Calculate the RMSE between predicted and observed values. Input your values and visualize the differences between them.

**Root Mean Squared Error (RMSE):**

## Understanding Root Mean Squared Error (RMSE)

The **Root Mean Squared Error (RMSE)** is a standard way to measure the error of a model in predicting quantitative data. It represents the square root of the average squared differences between predicted values \( P_i \) and observed values \( O_i \).

### Formula for RMSE

$$ \text{RMSE} = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (P_{i} - O_{i})^2 } $$

### Example Calculation

Let's calculate the RMSE for the following observed and predicted values:

- Observed values: [34, 37, 44, 47, 48]
- Predicted values: [37, 40, 46, 44, 46]

Following the formula:

- Find the differences between each pair of predicted and observed values:
- Difference 1: \( 37 - 34 = 3 \)
- Difference 2: \( 40 - 37 = 3 \)
- Difference 3: \( 46 - 44 = 2 \)
- Difference 4: \( 44 - 47 = -3 \)
- Difference 5: \( 46 - 48 = -2 \)
- Square each difference:
- \( 3^2 = 9 \)
- \( 3^2 = 9 \)
- \( 2^2 = 4 \)
- \( (-3)^2 = 9 \)
- \( (-2)^2 = 4 \)
- Sum the squared differences: \( 9 + 9 + 4 + 9 + 4 = 35 \)
- Divide by the number of observations: \( 35 / 5 = 7 \)
- Take the square root: \( \sqrt{7} \approx 2.646 \)

Therefore, the RMSE for this example is approximately **2.646**.