Foundations — R-squared and adjusted R-squared

Before you can trust the formula $R^2=1-\frac{S{S}_{res}}{S{S}_{tot}}$ you need to know what every single symbol in it means as a picture, not as a squiggle. Below we unpack them one at a time, each building on the last. Nothing is used before it is drawn.

1. The data points: $y_i$

Picture it: scatter a handful of dots on a graph. Each dot's height is one $y_i$. That's all a dataset is — a bunch of heights.

Why the topic needs it: these are the truths the model is trying to reproduce. Everything else — the average, the misses, the score — is built by comparing things to these dots.

2. The counter and the summation sign: $n$ and $\sum$

Why the topic needs it: $R^2$ is built from sums over every data point. Without $\sum$ we could not write "add up all the misses" compactly.

3. The average: $\bar{y}$

Picture it: draw one flat horizontal line through the middle of your dots. That line is $\bar{y}$. It's the "I know nothing, so I'll just guess the average for everyone" line.

4. The model's guesses: ${\hat{y}}_i$

Picture it: if the model is a straight line (as in Linear Regression), then ${\hat{y}}_i$ is the height of that sloped line directly above data point $i$.

Why the topic needs it: the whole point of $R^2$ is to compare the guesses ${\hat{y}}_i$ against the truths $y_i$. Both symbols must be crystal clear before we can talk about "error."

5. A single miss: the residual $y_i-{\hat{y}}_i$

Why we square it — squaring is the tool, and here's why that tool: if we just added raw residuals, the positive ones ($+10$) and negative ones ($-10$) would cancel and hide the errors, wrongly reporting "no error." Squaring turns every gap positive (a negative times a negative is positive) and punishes big misses more than small ones ($10^2=100$ vs $2^2=4$). This is exactly the idea behind Mean Squared Error.

6. Adding up the squared misses: $S{S}_{res}$ and $S{S}_{tot}$

Now we combine everything: the counter, the sum, the mean, the guesses, and the squaring.

Now the master formula reads in plain English: $R^2=1-\frac{S{S}_{res}}{S{S}_{tot}}=1-\frac{\text{model’s leftover error}}{\text{baseline’s error}}$ If the model kills most of the error, the fraction is near $0$, so $R^2$ is near $1$. If the model does no better than the average, $S{S}_{res}=S{S}_{tot}$, the fraction is $1$, and $R^2=0$.

7. The bookkeeping symbols for adjusted $R^2$: $p$ and degrees of freedom

How these foundations feed the topic

Equipment checklist

Test yourself — cover the right side and answer out loud.