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 ($)

5,3505,350

5,1035,103

4,8244,824

4,4374,437

5,0805,080

5,3585,358

5,8195,819

6,0916,091

5,8375,837

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 a three-period moving average, we'll proceed with the following steps:

### 1. **Calculate the Three-Period Moving Average:**

The three-period moving average is calculated by averaging the profits of the previous three months. Since the first three months (Jan, Feb, and Mar) do not have three previous months to average, we start calculating the moving average from the fourth month (Apr-12).

\[
\text{Moving Average for Apr-12} = \frac{(5,350 + 5,103 + 4,824)}{3} = \frac{15,277}{3} = 5,092.333333
\]

Similarly, we calculate for the other months:

- **May-12:**
\[
\frac{(5,103 + 4,824 + 4,437)}{3} = \frac{14,364}{3} = 4,788.000000
\]
- **Jun-12:**
\[
\frac{(4,824 + 4,437 + 5,080)}{3} = \frac{14,341}{3} = 4,780.333333
\]
- **Jul-12:**
\[
\frac{(4,437 + 5,080 + 5,358)}{3} = \frac{14,875}{3} = 4,958.333333
\]
- **Aug-12:**
\[
\frac{(5,080 + 5,358 + 5,819)}{3} = \frac{16,257}{3} = 5,419.000000
\]
- **Sep-12:**
\[
\frac{(5,358 + 5,819 + 6,091)}{3} = \frac{17,268}{3} = 5,756.000000
\]

### 2. **Calculate Errors:**

For each month starting from Apr-12, calculate the error, which is the difference between the actual profit and the moving average.

\[
\text{Error} = \text{Actual Profit} - \text{Moving Average}
\]

- **Apr-12:** $ 4,437 - 5,092.333333 = -655.333333 $
- **May-12:** $ 5,080 - 4,788.000000 = 292.000000 $
- **Jun-12:** $ 5,358 - 4,780.333333 = 577.666667 $
- **Jul-12:** $ 5,819 - 4,958.333333 = 860.666667 $
- **Aug-12:** $ 6,091 - 5,419.000000 = 672.000000 $
- **Sep-12:** $ 5,837 - 5,756.000000 = 81.000000 $

### 3. **Calculate MAD (Mean Absolute Deviation):**

MAD is the average of the absolute errors.

\[
\text{MAD} = \frac{| -655.333333 | + | 292.000000 | + | 577.666667 | + | 860.666667 | + | 672.000000 | + | 81.000000 |}{6} = \frac{3,138.666667}{6} = 523.1111
\]

**MAD ≈ 523.1**

### 4. **Calculate MSE (Mean Squared Error):**

MSE is the average of the squared errors.

\[
\text{MSE} = \frac{(-655.333333)^2 + (292.000000)^2 + (577.666667)^2 + (860.666667)^2 + (672.000000)^2 + (81.000000)^2}{6} = \frac{1,801,825.666667}{6} = 300304.2778
\]

**MSE ≈ 300304.3**

### 5. **Calculate MAPE (Mean Absolute Percentage Error):**

MAPE is the average of the absolute percentage errors. For each month:

\[
\text{MAPE} = \frac{| \frac{-655.333333}{4,437} | \times 100 + | \frac{292.000000}{5,080} | \times 100 + \cdots + | \frac{81.000000}{5,837} | \times 100}{6}
\]

\[
\text{MAPE} = \frac{14.767296 + 5.748032 + 10.793493 + 14.791356 + 11.033986 + 1.388363}{6} = 9.7534
\]

**MAPE ≈ 9.8%**

### Final Results:
- **MAD:** 523.1
- **MSE:** 300304.3
- **MAPE:** 9.8%

Would you like further details or have any questions?

### Related Questions:
1. What are the implications of a high MSE in this context?
2. How does MAPE differ from MAD and MSE in practical use?
3. Can you explain the advantages of using a three-period moving average?
4. What might cause large errors in the moving average predictions?
5. How would the calculations change if a four-period moving average was used?

### Tip:
Always check for outliers or significant changes in your data when calculating moving averages, as these can distort your results.

Learn how to calculate Mean Absolute Deviation (MAD), Mean Squared Error (MSE), and Mean Absolute Percentage Error (MAPE) for a three-period moving average using monthly profit data. Understand the application of statistical measures in time series analysis. Suitable for advanced high school students and anyone interested in statistical analysis.

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