Formula

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Examples
1) 3 year Weighted Moving Average forecast

year	1	2	3	4	5	6	7	8	9	10	11	12
Sales	5.2	4.9	5.5	4.9	5.2	5.7	5.4	5.8	5.9	6	5.2	4.8

Calculate 3 year Weighted Moving Average forecast with weight=1,2,1

Solution:
The value of table for `x` and `y`

x	1	2	3	4	5	6	7	8	9	10	11	12
y	5.2	4.9	5.5	4.9	5.2	5.7	5.4	5.8	5.9	6	5.2	4.8

The weights of the 3 years are respectively 1,2,1 and their sum is 4
Calculation of 3 year moving averages of the data

(1) year	(2) Sales	(3) 3 year weighted moving total	(4) 3 year weighted moving average `(3)-:4`
1	5.2
2	4.9	`1xx5.2+2xx4.9+1xx5.5=20.5`	`20.5-:4=5.125`
3	5.5	`1xx4.9+2xx5.5+1xx4.9=20.8`	`20.8-:4=5.2`
4	4.9	`1xx5.5+2xx4.9+1xx5.2=20.5`	`20.5-:4=5.125`
5	5.2	`1xx4.9+2xx5.2+1xx5.7=21`	`21-:4=5.25`
6	5.7	`1xx5.2+2xx5.7+1xx5.4=22`	`22-:4=5.5`
7	5.4	`1xx5.7+2xx5.4+1xx5.8=22.3`	`22.3-:4=5.575`
8	5.8	`1xx5.4+2xx5.8+1xx5.9=22.9`	`22.9-:4=5.725`
9	5.9	`1xx5.8+2xx5.9+1xx6=23.6`	`23.6-:4=5.9`
10	6	`1xx5.9+2xx6+1xx5.2=23.1`	`23.1-:4=5.775`
11	5.2	`1xx6+2xx5.2+1xx4.8=21.2`	`21.2-:4=5.3`
12	4.8

(1) year	(2) Sales	(3) 3 year weighted moving average	(4) Error	(5) |Error|	(6) `"Error"^2`	(7) `|%"Error"|`
1	5.2
2	4.9
3	5.5
4	4.9	5.125	`4.9-5.125=-0.225`	`0.225`	`0.0506`	`4.59%`
5	5.2	5.2	`5.2-5.2=0`	`0`	`0`	`0%`
6	5.7	5.125	`5.7-5.125=0.575`	`0.575`	`0.3306`	`10.09%`
7	5.4	5.25	`5.4-5.25=0.15`	`0.15`	`0.0225`	`2.78%`
8	5.8	5.5	`5.8-5.5=0.3`	`0.3`	`0.09`	`5.17%`
9	5.9	5.575	`5.9-5.575=0.325`	`0.325`	`0.1056`	`5.51%`
10	6	5.725	`6-5.725=0.275`	`0.275`	`0.0756`	`4.58%`
11	5.2	5.9	`5.2-5.9=-0.7`	`0.7`	`0.49`	`13.46%`
12	4.8	5.775	`4.8-5.775=-0.975`	`0.975`	`0.9506`	`20.31%`
13		5.3	Total	`3.525`	`2.1156`	`66.5%`

Forecasting errors

1. Mean absolute error (MAE), also called mean absolute deviation (MAD)
MAE`=1/n sum |e_i|=3.525/9=0.3917`


2. Mean squared error (MSE)
MSE`=1/n sum |e_i^2|=2.1156/9=0.2351`


3. Root mean squared error (RMSE)
RMSE`=sqrt(MSE)=sqrt(0.2351)=0.4848`


4. Mean absolute percentage error (MAPE)
MAPE`=1/n sum |e_i/y_i|=66.5/9=7.39`

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2) 3 year Weighted Moving Average forecast

year	1	2	3	4	5	6	7	8	9	10
Sales	30	25	35	25	20	30	35	40	30	45

Calculate 3 year Weighted Moving Average forecast with weight=1,2,1

Solution:
The value of table for `x` and `y`

x	1	2	3	4	5	6	7	8	9	10
y	30	25	35	25	20	30	35	40	30	45

The weights of the 3 years are respectively 1,2,1 and their sum is 4
Calculation of 3 year moving averages of the data

(1) year	(2) Sales	(3) 3 year weighted moving total	(4) 3 year weighted moving average `(3)-:4`
1	30
2	25	`1xx30+2xx25+1xx35=115`	`115-:4=28.75`
3	35	`1xx25+2xx35+1xx25=120`	`120-:4=30`
4	25	`1xx35+2xx25+1xx20=105`	`105-:4=26.25`
5	20	`1xx25+2xx20+1xx30=95`	`95-:4=23.75`
6	30	`1xx20+2xx30+1xx35=115`	`115-:4=28.75`
7	35	`1xx30+2xx35+1xx40=140`	`140-:4=35`
8	40	`1xx35+2xx40+1xx30=145`	`145-:4=36.25`
9	30	`1xx40+2xx30+1xx45=145`	`145-:4=36.25`
10	45

(1) year	(2) Sales	(3) 3 year weighted moving average	(4) Error	(5) |Error|	(6) `"Error"^2`	(7) `|%"Error"|`
1	30
2	25
3	35
4	25	28.75	`25-28.75=-3.75`	`3.75`	`14.0625`	`15%`
5	20	30	`20-30=-10`	`10`	`100`	`50%`
6	30	26.25	`30-26.25=3.75`	`3.75`	`14.0625`	`12.5%`
7	35	23.75	`35-23.75=11.25`	`11.25`	`126.5625`	`32.14%`
8	40	28.75	`40-28.75=11.25`	`11.25`	`126.5625`	`28.12%`
9	30	35	`30-35=-5`	`5`	`25`	`16.67%`
10	45	36.25	`45-36.25=8.75`	`8.75`	`76.5625`	`19.44%`
11		36.25	Total	`53.75`	`482.8125`	`173.88%`

Forecasting errors

1. Mean absolute error (MAE), also called mean absolute deviation (MAD)
MAE`=1/n sum |e_i|=53.75/7=7.6786`


2. Mean squared error (MSE)
MSE`=1/n sum |e_i^2|=482.8125/7=68.9732`


3. Root mean squared error (RMSE)
RMSE`=sqrt(MSE)=sqrt(68.9732)=8.305`


4. Mean absolute percentage error (MAPE)
MAPE`=1/n sum |e_i/y_i|=173.88/7=24.84`

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