The formula
How to calculate r squared
R squared is the proportion of variation in the outcome that a model accounts for. An R² of 0.61 means the model explains 61% of the variance and leaves 39% unexplained.
Adding predictors can only ever raise R², even when they are pure noise. Adjusted R² penalises each additional variable, so it can fall — which is what makes it the right measure for comparing models of different sizes.
The calculator asks for:
- Correlation coefficient
- Number of observations
- Number of predictors
The result updates on every keystroke. The URL updates too, which makes the filled-in version easy to bookmark or send to someone else.
Units matter more here than the arithmetic itself: the formula assumes a specific set of units for each input, stated next to the field, and converting into those units first is usually the difference between a correct result and one that is wrong by a clean power of ten.
Why r squared matters
This kind of calculation comes up in coursework, in a laboratory or field setting, and in professional practice, and the arithmetic is identical in every case — only the numbers being fed into it, and what is riding on getting them right, actually change.
Beyond a single check, the same calculation is worth rerunning whenever a measured input changes — a new reading, a corrected value, an updated assumption — since the result here always reflects exactly what is currently in the fields above rather than a value calculated once and then left stale.
A formula like this one is rarely the last step in a piece of work — the figure it produces usually feeds into a further calculation, a comparison against a published value, or a write-up that needs to state both the result and how confident it is. Getting this step right the first time, rather than propagating a small arithmetic slip through several more steps, is the main practical reason to check a manual calculation against a tool like this one before building on top of it.
Where a calculation like this one is part of a larger piece of work, it is generally worth running it with a round, easy-to-check set of numbers first — inputs of exactly 1, 10 or 100 — purely to confirm the formula is being applied correctly, before switching to the real measured values the actual result depends on.
Worked example
A concrete run-through, using the values already in the fields:
- Correlation coefficient: 0.78
- Number of observations: 30
- Number of predictors: 3
That gives:
- r squared: 0.6084
- Adjusted r squared: 0.56322
- Variance left unexplained: 39.16 %
The figures above are the calculator's own default values, shown purely so the working is visible rather than hidden — the same steps apply exactly to your own numbers, entered in the fields at the top of this page.
Reading the result
What counts as good depends entirely on the field. Physical sciences routinely expect above 0.9; models of human behaviour are doing well above 0.3. A very high R² on social data is more often a sign of overfitting than of insight.
Where this goes wrong. R² says nothing about whether the model is correctly specified. Anscombe's quartet is four datasets with identical regression statistics and completely different shapes — the residual plot tells you what R² cannot.
A result that is wrong by an exact factor of ten, a hundred or a similar round number is almost always a units error rather than a mistake in the formula itself — checking each input against the unit stated beside it is the fastest way to track it down.
It depends on the domain. Above 0.7 is strong in most applied settings, 0.3–0.5 is normal for social science, and anything above 0.95 on messy real-world data deserves suspicion.
Whenever comparing models with different numbers of predictors, which is most of the time in multiple regression. Plain R² always favours the larger model.
The answer it gives you is r squared. With 0.78 correlation coefficient, 30 number of observations and 3 number of predictors, that comes to 0.6084. Change any field and the figure moves with it.
Generally, no more than the least precise input justifies — a result reported to six decimal places from inputs measured to two significant figures is implying a precision the calculation does not actually have. The calculator shows full precision so you can round appropriately for your own use.
Yes — the equation shown in the formula section above is the standard form used in textbooks and reference material for this calculation, not a simplified or approximate version.
Yes, in the sense that it applies the correct standard formula and returns an accurate result for the inputs given — but check your own course or publication's requirements for how results should be rounded, presented and referenced, since those conventions vary and are not something a calculator can know on your behalf.