# What's the difference between the variance and the mean squared error?

I'm surprised this hasn't been asked before, but I cannot find the question on stats.stackexchange.

This is the formula to calculate the variance of a normally distributed sample:

$$\frac{\sum(X - \bar{X}) ^2}{n-1}$$

This is the formula to calculate the mean squared error of observations in a simple linear regression:

$$\frac{\sum(y_i - \hat{y}_i) ^2}{n-2}$$

What's the difference between these two formulas? The only difference I can see is that MSE uses $n-2$. So if that's the only difference, why not refer to them as both the variance, but with different degrees of freedom?

• What is it about the wikipedia page here that is not clear? – TrynnaDoStat Mar 5 '15 at 19:32
• Variance is the average of squared deviation of the observations from the mean. The MSE in contrast is the average of squared deviations of the predictions from the true values. – random_guy Mar 5 '15 at 19:38
• Both "variance" and "mean squared error" have multiple formulas and varying applications. To clarify your question, could you (a) describe what kind of data you are applying these concepts to and (b) give formulas for them? (It's likely that in so doing you will discover the answer to your question, too.) – whuber Mar 5 '15 at 19:41
• There's a more general formula, which both are special cases of: $\frac{\sum_i(y_i-\hat{y}_i)^2}{n-p}$ where $p$ is the number of parameters estimated in obtaining $\hat{y}$ – Glen_b Mar 6 '15 at 3:05

## 2 Answers

The mean squared error as you have written it for OLS is hiding something:

$$\frac{\sum_{i}^{n}(y_i - \hat{y}_i) ^2}{n-2} = \frac{\sum_{i}^{n}\left[y_i - \left(\hat{\beta}_{0} + \hat{\beta}_{x}x_{i}\right)\right] ^2}{n-2}$$

Notice that the numerator sums over a function of both $y$ and $x$, so you lose a degree of freedom for each variable, hence $n-2$. In the formula for the sample variance, the numerator is a function of a single variable, so you lose just one degree of freedom in the denominator.

However, you are on track in noticing that these are conceptually similar quantities. The sample variance measures the spread of the data around the mean (in squared units), while the MSE measures the vertical spread of the data around the regression line (in squared vertical units).

• @amoeba Hey! Thanks for the attention. Is there an official CV style guide that prompted this edit? If so I wanna learn of it. If not, well, Glen_b once rightly admonished me for being colonizing with my personal style preferences and edits to others Qs and As. What do you think? (And I ask this in a collegial tone: I think your edit does add something. Just wanna understand our editing values better.) – Alexis Mar 7 '15 at 15:10
• I don't think there is any official CV style guide making this suggestion, but in LaTeX there are inline formulas (marked with one dollar sign) that are rendered directly in the block of text, and displayed formulas (marked with two dollar signs) that are rendered on a separate line. Displayed formulas use different layout. Your formula was originally on a separate line but marked with one dollar sign; I don't think this makes sense. However, you are right about personal preferences, so feel free to roll back with apologies. The reason I edited was that I was fixing a typo in the Q anyway. – amoeba Mar 7 '15 at 15:23

In the variance formula, the sample mean approximates the population mean. The sample mean is calculated for a given sample with n data points. Knowing the sample mean leaves us with only n-1 independent data points as the nth data point is constrained by the sample mean, so (n-1) DOF in the denominator in the variance formula. To get the estimated value of y (= b0 + b1*x) in MSE formula, we need to estimate both b0 i.e. the intercept as well as b1 i.e. the slope so we lose 2 DOFs and so is the reason for (n-2) in the denominator in the MSE formula.