Formula

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Examples
1) 3 year Exponential 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 Exponential Moving Average forecast

Solution:
`alpha=2/(n+1)=2/(3+1)=0.5`

(1) year	(2) Sales	(3) Exponential Smoothing `(alpha=0.5)`
1	5.2	5.2
2	4.9	`0.5*5.2+0.5*5.2=5.2`
3	5.5	`0.5*4.9+0.5*5.2=5.05`
4	4.9	`0.5*5.5+0.5*5.05=5.275`
5	5.2	`0.5*4.9+0.5*5.275=5.0875`
6	5.7	`0.5*5.2+0.5*5.0875=5.1438`
7	5.4	`0.5*5.7+0.5*5.1438=5.4219`
8	5.8	`0.5*5.4+0.5*5.4219=5.4109`
9	5.9	`0.5*5.8+0.5*5.4109=5.6055`
10	6	`0.5*5.9+0.5*5.6055=5.7527`
11	5.2	`0.5*6+0.5*5.7527=5.8764`
12	4.8	`0.5*5.2+0.5*5.8764=5.5382`
13		`0.5*4.8+0.5*5.5382=5.1691`

(1) year	(2) Sales	(3) Exponential Smoothing	(4) Error	(5) |Error|	(6) `"Error"^2`	(7) `|%"Error"|`
1	5.2	5.2
2	4.9	5.2
3	5.5	5.05
4	4.9	5.275	`4.9-5.275=-0.375`	`0.375`	`0.1406`	`7.65%`
5	5.2	5.0875	`5.2-5.0875=0.1125`	`0.1125`	`0.0127`	`2.16%`
6	5.7	5.1438	`5.7-5.1438=0.5562`	`0.5562`	`0.3094`	`9.76%`
7	5.4	5.4219	`5.4-5.4219=-0.0219`	`0.0219`	`0.0005`	`0.41%`
8	5.8	5.4109	`5.8-5.4109=0.3891`	`0.3891`	`0.1514`	`6.71%`
9	5.9	5.6055	`5.9-5.6055=0.2945`	`0.2945`	`0.0867`	`4.99%`
10	6	5.7527	`6-5.7527=0.2473`	`0.2473`	`0.0611`	`4.12%`
11	5.2	5.8764	`5.2-5.8764=-0.6764`	`0.6764`	`0.4575`	`13.01%`
12	4.8	5.5382	`4.8-5.5382=-0.7382`	`0.7382`	`0.5449`	`15.38%`
13		5.1691	Total	`3.411`	`1.7648`	`64.19%`

Forecasting errors

1. Mean absolute error (MAE), also called mean absolute deviation (MAD)
MAE`=1/n sum |e_i|=3.411/9=0.379`


2. Mean squared error (MSE)
MSE`=1/n sum |e_i^2|=1.7648/9=0.1961`


3. Root mean squared error (RMSE)
RMSE`=sqrt(MSE)=sqrt(0.1961)=0.4428`


4. Mean absolute percentage error (MAPE)
MAPE`=1/n sum |e_i/y_i|=64.19/9=7.13`

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2) 3 year Exponential 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 Exponential Moving Average forecast

Solution:
`alpha=2/(n+1)=2/(3+1)=0.5`

(1) year	(2) Sales	(3) Exponential Smoothing `(alpha=0.5)`
1	30	30
2	25	`0.5*30+0.5*30=30`
3	35	`0.5*25+0.5*30=27.5`
4	25	`0.5*35+0.5*27.5=31.25`
5	20	`0.5*25+0.5*31.25=28.125`
6	30	`0.5*20+0.5*28.125=24.0625`
7	35	`0.5*30+0.5*24.0625=27.0312`
8	40	`0.5*35+0.5*27.0312=31.0156`
9	30	`0.5*40+0.5*31.0156=35.5078`
10	45	`0.5*30+0.5*35.5078=32.7539`
11		`0.5*45+0.5*32.7539=38.877`

(1) year	(2) Sales	(3) Exponential Smoothing	(4) Error	(5) |Error|	(6) `"Error"^2`	(7) `|%"Error"|`
1	30	30
2	25	30
3	35	27.5
4	25	31.25	`25-31.25=-6.25`	`6.25`	`39.0625`	`25%`
5	20	28.125	`20-28.125=-8.125`	`8.125`	`66.0156`	`40.62%`
6	30	24.0625	`30-24.0625=5.9375`	`5.9375`	`35.2539`	`19.79%`
7	35	27.0312	`35-27.0312=7.9688`	`7.9688`	`63.501`	`22.77%`
8	40	31.0156	`40-31.0156=8.9844`	`8.9844`	`80.719`	`22.46%`
9	30	35.5078	`30-35.5078=-5.5078`	`5.5078`	`30.336`	`18.36%`
10	45	32.7539	`45-32.7539=12.2461`	`12.2461`	`149.9668`	`27.21%`
11		38.877	Total	`55.0195`	`464.8548`	`176.22%`

Forecasting errors

1. Mean absolute error (MAE), also called mean absolute deviation (MAD)
MAE`=1/n sum |e_i|=55.0195/7=7.8599`


2. Mean squared error (MSE)
MSE`=1/n sum |e_i^2|=464.8548/7=66.4078`


3. Root mean squared error (RMSE)
RMSE`=sqrt(MSE)=sqrt(66.4078)=8.1491`


4. Mean absolute percentage error (MAPE)
MAPE`=1/n sum |e_i/y_i|=176.22/7=25.17`

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