A

6. MOVING AVERAGE EXAMPLE
Year Units
1994 2
1995 5
1996 2
1997 2
1998 7
1999 6
John is a building contractor with a record of a
total of 24 single family homes constructed
over a 6-year period. Provide John with a 3-
year moving average graph.

7. YEAR Units Sold Moving Avg.
1994 2
1995 5 3
1996 2 3
1997 2 3.67
1998 7 5
1999 6
2
5
2 2
7
6
3 3
3.67
5
0
2
4
6
8
1994 1995 1996 1997 1998 1999
Moving Avg.
Sales M.A

8. CALCULATION OF MOVING AVERAGE BASED ON PERIOD
• When period is odd-
example:-
Calculate the 3-yearly moving averages of the data given below:
yrs 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Sales
(million
of
rupees)
3 4 8 6 7 11 9 10 14 12

9. Year Sales (Million Rs.) y 3 Years Moving Total 3 Years Moving Avg.
1980 3
1981 4 15 15/3=5
1982 8 18 18/3=6
1983 6 21 7
1984 7 24 8
1985 11 27 9
1986 9 30 10
1987 10 33 11
1988 14 36 12
1989 12

10. 3
4
8
6
7
11
9
10
14
12
5
6
7
8
9
10
11
12
0
2
4
6
8
10
12
14
16
1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Rupees(Million)
Years
Moving Average when period is Odd
Actual Line Trend Linear (Trend)

11. CALCULATION OF MOVING AVERAGE BASED ON PERIOD
When period is even:-
Example :-
Compute 4-yearly moving averages from the following data:
year 1991 1992 1993 1994 1995 1996 1997 1998
Annual sale(Rs in
crores)
36 43 43 34 44 54 34 24

12. Years Annual Sales (y)
4 Year Moving
Total
4 Year Centered
M.T
4 Year Centered
M.A
1991 36
1992 43
156
1993 43 320 320/8=40
164
1994 34 339 339/8=42.375
175
1995 44 341 341/8=42.625
166
1996 54 322 322/8=40.25
156
1997 34
1998 24

13. 36
43 43
34
44
54
34
24
40
42.375 42.625
40.25
0
10
20
30
40
50
60
1991 1992 1993 1994 1995 1996 1997 1998
AnnualSales
Axis Title
Moving Average when period is Even
Annual Sales 4 Quarter Cented M.A

14. METHOD OF LEAST SQUARES
• This is the best method for obtaining trend values.
• It provides a convenient basis for obtaining the line of best fit
in a series.
• Line of the best fit is a line from which the sum of the
deviations of various points on its either side is zero.
• The sum of the squares of the deviation of various points
from the line of best fit is the least. – That is why this method
is known as method of least squares.

15. METHODS OF LEAST SQUARES
• Least squares, also used in regression analysis,
determines the unique trend line forecast which
minimizes the mean squares of deviations. The
independent variable is the time period and the
dependent variable is the actual observed value in
the time series
• equation of straight line trend:
Ŷ=a+bx

16. CONVERTING REGRESSION LINE INTO NORMAL
EQUATION
Regression Line is y= a+bx
Where
• y = dependent variable
• a = intercept
• b = slope
• x = independent variable
ᶺ
ᶺ

17. CONT.
y= a+bx
Now multiply with ∑ to convert it into normal equation:
y = a+bx
∑y = na + b∑x ………. (i)
Now multiply with ∑x:
∑xy = a∑x + b∑X2 ………. (ii)
ᶺ
ᶺ

18. BY USING FORMAULA
We can find the slope of straight line by
using formula as well:
That is:
b = ∑xy / ∑x2
a = ∑y / n

19. EXAMPLE:-
Determine trend line:-
Year: 2000 2001 2002 2003 2004
Sales(inRs
‘000)
35 56 79 80 40

20. CALCULATION
Year (x) Sales (Y) X
X2 XY
2000
2001
2002
2003
2004
35
56
79
80
40
-2
-1
0
1
2
4
1
0
1
4
-70
-56
0
80
80
TOTAL ∑Y=290 ∑X=0 ∑X2
=10 ∑XY=34

21. SOLUTION BY USING EQUATION
• From (i) ∑y = na + b∑x
290 = 5a + b (0)
290 = 5a
a = 290/5
a = 58
• From (ii) ∑xy = a∑x + b∑X2
34 = a (0) + b (10)
b = 34/10
b = 3.4

22. CALCULATION
• Now a = ∑ Y/ n = 290/ 5 = 58
and b = ∑XY/∑X2 = 34/10 = 3.4
Substituting these values in equation of trend line which is
Y=58+3.4X , with 2002=0
Year (x) X
=x-2002
Trend values
(Y=58+3.4X)
2000
2001
2002
2003
2004
-2
-1
0
1
2
58+3.4×(-2)=51.2
58+3.4×(-1)=54.6
58+3.4×(0)=58.0
58+3.4×(1)=61.4
58+3.4×(2)=64.8

23. 35
56
79 80
40
51.2
54.6
58
61.4
64.8
0
10
20
30
40
50
60
70
80
90
2000 2001 2002 2003 2004
Sales‘000
Year
Straight Trend Line
Actual Line Trend Line