Chapter 12
Forecasting
Lecture Outline
• Strategic Role of Forecasting in Supply Chain
Management
• Components of Forecasting Demand
• Time Series Methods
• Forecast Accuracy
• Time Series Forecasting Using Excel
• Regression Methods
Copyright 2011 John Wiley & Sons, Inc.
12-2
Forecasting
• Predicting the future
• Qualitative forecast methods
• subjective
• Quantitative forecast methods
• based on mathematical formulas
Copyright 2011 John Wiley & Sons, Inc.
12-3
Is it better to over-forecast or under-forecast
Effect
Cost
Schedule
Over-forecasting
Under-forecasting
<100%
100%
>100%
Target as a Percentage of Nominal Forecast
Qualitative Forecasting – Delphi Method
1.
2.
3.
4.
5.
6.
Use a group of experts
Address the forecast issue to them
Solicit their responses without discussion.
Average their responses, compute std. dev.
Report results to the group of experts
Return to step 3
Copyright 2011 John Wiley & Sons, Inc.
12-5
Assume you are an expert…
• In what year will the world have essentially
exhausted its oil reserves, at least from an
economic view point? Economics come into
play here because eventually it is no longer
economically cost-effective to extract what oil
remains. Just provide the year (like 2020), not a
range, please.
Copyright 2011 John Wiley & Sons, Inc.
12-6
Proven reserves of Oil--Worldwide
AMOUNT
(billions of barrels)
1400
1200
1000
800
AMOUNT
(billions of barrels)
600
400
200
0
1975
1980
1985
1990
1995
Year
Copyright 2011 John Wiley & Sons, Inc.
2000
2005
2010
The fallacy of forecasts
• In 1914, U.S. Bureau of Mines predicted U.S. oil
reserves would last only ten more years
• In 1939, the U.S. Dept. of the Interior predicted
that oil would last only 13 more years, and then
in 1951, when the oil shortage never occurred, it
predicted oil would run out in just 13 more years
Copyright 2011 John Wiley & Sons, Inc.
More fallacious forecasts
• In a book published in 1972 entitled Limits to
Growth, Dennis and Donella Meadows claimed
that only 550 billion barrels of oil remained in the
earth and that those barrels would all be
consumed by now
Copyright 2011 John Wiley & Sons, Inc.
Sasser’s National energy model—
wrong as well
RESOURCE/R
ESERVES
U.S. Natural
gas
U.S. Oil
U.S. Coal
Foreign Oil
Copyright 2011 John Wiley & Sons, Inc.
Sasser
Actual (2003)
forecast (2003)
All consumed 189 trillion
cubic feet
167 billion
21.9 billion
barrels
barrels
3.94 trillion
.271 trillion
tons
tons
1800 billion
1244 billion
barrels
barrels
11-12
System Dynamics models of energy
• Not a forecasting tool
• Enables understanding of the dynamics
• How such dynamical behavior is likely to play out,
given certain assumptions is key
• Enables cycles, structures, to be identified
• Enables policy implications to be discerned
Copyright 2011 John Wiley & Sons, Inc.
Commentary
• Are oil, gas and coal fossil fuels or are they of
abiotic origin?
• This is not just a scientific question…
Copyright 2011 John Wiley & Sons, Inc.
Evidence for abiotic origin
• Oil and gas are being found deep within the
Earth’s crust, especially the Russians have been
successful at this—how did decaying biomass
ever get seven to ten miles down, underneath
two miles of water?
• Oil in sedimentary rock contains traces of
material from rock below—especially the
Devonian and Cambrian rock formations
Copyright 2011 John Wiley & Sons, Inc.
More evidence for abiotic origin
• There is so much oil on planet earth—we really
don’t know how much, since we don’t know how
much unproven reserves there are
Copyright 2011 John Wiley & Sons, Inc.
The Oil Consumption Stack
Unproven
reserves
discovery rate
Proven
reserves
extraction rate
Crude oil
refinery rate
Refined oil
products
consumptioin rate
Copyright 2011 John Wiley & Sons, Inc.
Supply Chain Management
• Accurate forecasting determines inventory levels
in the supply chain
• Continuous replenishment
•
•
•
•
supplier & customer share continuously updated data
typically managed by the supplier
reduces inventory for the company
speeds customer delivery
• Variations of continuous replenishment
•
•
•
•
quick response
JIT (just-in-time)
VMI (vendor-managed inventory)
stockless inventory
Copyright 2011 John Wiley & Sons, Inc.
12-18
The Effect of Inaccurate Forecasting
Copyright 2011 John Wiley & Sons, Inc.
12-19
Forecasting
• Quality Management
• Accurately forecasting customer demand is a key to
providing good quality service
• Strategic Planning
• Successful strategic planning requires accurate
forecasts of future products and markets
Copyright 2011 John Wiley & Sons, Inc.
12-20
Types of Forecasting Methods
• Depend on
• time frame
• demand behavior
• causes of behavior
Copyright 2011 John Wiley & Sons, Inc.
12-21
Time Frame
• Indicates how far into the future is forecast
• Short- to mid-range forecast
• typically encompasses the immediate future
• daily up to two years
• Long-range forecast
• usually encompasses a period of time longer than
two years
Copyright 2011 John Wiley & Sons, Inc.
12-22
Demand Behavior
• Trend
• a gradual, long-term up or down movement of
demand
• Random variations
• movements in demand that do not follow a pattern
• Cycle
• an up-and-down repetitive movement in demand
• Seasonal pattern
• an up-and-down repetitive movement in demand
occurring periodically
Copyright 2011 John Wiley & Sons, Inc.
12-23
Demand
Demand
Forms of Forecast Movement
Random
movement
Time
(b) Cycle
Demand
Demand
Time
(a) Trend
Time
(c) Seasonal pattern
Copyright 2011 John Wiley & Sons, Inc.
Time
(d) Trend with seasonal pattern
12-24
Forecasting Methods
• Time series
• statistical techniques that use historical demand data
to predict future demand
• Regression methods
• attempt to develop a mathematical relationship
between demand and factors that cause its behavior
• Qualitative
• use management judgment, expertise, and opinion to
predict future demand
Copyright 2011 John Wiley & Sons, Inc.
12-25
Qualitative Methods
• Management, marketing, purchasing, and
engineering are sources for internal qualitative
forecasts
• Delphi method
• involves soliciting forecasts about technological
advances from experts
Copyright 2011 John Wiley & Sons, Inc.
12-26
Forecasting Process
1. Identify the
purpose of forecast
2. Collect historical
data
3. Plot data and identify
patterns
6. Check forecast
accuracy with one or
more measures
5. Develop/compute
forecast for period of
historical data
4. Select a forecast
model that seems
appropriate for data
7.
Is accuracy of
forecast
acceptable?
No
8b. Select new
forecast model or
adjust parameters of
existing model
Yes
8a. Forecast over
planning horizon
Copyright 2011 John Wiley & Sons, Inc.
9. Adjust forecast based
on additional qualitative
information and insight
10. Monitor results
and measure forecast
accuracy
12-27
Time Series
• Assume that what has occurred in the past will
continue to occur in the future
• Relate the forecast to only one factor - time
• Include
• moving average
• exponential smoothing
• linear trend line
Copyright 2011 John Wiley & Sons, Inc.
12-28
Moving Average
• Naive forecast
• demand in current period is used as next period’s
forecast
• Simple moving average
• uses average demand for a fixed sequence of periods
• stable demand with no pronounced behavioral
patterns
• Weighted moving average
• weights are assigned to most recent data
Copyright 2011 John Wiley & Sons, Inc.
12-29
Moving Average: Naïve Approach
MONTH
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Copyright 2011 John Wiley & Sons, Inc.
ORDERS
PER MONTH
120
90
100
75
110
50
75
130
110
90
-
FORECAST
120
90
100
75
110
50
75
130
110
90
12-30
Simple Moving Average
n
Di

i=1
MAn =
n
where
n = number of periods in
the moving average
Di = demand in period i
Copyright 2011 John Wiley & Sons, Inc.
12-31
3-month Simple Moving Average
MONTH
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
ORDERS
PER MONTH
120
90
100
75
110
50
75
130
110
90
-
Copyright 2011 John Wiley & Sons, Inc.
MOVING
AVERAGE
–
–
–
103.3
88.3
95.0
78.3
78.3
85.0
105.0
110.0
3
Di

i=1
MA3 =
=
3
90 + 110 + 130
3
= 110 orders for Nov
12-32
5-month Simple Moving Average
MONTH
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
ORDERS
PER MONTH
120
90
100
75
110
50
75
130
110
90
-
Copyright 2011 John Wiley & Sons, Inc.
MOVING
AVERAGE
–
–
–
–
–
99.0
85.0
82.0
88.0
95.0
91.0
5
Di

i=1
MA5 =
=
5
90 + 110 + 130+75+50
5
= 91 orders for Nov
12-33
Smoothing Effects
150 –
5-month
125 –
Orders
100 –
75 –
50 –
3-month
Actual
25 –
0–
|
Jan
|
Feb
|
Mar
|
Apr
|
May
|
|
June July
|
|
Aug Sept
|
Oct
|
Nov
Month
Copyright 2011 John Wiley & Sons, Inc.
12-34
Weighted Moving Average
• Adjusts moving average method to more closely
reflect data fluctuations
n
WMAn =
 Wi Di
i=1
where
Wi = the weight for period i,
between 0 and 100
percent
 Wi = 1.00
Copyright 2011 John Wiley & Sons, Inc.
12-35
Weighted Moving Average Example
MONTH
August
September
October
WEIGHT
DATA
17%
33%
50%
130
110
90
3
November Forecast
WMA3 =
Wi Di

i=1
= (0.50)(90) + (0.33)(110) + (0.17)(130)
= 103.4 orders
Copyright 2011 John Wiley & Sons, Inc.
12-36
Exponential Smoothing
•
•
•
•
Averaging method
Weights most recent data more strongly
Reacts more to recent changes
Widely used, accurate method
Copyright 2011 John Wiley & Sons, Inc.
12-37
Exponential Smoothing
Ft +1 = Dt + (1 - )Ft
where:
Ft +1 = forecast for next period
Dt = actual demand for present period
Ft = previously determined forecast for
present period
=
weighting factor, smoothing constant
Copyright 2011 John Wiley & Sons, Inc.
12-38
Effect of Smoothing Constant
0.0  1.0
If = 0.20, then Ft +1 = 0.20 Dt + 0.80 Ft
If = 0, then Ft +1 = 0 Dt + 1 Ft = Ft
Forecast does not reflect recent data
If = 1, then Ft +1 = 1 Dt + 0 Ft = Dt
Forecast based only on most recent data
Copyright 2011 John Wiley & Sons, Inc.
12-39
Exponential Smoothing (α=0.30)
PERIOD
1
2
3
4
5
6
7
8
9
10
11
12
MONTH
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Copyright 2011 John Wiley & Sons, Inc.
DEMAND
37
40
41
37
45
50
43
47
56
52
55
54
F2 = D1 + (1 - )F1
= (0.30)(37) + (0.70)(37)
= 37
F3 = D2 + (1 - )F2
= (0.30)(40) + (0.70)(37)
= 37.9
F13 = D12 + (1 - )F12
= (0.30)(54) + (0.70)(50.84)
= 51.79
12-40
Exponential Smoothing
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
13
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
37
40
41
37
45
50
43
47
56
52
55
54
–
Copyright 2011 John Wiley & Sons, Inc.
FORECAST, Ft + 1
( = 0.3)
( = 0.5)
–
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
51.79
–
37.00
38.50
39.75
38.37
41.68
45.84
44.42
45.71
50.85
51.42
53.21
53.61
12-41
Exponential Smoothing
70 –
60 –
 = 0.50
Actual
Orders
50 –
40 –
 = 0.30
30 –
20 –
10 –
0–
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
Month
Copyright 2011 John Wiley & Sons, Inc.
12-42
Adjusted Exponential Smoothing
AFt +1 = Ft +1 + Tt +1
where
T = an exponentially smoothed trend factor
Tt +1 = (Ft +1 - Ft) + (1 - ) Tt
where
Tt = the last period trend factor
= a smoothing constant for trend
0 ≤  ≤ 1
Copyright 2011 John Wiley & Sons, Inc.
12-43
Adjusted Exponential Smoothing (β=0.30)
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
37
40
41
37
45
50
43
47
56
52
55
54
Copyright 2011 John Wiley & Sons, Inc.
T3
= (F3 - F2) + (1 - ) T2
= (0.30)(38.5 - 37.0) + (0.70)(0)
= 0.45
AF3 = F3 + T3 = 38.5 + 0.45
= 38.95
T13 = (F13 - F12) + (1 - ) T12
= (0.30)(53.61 - 53.21) + (0.70)(1.77)
= 1.36
AF13 = F13 + T13 = 53.61 + 1.36 = 54.97
12-44
Adjusted Exponential Smoothing
PERIOD
MONTH
DEMAND
FORECAST
Ft +1
1
2
3
4
5
6
7
8
9
10
11
12
13
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
37
40
41
37
45
50
43
47
56
52
55
54
–
37.00
37.00
38.50
39.75
38.37
38.37
45.84
44.42
45.71
50.85
51.42
53.21
53.61
Copyright 2011 John Wiley & Sons, Inc.
TREND
Tt +1
ADJUSTED
FORECAST AFt +1
–
0.00
0.45
0.69
0.07
0.07
1.97
0.95
1.05
2.28
1.76
1.77
1.36
–
37.00
38.95
40.44
38.44
38.44
47.82
45.37
46.76
58.13
53.19
54.98
54.96
12-45
Adjusted Exponential Smoothing
Forecasts
70 –
Adjusted forecast ( = 0.30)
60 –
Actual
Demand
50 –
40 –
Forecast ( = 0.50)
30 –
20 –
10 –
0–
|
1
|
2
|
3
Copyright 2011 John Wiley & Sons, Inc.
|
4
|
5
|
|
6
7
Period
|
8
|
9
|
10
|
11
|
12
|
13
12-46
Linear Trend Line
y = a + bx
where
a = intercept
b = slope of the line
x = time period
y = forecast for
demand for period x
Copyright 2011 John Wiley & Sons, Inc.
xy - nxy
b =
x2 - nx2
a = y-bx
where
n = number of periods
x
x =
= mean of the x values
n
y
y = n = mean of the y values
12-47
Least Squares Example
x(PERIOD)
y(DEMAND)
xy
x2
1
2
3
4
5
6
7
8
9
10
11
12
73
40
41
37
45
50
43
47
56
52
55
54
37
80
123
148
225
300
301
376
504
520
605
648
1
4
9
16
25
36
49
64
81
100
121
144
78
557
3867
650
Copyright 2011 John Wiley & Sons, Inc.
12-48
Least Squares Example
x = 78 = 6.5
12
y = 557 = 46.42
12
b = xy - nxy =
x2 - nx2
3867 - (12)(6.5)(46.42) =1.72
650 - 12(6.5)2
a = y - bx
= 46.42 - (1.72)(6.5) = 35.2
Copyright 2011 John Wiley & Sons, Inc.
12-49
Linear trend line y = 35.2 + 1.72x
Forecast for period 13 y = 35.2 + 1.72(13) = 57.56 units
70 –
60 –
Actual
Demand
50 –
40 –
Linear trend line
30 –
20 –
10 –
|
1
|
2
|
3
Copyright 2011 John Wiley & Sons, Inc.
|
4
|
5
|
|
6
7
Period
|
8
|
9
|
10
|
11
|
12
|
13
12-50
Seasonal Adjustments
 Repetitive increase/ decrease in demand
 Use seasonal factor to adjust forecast
Seasonal factor = Si =
Copyright 2011 John Wiley & Sons, Inc.
Di
D
12-51
Seasonal Adjustment
YEAR
2002
2003
2004
Total
DEMAND (1000’S PER QUARTER)
1
2
3
4
Total
12.6
14.1
15.3
42.0
8.6
10.3
10.6
29.5
6.3
7.5
8.1
21.9
17.5
18.2
19.6
55.3
45.0
50.1
53.6
148.7
D1
42.0
S1 =
=
= 0.28
D 148.7
D3
21.9
S3 =
=
= 0.15
D 148.7
D2
29.5
S2 =
=
= 0.20
D 148.7
D4
55.3
S4 =
=
= 0.37
D 148.7
Copyright 2011 John Wiley & Sons, Inc.
12-52
Seasonal Adjustment
For 2005
y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17
SF1 = (S1) (F5) = (0.28)(58.17) = 16.28
SF2 = (S2) (F5) = (0.20)(58.17) = 11.63
SF3 = (S3) (F5) = (0.15)(58.17) = 8.73
SF4 = (S4) (F5) = (0.37)(58.17) = 21.53
Copyright 2011 John Wiley & Sons, Inc.
12-53
Forecast Accuracy
• Forecast error
• difference between forecast and actual demand
• MAD
• mean absolute deviation
• MAPD
• mean absolute percent deviation
• Cumulative error
• Average error or bias
Copyright 2011 John Wiley & Sons, Inc.
12-54
Mean Absolute Deviation (MAD)
 Dt - Ft 
MAD =
n
where
t = period number
Dt = demand in period t
Ft = forecast for period t
n = total number of periods
 = absolute value
Copyright 2011 John Wiley & Sons, Inc.
12-55
MAD Example
PERIOD
1
2
3
4
5
6
7
8
9
10
11
12
DEMAND, Dt
Ft ( =0.3)
(Dt - Ft)
|Dt - Ft|
37
40
41
37
45
50
43
47
56
52
55
54
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
–
3.00
3.10
-1.83
6.72
9.69
-0.20
3.86
11.70
4.19
5.94
3.15
–
3.00
3.10
1.83
6.72
9.69
0.20
3.86
11.70
4.19
5.94
3.15
49.31
53.39
557
Copyright 2011 John Wiley & Sons, Inc.
12-56
MAD Calculation
 Dt - Ft 
MAD =
n
53.39
=
11
= 4.85
Copyright 2011 John Wiley & Sons, Inc.
12-57
Other Accuracy Measures
Mean absolute percent deviation (MAPD)
|Dt - Ft|
MAPD =
Dt
Cumulative error
E = et
Average error
et
E= n
Copyright 2011 John Wiley & Sons, Inc.
12-58
Comparison of Forecasts
FORECAST
MAD
MAPD
E
(E)
Exponential smoothing (= 0.30)
Exponential smoothing (= 0.50)
Adjusted exponential smoothing
(= 0.50, = 0.30)
Linear trend line
4.85
4.04
3.81
9.6%
8.5%
7.5%
49.31
33.21
21.14
4.48
3.02
1.92
2.29
4.9%
–
–
Copyright 2011 John Wiley & Sons, Inc.
12-59
Forecast Control
• Tracking signal
• monitors the forecast to see if it is biased high or low
• 1 MAD ≈ 0.8 б
• Control limits of 2 to 5 MADs are used most frequently
Tracking signal =
Copyright 2011 John Wiley & Sons, Inc.
(Dt - Ft)
E
=
MAD
MAD
12-60
Tracking Signal Values
PERIOD
DEMAND
Dt
FORECAST,
Ft
1
2
3
4
5
6
7
8
9
10
11
12
37
40
41
37
45
50
43
47
56
52
55
54
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
TS3 =
Copyright 2011 John Wiley & Sons, Inc.
ERROR
Dt - Ft
–
3.00
3.10
-1.83
6.72
9.69
-0.20
3.86
11.70
4.19
5.94
3.15
E =
(Dt - Ft)
MAD
–
3.00
6.10
4.27
10.99
20.68
20.48
24.34
36.04
40.23
46.17
49.32
–
3.00
3.05
2.64
3.66
4.87
4.09
4.06
5.01
4.92
5.02
4.85
TRACKING
SIGNAL
–
1.00
2.00
1.62
3.00
4.25
5.01
6.00
7.19
8.18
9.20
10.17
6.10
= 2.00
3.05
12-61
Tracking Signal Plot
Tracking signal (MAD)
3 –
2 –
Exponential smoothing ( = 0.30)
1 –
0 –
-1 –
-2 –
Linear trend line
-3 –
|
0
|
1
|
2
|
3
Copyright 2011 John Wiley & Sons, Inc.
|
4
|
5
|
6
Period
|
7
|
8
|
9
|
10
|
11
|
12
12-62
Statistical Control Charts
=
(Dt - Ft)2
n-1
 Using  we can calculate statistical
control limits for the forecast error
 Control limits are typically set at  3
Copyright 2011 John Wiley & Sons, Inc.
12-63
Statistical Control Charts
18.39 –
UCL = +3
12.24 –
Errors
6.12 –
0–
-6.12 –
-12.24 –
-18.39 –
|
0
LCL = -3
|
1
|
2
|
3
Copyright 2011 John Wiley & Sons, Inc.
|
4
|
5
|
6
Period
|
7
|
8
|
9
|
10
|
11
|
12
12-64
Time Series Forecasting Using Excel
• Excel can be used to develop forecasts:
•
•
•
•
Moving average
Exponential smoothing
Adjusted exponential smoothing
Linear trend line
Copyright 2011 John Wiley & Sons, Inc.
12-65
Exponentially Smoothed and Adjusted
Exponentially Smoothed Forecasts
=B5*(C11-C10)+
(1-B5)*D10
=C10+D10
=ABS(B10-E10)
=SUM(F10:F20)
=G22/11
Copyright 2011 John Wiley & Sons, Inc.
12-66
Demand and Exponentially Smoothed
Forecast
Click on “Insert” then “Line”
Copyright 2011 John Wiley & Sons, Inc.
12-67
Data Analysis Option
Copyright 2011 John Wiley & Sons, Inc.
12-68
Forecasting With Seasonal Adjustment
Copyright 2011 John Wiley & Sons, Inc.
12-69
Forecasting With OM Tools
Copyright 2011 John Wiley & Sons, Inc.
12-70
Regression Methods
• Linear regression
• mathematical technique that relates a dependent
variable to an independent variable in the form of a
linear equation
• Correlation
• a measure of the strength of the relationship between
independent and dependent variables
Copyright 2011 John Wiley & Sons, Inc.
12-71
Linear Regression
y = a + bx
a = y-bx
xy - nxy
b =
x2 - nx2
where
a = intercept
b = slope of the line
x
x =
= mean of the x data
n
y
y = n = mean of the y data
Copyright 2011 John Wiley & Sons, Inc.
12-72
Linear Regression Example
x
(WINS)
y
(ATTENDANCE)
xy
x2
4
6
6
8
6
7
5
7
36.3
40.1
41.2
53.0
44.0
45.6
39.0
47.5
145.2
240.6
247.2
424.0
264.0
319.2
195.0
332.5
16
36
36
64
36
49
25
49
49
346.7
2167.7
311
Copyright 2011 John Wiley & Sons, Inc.
12-73
Linear Regression Example
49
= 6.125
8
346.9
y=
= 43.36
8
x=
xy - nxy2
b=
x2 - nx2
(2,167.7) - (8)(6.125)(43.36)
=
(311) - (8)(6.125)2
= 4.06
a = y - bx
= 43.36 - (4.06)(6.125)
= 18.46
Copyright 2011 John Wiley & Sons, Inc.
12-74
Linear Regression Example
60,000 –
Attendance, y
50,000 –
40,000 –
30,000 –
Linear regression line, y
= 18.46 + 4.06x
20,000 –
Attendance forecast for 7 wins
y = 18.46 + 4.06(7)
= 46.88, or 46,880
10,000 –
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
Wins, x
Copyright 2011 John Wiley & Sons, Inc.
12-75
Correlation and Coefficient of
Determination
• Correlation, r
• Measure of strength of relationship
• Varies between -1.00 and +1.00
• Coefficient of determination, r2
• Percentage of variation in dependent variable
resulting from changes in the independent variable
Copyright 2011 John Wiley & Sons, Inc.
12-76
Computing Correlation
r=
n xy -  x y
[n x2 - ( x)2] [n y2 - ( y)2]
(8)(2,167.7) - (49)(346.9)
r=
[(8)(311) - (49)2] [(8)(15,224.7) - (346.9)2]
r = 0.947
Coefficient of determination
r2 = (0.947)2 = 0.897
Copyright 2011 John Wiley & Sons, Inc.
12-77
Regression Analysis With Excel
=INTERCEPT(B5:B12,A5:A12)
=SUM(B5:B12)
Copyright 2011 John Wiley & Sons, Inc.
=CORREL(B5:B12,A5:A12)
12-78
Regression Analysis with Excel
Copyright 2011 John Wiley & Sons, Inc.
12-79
Regression Analysis With Excel
Copyright 2011 John Wiley & Sons, Inc.
12-80
Multiple Regression
Study the relationship of demand to two or more
independent variables
y = 0 + 1x1 + 2x2 … + kxk
where
0 = the intercept
1, … , k = parameters for the
independent variables
x1, … , xk = independent variables
Copyright 2011 John Wiley & Sons, Inc.
12-81
Multiple Regression With Excel
r2, the coefficient
of determination
Copyright 2011 John Wiley & Sons, Inc.
Regression equation
coefficients for x1 and x2
12-82
Multiple Regression Example
y = 19,094.42 + 3560.99 x1 + .0368 x2
y = 19,094.42 + 3560.99 (7) + .0368 (60,000)
= 46,229.35
Copyright 2011 John Wiley & Sons, Inc.
12-83
Copyright 2011 John Wiley & Sons, Inc.
All rights reserved. Reproduction or translation of this
work beyond that permitted in section 117 of the 1976
United States Copyright Act without express permission
of the copyright owner is unlawful. Request for further
information should be addressed to the Permission
Department, John Wiley & Sons, Inc. The purchaser
may make back-up copies for his/her own use only and
not for distribution or resale. The Publisher assumes no
responsibility for errors, omissions, or damages caused
by the use of these programs or from the use of the
information herein.
Copyright 2011 John Wiley & Sons, Inc.
12-84