How To Find Explained Variation On Calculator

Calculate Explained Variation (R²)

Enter the Total Sum of Squares (SST) and the Residual/Error Sum of Squares (SSE) to find the Explained Variation (R-squared).

Summary of Variation

Component	Value	Proportion of Total
Total Sum of Squares (SST)		1.000 (100%)
Regression Sum of Squares (SSR)
Residual Sum of Squares (SSE)

What is Explained Variation?

Explained Variation, most commonly known as the Coefficient of Determination or R-squared (R²), is a statistical measure that represents the proportion of the variance in a dependent variable that is predictable from the independent variable(s) in a regression model. It provides an indication of the goodness of fit of a model. In simpler terms, it tells you what percentage of the changes in the dependent variable can be explained by changes in the independent variable(s).

For example, if the R-squared of a model is 0.75, it means that 75% of the variation in the dependent variable can be explained by the independent variables included in the model, while the remaining 25% is due to other factors or random variability.

Anyone working with regression analysis, such as data scientists, statisticians, economists, researchers, and financial analysts, should understand and use Explained Variation to assess how well their models fit the data. It’s a key metric in evaluating the predictive power of a model.

A common misconception is that a high Explained Variation (R-squared) automatically means the model is good or that there’s a causal relationship. R-squared doesn’t indicate whether the coefficient estimates and predictions are biased, nor does it prove causation. It only measures the strength of the relationship between your model and the dependent variable based on the proportion of variance accounted for.

Explained Variation Formula and Mathematical Explanation

The Explained Variation (R-squared) is calculated using the sums of squares:

• Total Sum of Squares (SST): Measures the total variability in the dependent variable (Y) around its mean (Ȳ).
  SST = Σ(yi - ȳ)²
• Regression Sum of Squares (SSR): Measures the variability in Y that is explained by the regression model (the difference between the predicted values ŷi and the mean ȳ).
  SSR = Σ(ŷi - ȳ)²
• Residual (Error) Sum of Squares (SSE): Measures the variability in Y that is NOT explained by the regression model (the difference between the actual values yi and the predicted values ŷi).
  SSE = Σ(yi - ŷi)²

The fundamental relationship is: SST = SSR + SSE

The Explained Variation (R-squared) is then calculated as:

R² = SSR / SST

Or equivalently:

R² = 1 - (SSE / SST)

R-squared values range from 0 to 1 (or 0% to 100%). A value of 0 indicates that the model explains none of the variability, while a value of 1 indicates that the model explains all the variability.

Variables in Explained Variation Calculation

Variable	Meaning	Unit	Typical Range
yi	Observed value of the dependent variable for the i-th observation	Varies by context	Varies
ȳ	Mean of the observed values of the dependent variable	Varies by context	Varies
ŷi	Predicted value of the dependent variable for the i-th observation	Varies by context	Varies
SST	Total Sum of Squares	Squared units of Y	≥ 0
SSR	Regression Sum of Squares (Explained)	Squared units of Y	≥ 0
SSE	Residual Sum of Squares (Unexplained/Error)	Squared units of Y	≥ 0
R²	Coefficient of Determination (Explained Variation)	Dimensionless	0 to 1

Practical Examples (Real-World Use Cases)

Let’s look at how to find Explained Variation in practical scenarios.

Example 1: Advertising Spend and Sales

A company wants to understand how its advertising spend affects sales. They collect data and run a regression analysis.

• Total Sum of Squares (SST) for sales = 5000
• Residual Sum of Squares (SSE) from the model = 1000

Using the formula R² = 1 - (SSE / SST):

R² = 1 - (1000 / 5000) = 1 - 0.20 = 0.80

The Explained Variation (R²) is 0.80, or 80%. This means that 80% of the variation in sales can be explained by the variation in advertising spend according to their model.

Example 2: Study Hours and Exam Scores

A researcher is studying the relationship between hours spent studying and exam scores.

• Total Sum of Squares (SST) for exam scores = 1200
• Regression Sum of Squares (SSR) = 720

Using the formula R² = SSR / SST:

R² = 720 / 1200 = 0.60

The Explained Variation is 0.60, or 60%. So, 60% of the variation in exam scores can be explained by the number of hours students spent studying, according to the model.

How to Use This Explained Variation Calculator

1. Enter Total Sum of Squares (SST): Input the SST value, which you would typically get from the output of a regression analysis (like an ANOVA table).
2. Enter Residual Sum of Squares (SSE): Input the SSE (or Error Sum of Squares) value, also from your regression output.
3. View Results: The calculator automatically calculates and displays:
   • The primary result: Explained Variation as a percentage.
   • R-squared as a decimal.
   • Regression Sum of Squares (SSR).
   • A table summarizing SST, SSR, and SSE with proportions.
   • A pie chart visualizing the proportion of explained and unexplained variation.
4. Interpret: The percentage tells you how much of the dependent variable’s variance is accounted for by your model. Higher percentages suggest a better fit, but context is crucial.

When making decisions, consider that a high Explained Variation is desirable, but it doesn’t guarantee a good model. Look at other statistics like p-values for coefficients, residual plots, and the context of your study. For more on model fit, see our guide to hypothesis testing.

Key Factors That Affect Explained Variation Results

1. Number of Predictors: Adding more independent variables to a model, even irrelevant ones, can increase R-squared but may lead to overfitting. Adjusted R-squared is often preferred when comparing models with different numbers of predictors.
2. Goodness of Fit: How well the chosen model (e.g., linear, polynomial) actually represents the relationship between variables directly impacts R-squared. A poorly specified model will have low Explained Variation.
3. Outliers: Extreme values can disproportionately influence the sums of squares, either inflating or deflating the R-squared value.
4. Range of Data: A wider range of values for independent variables can sometimes lead to a higher R-squared, even if the underlying relationship strength is the same.
5. Linearity Assumption: If the relationship between variables is non-linear but a linear model is used, the R-squared will be lower than if a more appropriate model was used.
6. Sample Size: While R-squared itself doesn’t directly depend on sample size in its formula, smaller samples can lead to less stable and potentially inflated R-squared values that may not generalize well.

Understanding these factors helps in interpreting the Explained Variation correctly and building better predictive models.

Frequently Asked Questions (FAQ)

What is a good value for Explained Variation (R-squared)?

It depends heavily on the context and field of study. In some fields (like physics or chemistry with controlled experiments), R-squared values above 0.90 are common. In social sciences or fields with more inherent variability, values like 0.30 to 0.60 might be considered reasonable. There’s no single “good” value for Explained Variation.

Can R-squared be negative?

In standard linear regression, R-squared as calculated by 1 – SSE/SST ranges from 0 to 1. However, if a model is extremely poor and fits the data worse than a horizontal line (the mean), and R-squared is calculated using a formula that doesn’t constrain it, it could theoretically be negative, but this is rare in typical regression outputs.

What is Adjusted R-squared?

Adjusted R-squared is a modified version of R-squared that accounts for the number of predictors in the model. It increases only if the new term improves the model more than would be expected by chance. It’s often lower than R-squared and is preferred when comparing models with different numbers of independent variables.

Does a high R-squared mean the model is causally correct?

No. A high Explained Variation only indicates a strong correlation or goodness of fit within the sample data. It does not imply causation. Causal relationships must be established through experimental design or theoretical reasoning.

How does R-squared relate to correlation (r)?

In simple linear regression (one independent variable), R-squared is the square of the Pearson correlation coefficient (r) between the observed and predicted values, or between the independent and dependent variables. So, r = 0.8 means R² = 0.64.

Can I use R-squared to compare models with different dependent variables?

No. R-squared is relative to the total variance of the dependent variable. If the dependent variables are different (or transformed differently), their total variances will differ, making R-squared values incomparable.

What if my R-squared is very low?

A low Explained Variation suggests that your model does not explain much of the variability in the dependent variable. It could mean the relationship is weak, the wrong model was chosen, important variables are missing, or there’s a lot of inherent randomness.

How do I get SST and SSE values?

These values are typically found in the output of statistical software after running a regression analysis, often in an ANOVA (Analysis of Variance) table associated with the regression.