Consider the following data:



Monthly Profit of a GymMonth

Jan-12

Feb-12

Mar-12

Apr-12

May-12

Jun-12

Jul-12

Aug-12

Sep-12

Profit ($)

5,7505,750

5,5035,503

5,0645,064

4,7574,757

5,2005,200

5,6785,678

6,6196,619

6,1956,195

5,9255,925

Step 2 of 5 :  

What are the MAD, MSE and MAPE scores for the three-period moving average? Round any intermediate calculations, if necessary, to no less than six decimal places, and round your final answer to one decimal place.




To calculate the Mean Absolute Deviation (MAD), Mean Squared Error (MSE), and Mean Absolute Percentage Error (MAPE) for the three-period moving average, we'll follow these steps:

### Step 1: Calculate the Three-Period Moving Averages
First, we need the three-period moving averages for each month starting from April-12 to September-12:

\[
\text{Moving Average for Apr-12} = \frac{5750 + 5503 + 5064}{3} = \frac{16317}{3} = 5439.000000
\]
\[
\text{Moving Average for May-12} = \frac{5503 + 5064 + 4757}{3} = \frac{15324}{3} = 5108.000000
\]
\[
\text{Moving Average for Jun-12} = \frac{5064 + 4757 + 5200}{3} = \frac{15021}{3} = 5007.000000
\]
\[
\text{Moving Average for Jul-12} = \frac{4757 + 5200 + 5678}{3} = \frac{15635}{3} = 5211.666667
\]
\[
\text{Moving Average for Aug-12} = \frac{5200 + 5678 + 6619}{3} = \frac{17497}{3} = 5832.333333
\]
\[
\text{Moving Average for Sep-12} = \frac{5678 + 6619 + 6195}{3} = \frac{18492}{3} = 6164.000000
\]

### Step 2: Calculate the Errors
The errors are calculated as the difference between the actual profit and the moving average:

\[
\text{Error for Apr-12} = 4757 - 5439.000000 = -682.000000
\]
\[
\text{Error for May-12} = 5200 - 5108.000000 = 92.000000
\]
\[
\text{Error for Jun-12} = 5678 - 5007.000000 = 671.000000
\]
\[
\text{Error for Jul-12} = 6619 - 5211.666667 = 1407.333333
\]
\[
\text{Error for Aug-12} = 6195 - 5832.333333 = 362.666667
\]
\[
\text{Error for Sep-12} = 5925 - 6164.000000 = -239.000000
\]

### Step 3: Calculate MAD (Mean Absolute Deviation)
MAD is the average of the absolute errors:

\[
\text{MAD} = \frac{| -682.000000 | + | 92.000000 | + | 671.000000 | + | 1407.333333 | + | 362.666667 | + | -239.000000 |}{6} = \frac{682.000000 + 92.000000 + 671.000000 + 1407.333333 + 362.666667 + 239.000000}{6} = \frac{3454.000000}{6} = 575.6667 \approx 575.7
\]

### Step 4: Calculate MSE (Mean Squared Error)
MSE is the average of the squared errors:

\[
\text{MSE} = \frac{(-682.000000)^2 + (92.000000)^2 + (671.000000)^2 + (1407.333333)^2 + (362.666667)^2 + (-239.000000)^2}{6}
\]

\[
= \frac{465124.000000 + 8464.000000 + 450241.000000 + 1980589.444444 + 131527.111111 + 57121.000000}{6}
\]

\[
= \frac{3082066.555555}{6} = 513677.759259 \approx 513677.8
\]

### Step 5: Calculate MAPE (Mean Absolute Percentage Error)
MAPE is the average of the absolute percentage errors:

\[
\text{MAPE} = \frac{\frac{| -682.000000 |}{4757} + \frac{| 92.000000 |}{5200} + \frac{| 671.000000 |}{5678} + \frac{| 1407.333333 |}{6619} + \frac{| 362.666667 |}{6195} + \frac{| -239.000000 |}{5925}}{6}
\]

\[
= \frac{0.143399 + 0.017692 + 0.118178 + 0.212606 + 0.058544 + 0.040336}{6}
\]

\[
= \frac{0.590755}{6} = 0.098459 \approx 9.8\%
\]

### Summary of Results
- **MAD** = 575.7
- **MSE** = 513677.8
- **MAPE** = 9.8%

Would you like more details or have any other questions? Here are some related questions:

1. How would changing the number of periods in the moving average affect the MAD, MSE, and MAPE scores?
2. What is the significance of each error metric (MAD, MSE, MAPE) in evaluating forecast accuracy?
3. How can the errors be reduced in future predictions?
4. What would the errors be if we used a different moving average period (e.g., 4-period)?
5. How would you explain the difference between MAD and MAPE to someone unfamiliar with these concepts?

**Tip:** MAPE is especially useful in cases where you need to understand the accuracy of forecasts in percentage terms, making it easier to interpret in a business context.

Learn how to calculate Mean Absolute Deviation (MAD), Mean Squared Error (MSE), and Mean Absolute Percentage Error (MAPE) for a three-period moving average. Step-by-step solution with detailed calculations and interpretations.

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