Moving Average, MAD, Tracking
Signal Problems
Problems (short) 1-2
1. Given the following data, compute 3-period moving average
forecast for period 6?
Period
1
2
3
4
5
Demand
73
68
65
72
67
(65+72+67)/3 = 68
2. Monthly sales for the past five months were as follows: April
(15), May (20), June (18), July (22), August (20). Determine a
September forecast, using a 4-period moving average.
A) 16.5
B) 18.75
C) 19.1
D) 20.0
E) none of the above
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
2
Problems (short) 3-4
3. In order to increase the responsiveness of a forecast (i.e., respond
quickly to the data changes) made using the moving average
technique, the number of periods in the average should be:
A) decreased
B) increased
C) multiplied by a larger alpha
D) multiplied by a smaller alpha
E)none of the above
4. Given the following data, Compute MAD and TS at the end of
period 4.
Sum |A-F| = 15
Period Actual Forecast A-F |A-F|
MAD = Sum |A-F| /n
1
15
15
0
0
=15/4 = 3.75
2
10
6
4
4
Sum (A-F) = 9
3
16
8
8
8
TS = Sum(A-F)/MAD
4
9
12
-3
3
TS = 9/3.75 = 2.4
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
3
Problems (short) 5-6
5. Given forecast errors of 4, 8, and - 3, what is the MAD?
A) 4
B) 3
C) 5
D) 6
E) 12
6. The sum of the forecast errors (SFE) and the mean absolute
deviation (MAD) are calculated in each period. The values of SFE
and MAD in the last period to be 46 and 21, respectively. Which of
the following is the value of tracking signal in the last period?
A) 0.6
B) 1.8
C) 2
D) 2.2
E) 2.5
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
4
Problems 7-8
7. You must choose between two alternative forecasting models.
Both models have been used to prepare forecasts for a six-month
period. Compute mean absolute deviation (MAD) for the
forecasting model 2.
Month
Sales
Forecasts
A) 0.33
B) 2.0
C) 5.67
D) 9.51
E) none of the above
1
2
3
4
5
6
492
470
485
493
498
492
Model 1
488
484
480
490
497
493
Model 2
495
482
478
488
492
493
8. Which forecasting model would you recommend? What is the
MAD for the recommended forecasting model?
A) Model 1 with MAD of 4.67
B) Model 2 with MAD of 0.33
C) Model 1 with MAD of 3.39
D) Model 2 with MAD of 2.0
E) none of the above
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
5
Problem 9
9.
I- to select the best forecasting technique
II- to estimate the standard deviation of the forecast.
III- to see if the forecast is within control limits
IV- to see if the forecast does not show any specific pattern.
A) the main two applications of MAD are I and II. The main
two applications of Tracking Signal are III and IV.
B) the main two applications of MAD are I and III. The main
two applications of Tracking Signal are II and IV.
C) the main two applications of MAD are I and IV. The main
two applications of Tracking Signal are II and III.
D) the main two applications of MAD are II and III. The main
two applications of Tracking Signal are I and IV.
E) none of the above
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
6
Problem 10
10. Given the following tracking signal graph
A) the forecasting method overestimates the demand
B) the forecasting method underestimates the demand
C) the demand is very seasonal
D) the forecasting method is moving average
E) the forecasting method is exponential smoothing
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
7
Problem 11
11. The 5-period moving average in month 6 was 150 units. Actual
demand in month 7 is 180 units. What is 6 period moving average
in month 7?
MA56 = (A6+A5+A4+A3+A2)/5
MA67 = (A7+A6+A5+A4+A3+A2)/6
MA56 = (A6+A5+A4+A3+A2)/5 = 150
A6+A5+A4+A3+A2 = 750
A7 = 180
MA67 = (A7+A6+A5+A4+A3+A2)/6
MA67 = (180+750)/6 = 155
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
8
Problem 12
12. Actual demand in month 8 is 160 units. The 4-period moving
average in month 7 was 110 units. What is 5-period moving
average in month 8?
A) 100
MA47 = (A7+A6+A5+A4)/4 = 110
B) 110
C) 120
A7+A6+A5+A4 = 440
D) 140
A8 = 160
E) 150
MA58 = (A8+ A7+A6+A5+A4)/5
MA58 = (160+440)/5 = 120
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
9
Problem 13
13. Suppose the 5-period moving average in period 20 is equal to
800. Suppose period 16 demand is 850. Also suppose the demand
for period 21 is 900. Compute 5-period moving average for
period 21.
MA520 = (A20+A19+A18+A17+A16)/5 =800 = (
=(
+850)/5 =800 
+850 =4000

MA521 = (A21+ A20+A19+A18+A17)/5 =??? = (
=(
+A16)/5
=3150
+A21)/5
+900)/5 = (3150+900)/5 =810
MA521 = MA520 +(A21- A16)/5
MA521 = 800 +(900- 850) /5=810
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
10
Moving Average, from period t to t+1
Using the following data you can compute 4-period and 7-period
moving averages in period 20.
t
14
15
16
17
18
19
20
At
658 864 1110 634 855 738 910
(658+864+1110+634+855+738+910)/7 = 824.14
(634+855+738+910)/4 = 784.25
Now suppose you do not have the actual data.
You only the demand for period 21 to be 800, 4-period moving
average in period 20 to be 784.25, and 7-period moving average
in period 20 to be 824.14.
Can you compute 7-period moving average and 4 period moving
average in period 21 without using the original data?
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
11
Problem 14: n-Period Moving Average from t to t+1
Let us first check how do we compute 7-period moving average
using all the available data
MA720 = (A14+A15+A16+A17+A18+A19+A20)/7
MA720 = (658+864+1110+634+855+738+910)/7 = 824.14
Demand for period 21 is 800
MA721 = (864+1110+634+855+738+910+800)/7 = 844.43
MA720 = (864+1110+634+855+738 +910)/7 + (658)/7 = 824.14
MA721 = (800 )/7 + ( 864+1110+634+855+738+900)/7=844.43
Therefore, we can compute MA721 , in the following simple way
MA721 = MA720 +(A21- A14)/7
MA721 = 824. 14 +(800- 658) /7=844.43
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
12
Problem 15: n-Period Moving Average from t to t+1
Suppose 4-Period moving average in month 20 is 784.25, the
actual demand for period 17 is 634, and the demand for period 21
is 800. Compute 4-period moving average for period 21 without
using other data?
MA4 20 = (634+855+738+910)/4 = 784.25
MA421 = (855+738+910+800)/4 = 825.75
Therefore, we can compute MA421 , in the following simple way
MA421 = MA420 +(A21- A17)/4
MA421 = 784.25 +(800 - 634) /4=825.75
In general ( the data of period t minus the data of period t-n) / n
In Moving Average forecasting always
F(t+1) = MAt  F22= MA21
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
13
Problem 16: MAD from t to t+1
Actual and forecast data are available from period 1 to period 10.
In period 10: MAD = 110 and TS = 2.2, Forecast for period 11 is
1210 (F11=1210) , Actual demand in Period 11 is 1100 (A11=1100).
Compute MAD in period 11.
First Lets look at MAD in Period 10
MAD = Sum |A-F| /n
110 = Sum |A-F| /10
Sum |A-F| = 1100  Sum |A-F| from period 1 to 10 = 1100
In period 11, A-F = 1100 -1210 = -110
Sum |A-F| from period 1 to 11 = 1100 + 110 = 1210
MAD in period 11 = Sum |A-F| /11
MAD in period 11 = 1210/11 = 110
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
14
Problem 17: TS from t to t+1
Actual and forecast data are available from period 1 to period 10.
In period 10: MAD = 110 and TS = 2.2, Forecast for period 11 is
1210 (F11=1210) , Actual demand in Period 11 is 1100 (A11=1100).
MAD in Period 11 = 110. Compute TS in period 11.
First Lets look at TS in Period 10
TS = Sum(A-F)/MAD
2.2 = Sum(A-F)/110  Sum (A-F) from period 1 to 10 = 242
(A-F) in period 11 = 1100-1210 = -110
Sum (A-F) from period 1 to 11 = 242 -110 = 132
TS = Sum(A-F)/MAD
Sum(A-F) in Period 11 = 132
MAD in Period 11 = 110
TS in Period 11 = 132/110 = 1.2
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
15
Problem 18: TS UCL and LCL
What are the reasonable values for UCL and LCL in Tracking
Signal?
At is Actual and Ft is forecast of a random variable such as
demand.
Forecast error (A random Variable) Et =At-Ft has mean of 0.
MAD provides an estimate for the standard deviation of Et.
StdDev (Et) = 1.25 MAD. See for example
http://www.estepsoftware.com/papers/madrsquare.pdf
Et = Normal (0,1.25MAD)
If x = Normal(µ,σ)  Sum (x) = Normal(µ, √N σ)
StdDev [Sum(Et)] = √N StdDev (Et)
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
16
Problem 18: TS UCL and LCL
Et = Normal (0,1.25MAD)
Sum (Et) = Normal (0, √N 1.25MAD)
3 ≥ (Et -0)/(N 1.25 MAD) ≥ -3.
+ 3n 1.25 ≥ (Et -0)/MAD ≥ - 3N 1.25.
+ 3.75N ≥ (Et -0)/MAD ≥ - 3.75N.
Tracking Signal TS= Et/MAD with samples of size N is
distributed normally around 0
with UCL = 3.75N
and LCL =-3.75N
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
17
Problem 18: TS UCL and LCL
Et = Normal (0,1.25MAD)
Sum (Et) = Normal (0, √N 1.25MAD)
3 ≥ (Et -0)/N 1.25 MAD ≥ -3.
+ 3n 1.25 ≥ (Et -0)/MAD ≥ - 3N 1.25.
+ 3.75N ≥ (Et -0)/MAD ≥ - 3.75N.
Tracking Signal TS= Et/MAD with samples of size N is
distributed normally around 0
with UCL = 3.75N
and LCL =-3.75N
MA, MAD, TS Problems
Ardavan Asef-Vaziri
Jan-2015
18