How To Calculate Variance Explained By Variable Regression

Use this interactive calculator to estimate explained variance in regression using correlation, sums of squares, or the added contribution of a predictor in a hierarchical model. The tool reports R-squared, explained variance percentage, unexplained variance, and incremental contribution.

Expert Guide: How to Calculate Variance Explained by Variable Regression

Variance explained is one of the most important ideas in regression analysis because it tells you how much of the variability in an outcome can be accounted for by one predictor or by an entire set of predictors. When people ask how to calculate variance explained by variable regression, they are usually referring to one of three closely related quantities: the overall R-squared of a model, the R-squared derived from a correlation in simple regression, or the incremental variance explained by adding a single predictor to a model.

In practice, all three concepts are useful. If you are running a simple regression with one predictor, variance explained is often just the square of the Pearson correlation coefficient. If you are working from ANOVA output or regression diagnostics, you can calculate it from sums of squares. If you want to know what one variable adds beyond the others, you compare the full model to a reduced model and calculate the change in R-squared. This is especially common in psychology, education, epidemiology, economics, and business analytics.

What does variance explained mean?

Variance explained refers to the proportion of total variation in the dependent variable that is captured by the regression model. Suppose your outcome is exam score and your predictor is study hours. If your model has an R-squared of 0.49, then 49% of the variance in exam scores is explained by study hours, while 51% remains unexplained by that model. The unexplained portion may reflect omitted variables, measurement error, random noise, nonlinearity, or simply the natural complexity of the outcome.

This statistic is useful because it gives a standardized summary of model fit. A raw slope coefficient tells you direction and magnitude in the original units, but variance explained tells you how much of the overall pattern in the data the model captures.

The three main formulas you need

1. Simple regression from correlation: R-squared = r-squared
2. From sums of squares: R-squared = SSR / SST
3. Unique contribution of an added variable: Delta R-squared = R-squared full model – R-squared reduced model

These formulas are mathematically connected. In simple linear regression with one predictor, the squared correlation between the predictor and outcome equals the same R-squared you would get from sums of squares. In multiple regression, the contribution of one variable is usually evaluated through a model comparison rather than through squaring its simple correlation, because predictors often overlap with one another.

Method 1: Calculate variance explained from the correlation coefficient

In simple linear regression with one predictor, the explained variance is the square of the Pearson correlation coefficient. This is the fastest method when you already know r.

• If r = 0.70, then R-squared = 0.49, meaning 49% of the variance is explained.
• If r = -0.40, then R-squared = 0.16, meaning 16% of the variance is explained.
• The sign of r tells you the direction of association, but after squaring, R-squared is always nonnegative.

This is important because a strong negative relationship can explain just as much variance as a strong positive relationship. For example, if sleep deprivation and cognitive performance have a correlation of -0.80, then the variance explained is 0.64 or 64%. The predictor is highly informative even though the relationship is negative.

Method 2: Calculate variance explained from sums of squares

If you are using regression output from statistical software, you may be given the regression sum of squares and the total sum of squares. In that case:

R-squared = SSR / SST

Here, SSR is the variation explained by the regression model, while SST is the total variation in the dependent variable. If the model explains 185 units of variability out of 250 total units, then:

R-squared = 185 / 250 = 0.74

This means the model explains 74% of the variance. The remaining 26% is residual variance, often represented through SSE, the error sum of squares. Since SST = SSR + SSE, you can also calculate unexplained variance as SSE / SST.

Scenario	Statistic Given	Computation	Explained Variance
Simple regression, moderate relationship	r = 0.45	0.45 x 0.45 = 0.2025	20.25%
Simple regression, strong negative relationship	r = -0.78	0.78 x 0.78 = 0.6084	60.84%
Model ANOVA output	SSR = 320, SST = 500	320 / 500 = 0.64	64.00%
Model ANOVA output	SSR = 96, SST = 240	96 / 240 = 0.40	40.00%

Method 3: Calculate the variance explained by one variable in multiple regression

In multiple regression, the phrase variance explained by a variable usually means the additional variance captured when that variable is added to a model that already contains other predictors. This is also called incremental variance explained, change in R-squared, or Delta R-squared.

The formula is:

Delta R-squared = R-squared full model – R-squared reduced model

For example, suppose a reduced model with age, income, and education has R-squared = 0.31. After adding job satisfaction, the full model has R-squared = 0.44. The added variable contributes:

0.44 – 0.31 = 0.13

So job satisfaction explains an additional 13% of the variance beyond the other predictors. This is often the best answer when you need the contribution of one variable specifically, because predictors can share overlapping explanatory power.

Why not just square the variable’s correlation in multiple regression?

Because the simple correlation between one predictor and the outcome ignores overlap with other predictors. A variable may have a high zero-order correlation but add little unique variance if another predictor already captures the same information. Conversely, a variable with a modest simple correlation can still add valuable unique information once the other predictors are held constant.

Reduced Model R-squared	Full Model R-squared	Delta R-squared	Interpretation
0.28	0.34	0.06	The added predictor explains 6% additional variance.
0.41	0.43	0.02	The added predictor contributes a small unique increment.
0.52	0.67	0.15	The added predictor materially improves model fit.
0.71	0.72	0.01	The added predictor explains little unique variance.

Step by step example

Example A: Simple regression using correlation

1. Measure the correlation between advertising spend and sales growth.
2. Suppose the correlation is r = 0.62.
3. Square the value: 0.62 x 0.62 = 0.3844.
4. Convert to a percentage: 38.44%.
5. Interpretation: advertising spend explains about 38.44% of the variance in sales growth in this simple regression setting.

Example B: Regression ANOVA output

1. Suppose your model output reports SSR = 210 and SST = 300.
2. Compute 210 / 300 = 0.70.
3. Convert to a percentage: 70%.
4. Interpretation: the model explains 70% of the variance in the dependent variable.

Example C: Added contribution of one variable

1. Estimate a reduced model with demographics only. Let R-squared = 0.36.
2. Add work experience to the model. Let the full model have R-squared = 0.49.
3. Compute the difference: 0.49 – 0.36 = 0.13.
4. Interpretation: work experience explains an additional 13% of the variance after controlling for demographics.

How to interpret small, moderate, and large explained variance

There is no universal cutoff for what counts as good R-squared, because acceptable values depend on the field, measurement quality, theory, and complexity of the outcome. Human behavior and health outcomes often have lower R-squared values than tightly controlled physical systems. In social science, a model explaining 15% to 30% of the variance may still be meaningful. In engineering or calibration settings, analysts may expect much higher values.

• Low explained variance: the model captures only a small portion of variability, though effects may still be statistically important.
• Moderate explained variance: the model provides useful predictive structure but leaves substantial variance unexplained.
• High explained variance: the model tracks the outcome closely, though you still need to check assumptions and overfitting risk.

Common mistakes when calculating variance explained

• Confusing correlation with explained variance: if r = 0.50, explained variance is not 50%, it is 25% because you must square the correlation.
• Using a simple correlation in a multiple regression question: when the question asks for variance explained by one variable after controlling for others, use Delta R-squared from nested models.
• Ignoring the sign issue: negative correlations still produce positive R-squared values because the coefficient is squared.
• Interpreting R-squared as causation: a model can explain variance without establishing a causal mechanism.
• Overvaluing high R-squared: a very high R-squared can still come from overfitting, data leakage, or model misspecification.

Related statistics worth knowing

Adjusted R-squared

Adjusted R-squared penalizes the addition of unnecessary predictors. In multiple regression, it can be more informative than raw R-squared when comparing models with different numbers of variables.

Partial and semipartial correlations

These quantify unique relationships after controlling for other variables. In many settings, the squared semipartial correlation corresponds to the unique variance explained by a predictor.

Cohen’s f-squared

For model comparison, a useful effect size is f-squared = Delta R-squared / (1 – R-squared full model). This expresses the incremental contribution of the added variable relative to unexplained variance remaining in the full model.

When should you report variance explained?

You should report explained variance whenever you want readers to understand practical model fit, not just whether a coefficient is statistically significant. A clear reporting format might be:

• The full model explained 44% of the variance in job performance, R-squared = 0.44.
• Adding conscientiousness increased explained variance by 7 percentage points, Delta R-squared = 0.07.
• The predictor showed a moderate unique effect after accounting for prior covariates.

Authoritative references for deeper study

• Penn State Eberly College of Science: Regression Methods
• NIST Engineering Statistics Handbook
• UCLA Statistical Methods and Data Analytics

Bottom line

To calculate variance explained by variable regression, first determine which context you are in. If you have a single predictor and a correlation coefficient, square the correlation. If you have sums of squares from regression output, divide SSR by SST. If you need the contribution of one variable in multiple regression, compare the full and reduced models and calculate Delta R-squared. That last approach is usually the most accurate answer when the phrase by variable refers to the unique explanatory value of a specific predictor.

Use the calculator above to handle all three methods quickly. It will convert your input into R-squared, explained variance percentage, unexplained variance, and a clean visual chart so you can interpret the result immediately.