The coefficient of determination is a number between 0 and 1 that measures how well a statistical model predicts an outcome.

Coefficient of determination (R^{2}) |
Interpretation |
---|---|

0 | The model does not predict the outcome. |

Between 0 and 1 | The model partially predicts the outcome. |

1 | The model perfectly predicts the outcome. |

The coefficient of determination is often written as *R*^{2}, which is pronounced as “r squared.” For simple linear regressions, a lowercase *r* is usually used instead (*r*^{2}).

## What is the coefficient of determination?

The coefficient of determination (*R*²) measures how well a statistical model predicts an outcome. The outcome is represented by the model’s dependent variable.

The lowest possible value of *R*² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its *R*² will be to 1.

More technically, *R*^{2} is a measure of goodness of fit. It is the proportion of variance in the dependent variable that is explained by the model.

Graphing your linear regression data usually gives you a good clue as to whether its *R*^{2} is high or low. For example, the graphs below show two sets of simulated data:

- The observations are shown as dots.
- The model’s predictions (the line of best fit) are shown as a black line.
- The distance between the observations and their predicted values (the residuals) are shown as purple lines.

You can see in the first dataset that when the *R*^{2} is high, the observations are close to the model’s predictions. In other words, most points are close to the line of best fit:

In contrast, you can see in the second dataset that when the *R*^{2} is low, the observations are far from the model’s predictions. In other words, when the *R*^{2} is low, many points are far from the line of best fit:

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**Calculating the coefficient of determination**

You can choose between two formulas to calculate the coefficient of determination (*R*²) of a simple linear regression. The first formula is specific to simple linear regressions, and the second formula can be used to calculate the *R*² of many types of statistical models.

### Formula 1: Using the correlation coefficient

### Formula 2: Using the regression outputs

## Interpreting the coefficient of determination

You can interpret the coefficient of determination (*R*²) as the proportion of variance in the dependent variable that is predicted by the statistical model.

Another way of thinking of it is that the *R*² is the proportion of variance that is shared between the independent and dependent variables.

You can also say that the *R*² is the proportion of variance “explained” or “accounted for” by the model. The proportion that remains (1 − *R*²) is the variance that is not predicted by the model.

If you prefer, you can write the *R*² as a percentage instead of a proportion. Simply multiply the proportion by 100.

*R*² as an effect size

Lastly, you can also interpret the *R*² as an effect size: a measure of the strength of the relationship between the dependent and independent variables. Psychologist and statistician Jacob Cohen (1988) suggested the following rules of thumb for simple linear regressions:

Minimum coefficient of determination (R²) value |
Effect size interpretation |
---|---|

.01 | Small |

.09 | Medium |

.25 | Large |

Be careful: the *R*² on its own can’t tell you anything about causation.

## Reporting the coefficient of determination

If you decide to include a coefficient of determination (*R*²) in your research paper, dissertation or thesis, you should report it in your results section. You can follow these rules if you want to report statistics in APA Style:

- You should use “
*r*²” for statistical models with one independent variable (such as simple linear regressions). Use “*R*²” for statistical models with multiple independent variables. - You don’t need to provide a reference or formula since the coefficient of determination is a commonly used statistic.
- You should italicize
*r*² and*R*² when reporting their values (but don’t italicize the ²). - You shouldn’t include a leading zero (a zero before the decimal point) since the coefficient of determination can’t be greater than one.
- You should provide two significant digits after the decimal point.
- Very often, the coefficient of determination is provided alongside related statistical results, such as the
*F*value, degrees of freedom, and*p*value.