Mean Squared Error Vs Sum Squared Error What is the difference between Mean Squared Error and Sum Squared Error in Linear regression?
 A: Let there be the following regression equation: $
\hat y = X\hat\beta
$.
The true values are $y = (y_1,\dots,y_N)$. Using analogous notation, the predicted values are $\hat y = (\hat y_1,\dots,\hat y_N)$.
There are $N$ observations.
$$
SSE = \sum_{i = 1}^N\big(y_i - \hat y_i\big)^2
$$
$$
MSE = \dfrac{1}{N}\sum_{i = 1}^N\big(y_i - \hat y_i\big)^2
$$
The names describe the quantities quite well. "Sum of squared errors" (SSE) is the sum of the squared errors (really residuals, but this is just a slang), and the "mean squared error" (MSE) is the mean of the squared errors (again, really residuals, but "errors" is an acceptable slang in many circles).
You can get into some funky numbers out front of the $MSE$, not just $\dfrac{1}{N}$. Among these are $\dfrac{1}{N-1}$, $\dfrac{1}{N-2}$, $\dfrac{1}{N-p}$, and$\dfrac{1}{2N}$. The first three have to do with creating an unbiased estimator of the variance of the error term (de-emphasized in machine-learning, for better or for worse), and the final one has to do with mathematical convenience when we take derivatives.
A: As noticed in the comments and in the answer by Dave, the difference is about dividing the sum by the sample size. The practical consequence is that you can use one form over the other to simplify some math, e.g.

*

*When you partition the sum of squared errors the math is less clumsy than if you put $\tfrac{1}{N}$ everywhere.

*On another hand, when you are computing mean squared error on different samples of data, it will not be influenced by sample size, while the sum of squared errors would, e.g. 2 + 2 + 2 = 6, while (2 + 2 + 2)/3 = 2, and when you increase the number of elements 2 + 2 + 2 + 2 + 2 = 10, and (2 + 2 + 2 + 2 + 2)/5 = 2. It can make the metric easier to interpret.

