R-squared in linear regression is a statistic that tells you what fraction of the variation in the dependent variable is explained by your regression line or model.

Quick Scoop: Core Idea

  • R-squared (also called the coefficient of determination) measures how well your regression line fits the data.
  • It is usually between 0 and 1 in standard linear regression setups.
  • Closer to 1 ⇒ the model captures most of the variability in the outcome; closer to 0 ⇒ it captures very little.

In simple linear regression with one predictor, R-squared is literally the square of the correlation between X and Y, which is where the name comes from.

How R-squared Is Calculated

Conceptually, R-squared compares how much error your model makes vs. a naive model that always predicts the mean of Y.

  • Total variation in Y is measured by the total sum of squares (TSS), the squared deviations of Y from its mean.
  • Unexplained variation is measured by the residual sum of squares (RSS), the squared differences between actual and predicted values.
  • R-squared is the fraction of total variation that is “explained” by the model (1 − RSS/TSS).

So if R-squared = 0.80, you’d say “the model explains about 80% of the variance in the dependent variable.”

Interpreting R-squared in Practice

R-squared is a goodness-of-fit measure, but “good” depends on context.

  • High R-squared (e.g., 0.9) often appears in physical or engineered systems where relationships are tight and noise is low.
  • Moderate or low R-squared can be normal in fields like social science or finance, where outcomes are noisy.
  • A higher R-squared means the fitted values lie closer to the regression line, i.e., smaller residuals on average.

R-squared does not prove causality; it only measures how strongly the model’s predictions move with the observed data.

Limitations and Pitfalls

Even though R-squared is popular, it has some important limitations.

  • Adding more predictors will never decrease R-squared; it usually goes up, even if the new variables are useless.
  • A very high R-squared can still come from a badly specified model (e.g., missing nonlinearity, overfitting, or omitted variables).
  • R-squared alone does not tell you whether coefficients are statistically significant or whether assumptions (linearity, homoscedasticity, etc.) hold.

That’s why adjusted R-squared, residual diagnostics, and domain knowledge are typically used alongside plain R-squared.

Simple Example (Story Style)

Imagine you are trying to predict house prices from square footage in a single city.

  • Model A: You just draw a flat line at the average house price. This is the “no model” baseline.
  • Model B: You fit a linear regression: price = a + b¡(square footage).

If Model B’s R-squared is 0.75, it means that, compared to Model A, the regression explains about 75% of the variation in prices across houses just by using square footage.

In other words: R-squared is a score from 0 to 1 that tells you “how much better than just predicting the average” your regression line really is.

TL;DR: R-squared in linear regression is the proportion of variance in the dependent variable that your model explains, comparing it to a baseline that always predicts the mean; higher values mean a tighter fit, but it doesn’t guarantee a correct or causal model.

Information gathered from public forums or data available on the internet and portrayed here.