Physics / Engineering
0.95 - 1.00
0.95 - 1.00Typical range
Relationships are often governed by physical laws and low measurement noise.
R-Squared
R-Squared (R²)What Percentage of the Story Does Your Correlation Tell?
Imagine you are predicting weight from height. If you know nothing, you guess the average. If you know height, your guess gets better. R² tells you exactly how much better.
Featured snippet answer
In one sentence: R² is the percentage of variation in one variable that is explained by the other.
Height → Weight
R² = 64.0%
Narrow ellipse
Height explains 64% of the variation in weight. The rest is up to diet, muscle mass, genetics, and other factors.
Explained variance
44.9%
The rest is explained by other factors.
Total variance100%
Explained
44.9%
44.9%
Unexplained55.1% of the variation
Model signalr² grows as r gets stronger
Your variable explains 44.9% of the variation. This is meaningful, but more than half may still come from factors outside this relationship.
Better prediction than guessing the average is the core idea here: R² shows how much that improvement comes from your variable.
Current band: Moderate explanatory power. In this range, the effect is usually described as moderate explanatory power.
What is R²?
Formal definition and plain English
Math language
R² is the proportion of variance in the dependent variable that is predictable from the independent variable.
Human translation
If you use X to predict Y, R² tells you how much better your prediction is than guessing the average.
Formula
R² = r² = Explained Variance / Total Variance
Squaring removes direction. r can be positive or negative, but explanatory power cannot be negative. That is why r = 0.80 and r = -0.80 both give R² = 0.64.
r vs R²
The ah-ha section
This is where most readers realize why r = 0.50 does not mean "half the story." R² translates that feeling into a real percentage.
±1.00
100.0%
Perfect prediction; every point sits on a line.
A straight line
±0.90
81.0%
Extremely strong; predictions are very accurate.
Very narrow ellipse
±0.80
64.0%
Strong; most variation is explained.
Narrow ellipse
±0.70
49.0%
Moderately strong; about half the story is explained.
Medium ellipse
±0.60
36.0%
Moderate; clear trend with substantial noise.
Wide ellipse
±0.50
25.0%
Moderate-looking r, but only one quarter explained.
Very wide ellipse
±0.30
9.0%
Weak; most variation remains unexplained.
Almost circular
±0.10
1.0%
Very weak; usually negligible in practice.
Nearly random cloud
The "r = 0.5 feels moderate" trap
When you see r = 0.50, it sounds like "half strength." But R² = 0.25 - meaning your variable only explains 25% of the variation. The other 75% is unexplained.
This is why researchers who only report r can accidentally oversell their findings. R² gives you the honest picture.
How to interpret
R² calculator and interpreter
R²
0.449
44.9%
Your variable explains 44.9% of the variation. This is meaningful, but more than half may still come from factors outside this relationship.
By field
What is a good R²?
The most important rule is simple: always compare within your field. The same R² can be weak in engineering and excellent in social psychology.
Medicine / Physiology
0.60 - 0.90
0.60 - 0.90Typical range
Biological systems are complex, but strong core variables can often be measured.
Psychology
0.10 - 0.40
0.10 - 0.40Typical range
Human behavior is shaped by many unmeasured or hard-to-measure factors.
Social Science
0.05 - 0.30
0.05 - 0.30Typical range
Social outcomes are noisy, multi-causal, and context-sensitive.
Finance / Economics
0.02 - 0.20
0.02 - 0.20Typical range
Markets are stochastic; very high R² can signal leakage or overfitting.
Education Research
0.10 - 0.35
0.10 - 0.35Typical range
Learning outcomes depend on prior knowledge, home context, teachers, and motivation.
An R² of 0.15 would be embarrassing in an engineering study but would be meaningful in social psychology. A financial model with R² = 0.80 should make you suspicious - markets are not that predictable.
Calculator
What the number can and cannot do
How to use it
Move the r input and watch the explained area grow or shrink. This turns an abstract percentage into something the eye can read instantly.
The tool is intentionally simple: no dependencies, no chart library, and no hidden math. Just r, R², and the share of variation explained.
Current output
r = 0.67 → R² = 44.9%
Your variable explains 44.9% of the variation. This is meaningful, but more than half may still come from factors outside this relationship.
Limitations
Where R² fails
Nonlinear relationships
R² assumes a linear relationship. If the true pattern is curved, Pearson r and R² can understate the real relationship.
Spearman correlationStart with the scatter plot and ask whether the pattern is actually linear before trusting R².
R² is not causation
A high R² only says the variables move together in a way your model can capture. It does not prove one caused the other.
Correlation does not imply causationMore variables can inflate R²
In multiple regression, adding more predictors usually pushes R² upward even if they are not meaningful. That is why adjusted R² exists.
High R² can still overfit
A model can fit existing data extremely well and still fail on new data. R² tells you about fit to current data, not future prediction by itself.
In analysis
What role does R² play?
Bivariate correlation analysis
Measures the shared variation between two variables.
Pearson correlation calculator
Pearson correlation calculatorSimple linear regression
The main goodness-of-fit measure for one predictor.
Simple regression guide
Simple regression guideMultiple linear regression
Useful, but should be paired with adjusted R².
Adjusted R² guide
Adjusted R² guidePartial correlation analysis
Shows explanatory power after controlling a third variable.
Partial correlation
Partial correlationIn simple correlation, R² is the shared variation ratio. In regression, it becomes the fit score. In partial correlation, it reflects explanatory power after controlling a third variable.
FAQ
R² FAQ
What does R-squared mean in correlation?+
R-squared in correlation tells you what percentage of the variation in one variable is explained by the other. For example, r = 0.70 gives R² = 0.49, meaning one variable explains 49% of the variation in the other.
What is a good R-squared value?+
There is no universal good R-squared because it depends entirely on your field. In physics, R² above 0.95 may be expected, while in psychology or social science, R² of 0.10 to 0.30 can be meaningful.
What is the difference between r and R-squared?+
The correlation coefficient r measures direction and linear strength from -1 to +1. R-squared equals r², ranges from 0 to 1, and measures the proportion of variance explained.
Can R-squared be negative?+
In simple correlation analysis, R² = r² and is always between 0 and 1. In regression software, a negative R² can appear when a model fits worse than a flat average line.
Is R-squared the same as correlation?+
No. R-squared is derived from correlation, but they answer different questions: r shows direction and strength, while R² shows the percentage of variance explained.
Why is my R-squared so low?+
A low R² can mean the relationship is weak, important variables are missing, the relationship is non-linear, or measurement error is high. In some fields, low R² values are normal and still useful.
Next steps
Primary CTA
Calculate correlation and significance
Use the Pearson calculator when you need r, p-value, and interpretation together.
Learn next
Understand statistical significance
Go from explained variation to statistical evidence.
Method choice
Not sure which correlation to use?
Compare Pearson and Spearman before choosing a method for your data.