MSE Calculator

What Is Mean Squared Error (MSE)?

Mean squared error measures how far predicted values sit from observed values. It averages the squared residuals, so large errors are penalized more than small ones and positive and negative residuals cannot cancel. Related tools: Standard Deviation Calculator and Linear Regression Calculator.

MSE and SSE Formulas

For observed values $x_i$ and predicted values $y_i$:

$$\text{SSE} = \sum_{i=1}^{n}(x_i - y_i)^2$$

$$\text{MSE} = \frac{1}{n}\sum_{i=1}^{n}(x_i - y_i)^2 = \frac{\text{SSE}}{n}$$

$$\text{RMSE} = \sqrt{\text{MSE}}$$

This calculator can set every prediction to the sample mean, or accept custom predicted values of the same length.

Example

For the sample $3, 15, 6, 3, 44, 8, 15, 9, 7, 25, 24, 5, 88, 44, 3, 21$, the mean is $20$. Squared deviations sum to SSE $= 7590$, so:

$$\text{MSE} = 7590 / 16 = 474.375, \quad \text{RMSE} \approx 21.78$$

Why Square the Errors?

Without squaring (or taking absolute values), positive and negative residuals can cancel and hide poor fit. Squaring makes every residual contribute positively. RMSE restores the original units after averaging squares.