Coefficient of Determination (R²) Equation, Derivation & Rearrangements

Core idea

Overview

The coefficient of determination represents the proportion of the variance in the dependent variable that is predictable from the independent variable. In simple linear regression, it is calculated by squaring the Pearson correlation coefficient to quantify the strength of the linear relationship.

When to use: Use this formula when you need to determine the 'effect size' or the shared variance between two continuous variables in psychological research. It is specifically applicable after calculating a Pearson correlation to understand how much of the data's variability is explained by the model.

Why it matters: It transforms abstract correlation coefficients into a more intuitive percentage of explained variance, which is crucial for evaluating the practical significance of a finding. For example, a correlation of 0.7 sounds high, but it only explains 49 percent of the variance, leaving over half to other factors.

Symbols

Variables

R^2 = R-Squared, r = Correlation

R^{2}

R-Squared

V a r iab l e

r

Correlation

V a r iab l e

Walkthrough

Derivation

Formula: Coefficient of Determination (R²)

R² is the proportion of variance in the dependent variable explained by the independent variable(s).

A linear regression model has been fitted.

1

Define total and residual sums of squares:

$S S_{t}$ otal is total variance; $S S_{r}$ esidual is unexplained variance.

S S_{total} = \sum (y_{i} - \overset{y}{ˉ})^{2}, S S_{residual} = \sum (y_{i} - \overset{y}{^}_{i})^{2}

2

Compute R²:

R² ranges from 0 (no variance explained) to 1 (all variance explained).

R^{2} = 1 - \frac{S S _{residual}}{S S _{total}}

Note: For simple linear regression, R² = r².

Result

R^{2} = 1 - \frac{S S _{residual}}{S S _{total}}

Source: University Psychology — Statistics

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a parabolic curve representing the square of the correlation coefficient. Because the independent variable is squared, the output remains non-negative, resulting in a U-shaped curve that reaches a minimum value of zero at the origin.

Graph type: parabolic

Why it behaves this way

Intuition

Imagine a scatter plot where data points represent observations. R2 quantifies how much of the vertical spread of these points (variance of the dependent variable)

R2

The proportion of the total variance in the dependent variable that is explained by the independent variable(s) in a regression model.

It tells you how much of the 'differences' or 'spread' in what you're trying to predict can be accounted for by the predictor(s). A value of 0.75 means 75% of the variation is explained.

r

The Pearson product-moment correlation coefficient, measuring the strength and direction of a linear relationship between two continuous variables.

It indicates how closely two variables move together in a straight line. A value near +1 or -1 means a strong relationship, while 0 means no linear relationship.

Signs and relationships

r2: Squaring the Pearson correlation coefficient (r) converts its range from [-1, 1] to [0, 1]. This operation removes the directionality (positive or negative association)

Free study cues

Insight

Canonical usage

The coefficient of determination (R2) is a dimensionless quantity representing the proportion of variance in the dependent variable explained by the independent variable(s), typically reported as a decimal or percentage.

Common confusion

A common confusion is misinterpreting R2 as a direct measure of the strength of the relationship rather than the proportion of explained variance, or failing to convert it to a percentage for easier interpretation when

Dimension note

R2 is a dimensionless ratio representing the proportion of variance in the dependent variable that is predictable from the independent variable(s). It has no physical units.

Unit systems

$R 2$ dimensionless · Represents the proportion of variance explained, expressed as a value between 0 and 1.

$r$ dimensionless · The Pearson correlation coefficient, also dimensionless, ranging from -1 to 1.

One free problem

Practice Problem

A clinical psychologist finds a Pearson correlation of 0.60 between hours of mindfulness practice and a reduction in anxiety scores. What is the coefficient of determination for this relationship?

Correlation0.6

Solve for: $r 2$

Hint: Square the correlation coefficient to find the coefficient of determination.

The full worked solution stays in the interactive walkthrough.

Study smarter

Tips

Always convert R² to a percentage by multiplying by 100 for easier reporting.
Remember that R² loses the directional information (positive or negative) of the original correlation.
A high R² indicates a good fit, but it does not prove a causal relationship exists.
Note that R² ranges from 0 to 1, regardless of whether the correlation was negative or positive.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

R² is the proportion of variance in the dependent variable explained by the independent variable(s).

Use this formula when you need to determine the 'effect size' or the shared variance between two continuous variables in psychological research. It is specifically applicable after calculating a Pearson correlation to understand how much of the data's variability is explained by the model.

It transforms abstract correlation coefficients into a more intuitive percentage of explained variance, which is crucial for evaluating the practical significance of a finding. For example, a correlation of 0.7 sounds high, but it only explains 49 percent of the variance, leaving over half to other factors.

Always convert R² to a percentage by multiplying by 100 for easier reporting. Remember that R² loses the directional information (positive or negative) of the original correlation. A high R² indicates a good fit, but it does not prove a causal relationship exists. Note that R² ranges from 0 to 1, regardless of whether the correlation was negative or positive.

References

Sources

Wikipedia: Coefficient of determination
Statistical Methods for Psychology by David C. Howell
Britannica: Coefficient of determination
Discovering Statistics Using IBM SPSS Statistics by Andy Field
Statistics for the Behavioral Sciences by Frederick J Gravetter and Larry B Wallnau
Wikipedia: Pearson correlation coefficient
Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). Sage Publications.
University Psychology — Statistics

Coefficient of Determination (R²)