# Does $R^2$ interpretable as the proportion of *variation* explained, or the proportion of *variance* explained?

What is the correct term to be used in the title phrase? Wikipedia says:

$\dots$ The coefficient of determination $R^2$ is a measure of the global fit of the model. Specifically, $R^2$ is an element of $[0, 1]$ and represents the proportion of variability in $Y_i$ that may be attributed to some linear combination of the regressors (explanatory variables) in X.

$R^2$ is often interpreted as the proportion of response variation "explained" by the regressors in the model. Thus, $R^2 = 1$ indicates that the fitted model explains all variability in y, while $R^2 = 0$ indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope = 0, intercept = $\bar{y}$) between the response variable and regressors). An interior value such as $R^2 = 0.7$ may be interpreted as follows: "Seventy percent of the variance in the response variable can be explained by the explanatory variables. The remaining thirty percent can be attributed to unknown, lurking variables or inherent variability."

Are any of variability, variance, or variation malapropisms? Which is standard terminology?

• Variance is formal and it is a statistic with the precise definition (mean squared deviation). Variability and variation are not formal and may have different statistics to express them. Variability seems to be a bit closer to variance by meaning than variation. Quite often variability or variation are used as the substitutes, synonyms of variance, but the context lets know about it. Sum squared deviation (the scatter) directly tied with variance is especially often called variability/variation. Jan 20, 2015 at 0:47

I've heard all three but I don't particularly like any of them (with variance being at the bottom of the list ). Really, I prefer calling $R^{2}$ a measure of fit for our model to the data (and if you really want to use $R^{2}$, use its adjusted version instead.)

The reason I don't like calling it a variance is because only one term in the expression for it proportional to the variance. It's form is $1-\frac {SS_{E}}{SS_{T}}$, right? In that expression only $SS_{T}$ is proportional to a variance. Considering, as the comment claimed, that variance is a well defined formal quantity it doesn't seem accurate to call what we get from $R^{2}$ a proportion of variance explained.

• SSE is proportional to residual variance, isn't it? And $R^2$ is SSFitted/SSTotal, where SSFitted is proportional to the variance of fitted values. Mar 6, 2017 at 19:34

I like to explain R-squared to clients as follows - it's just the squared correlation between the predicted values of the Dependent Variable and the actual values of the Dependent Variable.

if you plot the former against the latter as a scatter chart, and fit a regression line, then the R-squared of that regression line is the same as the R-squared of your original regression.

Plotting actual vs predicted is also a rather good way of showing how bad an R-squared of (say) 70% really is - there is lots of scatter even at what might be thought, at first sight, to be a relatively high R-squared value.

• Maybe I'm wrong, but I'm not totally convinced that R2 is a correlation between the predicted and actual values. Can someone else confirm it?
– ABCD
Jun 2, 2015 at 8:24
• @StudentT This statistic is called $R^2$ precisely because it is the square of $R$, the (sample) correlation coefficient between actual and predicted values. See stats.stackexchange.com/questions/87963 for explanations and the conditions under which this is true.
– whuber
Jun 2, 2015 at 13:26

Informally, R2 is a measure for the variation or variability. In this context, variation measures the size of the residuals. Clearly, variation isn't equal to the variance. Variance is simply a measure of the deviation from the sample mean.

• This is not correct. $R^2$ is a ratio of two "variabilities": SSFitted and SSTotal. Describing $R^2$ as a measure of variability could be misleading. A measure of variability of what? Mar 6, 2017 at 19:41

In the context of linear regression, $$R^2$$ is a measure (estimate?) of the proportion of variance explained by the model. It seems to me that if we want to be precise, variance is the only word that can be used. To say that it is a measure of proportion of response variation is at best informal; the quote from Wikipedia is misleading, as this answer to my question In linear regression, does $R^2$ really measure the fraction of explained variation? explains. Someone should rewrite the Wikipedia article!

“Variation” and “variability” are English words that have colloquial meanings that basically everyone more-or-less understands: the extent to which the values are different. If there is minimal variability, the values are close together; is there is extensive variation, the values are far apart.

“Variance” is technical nomenclature in statistics that refers to a specific way of quantifying the colloquial notion of variability or variation.

In this regard, the standard definition of $$R^2$$ addresses the degree to which the variability/variation in the data is explained by the model, using variance as the mathematical definition of that variability.

Nothing stops us from using alternative measures of variability, such as comparing the IQRs of the raw data and the residuals or the mean absolute errors, except that this particular way of measuring the degree to which the variability is explained aligns with some other nice ideas related to model fit that I discuss here.