more details on how you got 81 and the rest on Square the Deviations

Let's delve into more detail on how to calculate each squared deviation:

### Steps for Squaring the Deviations

After finding the mean, which is \(-4\), the next step is to calculate the deviation of each data point from the mean and then square each of these deviations. Here's how it works:

1. **Data Point:** \( 5 \)
   - **Deviation:** \( 5 - (-4) = 5 + 4 = 9 \)
   - **Squared Deviation:** \( 9^2 = 81 \)

2. **Data Point:** \( -12 \)
   - **Deviation:** \( -12 - (-4) = -12 + 4 = -8 \)
   - **Squared Deviation:** \( (-8)^2 = (-8) \times (-8) = 64 \)

3. **Data Point:** \( 5 \)
   - **Deviation:** \( 5 - (-4) = 9 \)  (same calculation as for the first data point)
   - **Squared Deviation:** \( 9^2 = 81 \)

4. **Data Point:** \( -12 \)
   - **Deviation:** \( -12 - (-4) = -8 \)  (same as for the second data point)
   - **Squared Deviation:** \( (-8)^2 = 64 \)

5. **Data Point:** \( 5 \)
   - **Deviation:** \( 5 - (-4) = 9 \)  (same as for the first and third data points)
   - **Squared Deviation:** \( 9^2 = 81 \)

6. **Data Point:** \( -12 \)
   - **Deviation:** \( -12 - (-4) = -8 \)  (same as for the second and fourth data points)
   - **Squared Deviation:** \( (-8)^2 = 64 \)

7. **Data Point:** \( -7 \)
   - **Deviation:** \( -7 - (-4) = -7 + 4 = -3 \)
   - **Squared Deviation:** \( (-3)^2 = (-3) \times (-3) = 9 \)

### Summary of Squared Deviations:
- **81** (from 5)
- **64** (from -12)
- **81** (from 5)
- **64** (from -12)
- **81** (from 5)
- **64** (from -12)
- **9** (from -7)

Each squared deviation is found by subtracting the mean from the original data point, then squaring the result. This squaring process ensures that all deviations are positive, which is important when calculating statistical measures like variance.

Do you have any further questions or need more details?

Here are 5 related questions you might find interesting:

1. How would you calculate the sum of all squared deviations for this dataset?
2. What happens to the variance if all data points are multiplied by a constant factor?
3. How does the concept of squared deviations relate to the least squares method used in regression analysis?
4. Can you find the standard deviation of the dataset given the squared deviations?
5. How do you interpret the squared deviations in terms of data spread or variability?

**Tip:** Squaring deviations emphasizes larger differences from the mean, which can be useful for identifying outliers in a dataset.

Explore the detailed process of calculating squared deviations from the mean using a dataset, with step-by-step examples and explanations. Learn how to apply this concept in statistical analysis to measure data variability.

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