What is mean squared error (MSE)?

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🔹 Formula

$$MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$

Where:

• $n$ = number of data points

• $y_i$ = actual value

• $\hat{y}_i$ = predicted value

🔹 Key Points

• Squaring the errors makes all differences positive (avoids cancellation).

• Larger errors get penalized more (because of squaring).

• Lower MSE = better model fit.

• MSE = 0 means perfect predictions.

🔹 Example

Actual values ($y$): [3, 5, 2]
Predicted values ($\hat{y}$): [2.5, 4.8, 2.2]

$$MSE = \frac{(3-2.5)^2 + (5-4.8)^2 + (2-2.2)^2}{3}$$ $$= \frac{0.25 + 0.04 + 0.04}{3} = 0.11$$

👉 The model’s MSE = 0.11, meaning predictions are close to actual values.

🔹 Pros & Cons

✅ Advantages

• Simple to calculate and widely used.

• Strongly penalizes large errors.

⚠️ Disadvantages

• Squared scale → error units are not interpretable (e.g., predicting prices in dollars → MSE in dollars²).

• Sensitive to outliers (a single large error dominates).

🔹 Relation to Other Metrics

• RMSE (Root Mean Squared Error) → square root of MSE, brings error back to original units.

• MAE (Mean Absolute Error) → less sensitive to outliers (uses absolute errors).

✅ In short:
MSE is the average squared error between predicted and actual values. It’s useful for comparing regression models but is best interpreted alongside RMSE and MAE.

Read more :

What is log loss?

What is R² score in regression?

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