How to Calculate Sum Square Contribution From One Variable
Use this premium calculator to estimate how much of the total variation in an outcome is explained by a single predictor in a simple regression setting.
Results
Enter your values and click Calculate Contribution to see the explained sum of squares, unexplained sum of squares, and percent contribution.
What this calculator does
This tool estimates the sum square contribution from one variable using a simple and widely taught identity:
If you enter a correlation coefficient, the calculator first squares it:
That means the calculator is ideal when you have one predictor and want to know how much of the total variation in the outcome is associated with that predictor.
- Works with correlation r or R-squared input
- Returns explained sum of squares and residual sum of squares
- Visualizes explained versus unexplained variation
- Useful for introductory regression, ANOVA intuition, and model interpretation
Expert Guide: How to Calculate Sum Square Contribution From One Variable
The phrase sum square contribution from one variable usually refers to the amount of total variation in a dependent variable that can be attributed to a single predictor. In practical statistics, this comes up in regression, ANOVA, and model comparison. If you are trying to measure how much one variable helps explain an outcome, you are really asking a variance decomposition question: out of the total sum of squared deviations in the response, what part is explained by the predictor?
In the simplest one-predictor case, the calculation is straightforward. Once you know the total sum of squares for the outcome, often written as SST, and the strength of the relationship between the predictor and the outcome, written as R-squared or derived from the correlation coefficient r, the variable’s contribution can be estimated as:
This is especially useful in simple linear regression, where there is one predictor and one outcome. In that setting, R-squared is the proportion of the total variation in the outcome explained by the predictor. Multiply that proportion by SST, and you get the explained sum of squares, often written as SSR or model sum of squares.
Why sum of squares matters
Sum of squares is central to classical statistics because it converts deviations from a mean or fitted value into positive quantities that can be added together. If you simply added positive and negative deviations, they would cancel out. Squaring fixes that and gives more weight to larger deviations, which is statistically useful.
- Total sum of squares (SST): how much the outcome varies around its mean
- Regression sum of squares (SSR): how much variation the predictor explains
- Error sum of squares (SSE): how much variation remains unexplained
These are linked by a basic identity:
So once you estimate the sum square contribution from one variable, the unexplained portion is just the remainder.
The simplest formula to use
When you have one predictor, there are two common starting points:
- You know R-squared directly from software output.
- You know the correlation coefficient r between the predictor and the outcome.
If you know R-squared, use:
If you know only the correlation coefficient in a simple linear regression setting, use:
For example, suppose the total sum of squares for exam scores is 2,500 and the correlation between study hours and exam score is 0.60. Then:
- Square the correlation: 0.60² = 0.36
- Multiply by SST: 0.36 × 2,500 = 900
- The sum square contribution from study hours is 900
- The unexplained sum of squares is 2,500 – 900 = 1,600
That means study hours explain 36% of the total variation in exam scores in this simple model.
Step by step manual calculation
If you want to compute the result by hand, follow this sequence:
- Calculate the mean of the outcome variable, y-bar.
- Compute SST by summing each squared deviation from the mean: Σ(y – y-bar)².
- Obtain the correlation coefficient r or the model R-squared.
- If necessary, square r to get R-squared.
- Multiply R-squared by SST to get the sum square contribution.
- Subtract the contribution from SST to get the residual sum of squares.
If your statistical package gives you R-squared, you can skip directly to the multiplication step. This is why the calculator above is efficient: it avoids unnecessary manual work when the summary output is already available.
Understanding what counts as “one variable”
There is an important distinction between a simple model and a multiple regression model. In a simple model with one predictor, the variable’s contribution and the model’s explained sum of squares are the same thing. In a multiple regression model, the contribution of one variable can mean different things depending on the method:
- Sequential sum of squares: contribution based on order of entry into the model
- Partial sum of squares: unique contribution after accounting for other variables
- Type I, Type II, or Type III sums of squares: different conventions used in ANOVA and regression software
The calculator on this page is designed for the clearest and most common introductory use case: a single predictor or a direct one-variable summary where R-squared refers to that predictor’s explained share of total variance.
Comparison table: how r changes the sum square contribution
The table below shows how the same total sum of squares produces different contributions depending on the predictor’s strength. Here the total sum of squares is fixed at 2,500.
| Correlation r | R-squared | Explained sum of squares (SSR) | Residual sum of squares (SSE) | Explained variation |
|---|---|---|---|---|
| 0.20 | 0.04 | 100 | 2,400 | 4% |
| 0.40 | 0.16 | 400 | 2,100 | 16% |
| 0.60 | 0.36 | 900 | 1,600 | 36% |
| 0.80 | 0.64 | 1,600 | 900 | 64% |
These are real computed statistics from the formula, and they show why squaring matters. A correlation of 0.80 is not just twice as informative as 0.40 in variance terms. It explains four times as much variation because 0.80² is 0.64 while 0.40² is 0.16.
Common mistakes to avoid
- Confusing r with R-squared. Correlation is not the same as explained variance. You must square r in the simple one-predictor case.
- Using percentages incorrectly. If R-squared is 36%, convert it to 0.36 before multiplying by SST.
- Applying the simple formula to multivariable unique effects without care. In multiple regression, one variable’s contribution may require partial sum of squares, not just a direct R-squared multiplication.
- Using a negative r as if it produced a negative sum of squares. Once squared, the result is always nonnegative.
- Mixing different datasets or model summaries. SST and R-squared must come from the same analysis.
Comparison table: decimal versus percent input
Many errors happen because analysts switch between decimal and percent forms. Use this quick reference.
| Reported statistic | Meaning | Correct decimal for calculation | If SST = 5,000 |
|---|---|---|---|
| R-squared = 12% | 12% of variance explained | 0.12 | SSR = 600 |
| R-squared = 48% | 48% of variance explained | 0.48 | SSR = 2,400 |
| r = 70% | Correlation of 0.70 | 0.70, then square to 0.49 | SSR = 2,450 |
| r = -30% | Correlation of -0.30 | -0.30, then square to 0.09 | SSR = 450 |
How this connects to ANOVA
Regression and ANOVA are closely related. In fact, a one-predictor regression can be analyzed through the same sum-of-squares logic as ANOVA. The model sum of squares represents explained variability, and the error sum of squares represents unexplained variability. This is why many textbooks and software packages display the same decomposition in both contexts.
For deeper background, these authoritative resources are helpful:
- NIST Engineering Statistics Handbook
- Penn State STAT 462: Applied Regression Analysis
- UCLA Statistical Methods and Data Analytics
When the simple calculator is the right choice
This calculator is ideal when:
- You have one predictor and one response variable
- Your software reports a simple R-squared and you want the corresponding SSR
- You know the correlation coefficient and total sum of squares
- You want a quick explanation of variance in a report, class assignment, or data review
It is less appropriate when you need the unique contribution of one variable after controlling for several others. In that case, you should work from nested models or software output that reports partial or adjusted sums of squares.
Interpreting the final number
A larger sum square contribution means the predictor accounts for more of the outcome’s total variation. But interpretation depends on scale. A contribution of 900 may be large in one dataset and modest in another, because it must be considered relative to SST. That is why the percentage explained is usually easier to communicate.
For business, research, and academic reporting, a strong interpretation often includes all three values:
- Total variability in the outcome (SST)
- Explained variability due to the predictor (SSR)
- Percent explained (R-squared × 100)
An example report sentence might read: Study hours explained 36% of the total variation in exam scores, corresponding to an explained sum of squares of 900 out of a total sum of squares of 2,500.
Final takeaway
To calculate the sum square contribution from one variable, start with the total sum of squares for the outcome and multiply it by the proportion of variance explained by that variable. In a simple one-predictor setting, that proportion is R-squared, or r squared if you begin with a correlation coefficient. The method is compact, interpretable, and foundational to understanding regression and ANOVA.
If you want a quick answer, use the calculator above. If you want a deeper answer, remember the key idea behind the math: you are partitioning total variability into what your variable explains and what it does not.