# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/statistics-regression/coefficient-of-determination Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/statistics-regression/coefficient-of-determination/en.mdx Learn how r² measures how well your regression line explains data variation. Learn coefficient of determination with visual examples and calculations. --- ## What is the Coefficient of Determination? After finding the best-fit linear regression line for our data, the next question is: **how well does that line actually represent or explain our data?** The measure that answers this question is the **Coefficient of Determination**, denoted as **$$r^2$$** (read: r-squared). Visible text: The measure that answers this question is the **Coefficient of Determination**, denoted as **** (read: r-squared). Simply put, $$r^2$$ tells us the **proportion or percentage** of the variation (ups and downs in values) in the dependent variable (Y) that **can be explained** by the variation in the independent variable (X) using our linear regression model. Visible text: Simply put, tells us the **proportion or percentage** of the variation (ups and downs in values) in the dependent variable (Y) that **can be explained** by the variation in the independent variable (X) using our linear regression model. ## Coefficient of Determination from a Scatter Diagram The value of $$r^2$$ is closely related to how tightly the data points cluster around the regression line: Visible text: The value of is closely related to how tightly the data points cluster around the regression line: 1. **High $$r^2$$ (approaching $$1$$ or $$100\%$$)** High $$r^2$$ } description="Data points are very close to the regression line." xAxisLabel="Variable X" yAxisLabel="Variable Y" datasets={[ { name: "Data", color: "var(--chart-1)", points: [ { x: 1, y: 2 }, { x: 2, y: 3.1 }, { x: 3, y: 3.9 }, { x: 4, y: 5.2 }, { x: 5, y: 6.1 }, { x: 6, y: 6.8 }, { x: 7, y: 8.1 }, { x: 8, y: 9.0 }, ], }, ]} calculateRegressionLine showResiduals regressionLineStyle={{ color: "var(--chart-4)" }} /> See how the data points above are very tightly packed and close to the regression line? This indicates a high $$r^2$$ value (for example, maybe around $$0.95$$ or $$95\%$$). This means that most of the variation in $$Y$$ values _can be explained_ well by the regression line (or by variable $$X$$). 2. **Low $$r^2$$ (approaching $$0$$ or $$0\%$$)** Low $$r^2$$ } description="Data points are scattered far from the regression line." xAxisLabel="Variable X" yAxisLabel="Variable Y" datasets={[ { name: "Data", color: "var(--chart-2)", points: [ { x: 1, y: 1 }, { x: 2, y: 4 }, { x: 3, y: 2 }, { x: 4, y: 6 }, { x: 5, y: 5 }, { x: 6, y: 8 }, { x: 7, y: 6 }, { x: 8, y: 9 }, ], }, ]} calculateRegressionLine showResiduals regressionLineStyle={{ color: "var(--chart-4)" }} /> Compare this with this diagram. The points are more spread out from the regression line (the residual lines are longer). This indicates a low $$r^2$$ value (for example, maybe around $$0.40$$ or $$40\%$$). This means that this regression line is _not very good_ at explaining the variation in $$Y$$ values; only a small portion of the variation in $$Y$$ can be explained by $$X$$ through this model. Visible text: 1. **High (approaching or )** High } description="Data points are very close to the regression line." xAxisLabel="Variable X" yAxisLabel="Variable Y" datasets={[ { name: "Data", color: "var(--chart-1)", points: [ { x: 1, y: 2 }, { x: 2, y: 3.1 }, { x: 3, y: 3.9 }, { x: 4, y: 5.2 }, { x: 5, y: 6.1 }, { x: 6, y: 6.8 }, { x: 7, y: 8.1 }, { x: 8, y: 9.0 }, ], }, ]} calculateRegressionLine showResiduals regressionLineStyle={{ color: "var(--chart-4)" }} /> See how the data points above are very tightly packed and close to the regression line? This indicates a high value (for example, maybe around or ). This means that most of the variation in values _can be explained_ well by the regression line (or by variable ). 2. **Low (approaching or )** Low } description="Data points are scattered far from the regression line." xAxisLabel="Variable X" yAxisLabel="Variable Y" datasets={[ { name: "Data", color: "var(--chart-2)", points: [ { x: 1, y: 1 }, { x: 2, y: 4 }, { x: 3, y: 2 }, { x: 4, y: 6 }, { x: 5, y: 5 }, { x: 6, y: 8 }, { x: 7, y: 6 }, { x: 8, y: 9 }, ], }, ]} calculateRegressionLine showResiduals regressionLineStyle={{ color: "var(--chart-4)" }} /> Compare this with this diagram. The points are more spread out from the regression line (the residual lines are longer). This indicates a low value (for example, maybe around or ). This means that this regression line is _not very good_ at explaining the variation in values; only a small portion of the variation in can be explained by through this model. ## Calculating the Coefficient of Determination The easiest way to calculate $$r^2$$ is by **squaring the Correlation Coefficient ($$r$$)** that we learned about earlier. Visible text: The easiest way to calculate is by **squaring the Correlation Coefficient ()** that we learned about earlier. ```math r^2 = (r)^2 ``` So, if you've already calculated the value of $$r$$, just square it! Visible text: So, if you've already calculated the value of , just square it! Since the value of $$r$$ is always between $$-1$$ and $$+1$$ ($$-1 \le r \le 1$$), the value of $$r^2$$ will always be between $$0$$ and $$1$$. Visible text: Since the value of is always between and (), the value of will always be between and . ```math 0 \le r^2 \le 1 ``` **Mathematically (using Sum of Squares):** The value of $$r^2$$ can also be calculated directly using the Sum of Squares values used to calculate $$r$$: Visible text: The value of can also be calculated directly using the Sum of Squares values used to calculate : ```math r^2 = \frac{(SS_{xy})^2}{SS_{xx} SS_{yy}} ``` ## Interpretation as a Percentage The value of $$r^2$$ is often converted into a percentage (by multiplying by $$100$$) for easier interpretation. Visible text: The value of is often converted into a percentage (by multiplying by ) for easier interpretation. - If $$r^2 = 0.81$$, it means that **$$81\%$$** of the total variation in variable $$Y$$ can be explained by the variation in variable $$X$$ through the linear regression model. - The remaining variation ($$1 - r^2$$ or $$19\%$$ in this example) is explained by other factors not included in the model (could be other variables, or _random error_). Visible text: - If , it means that **** of the total variation in variable can be explained by the variation in variable through the linear regression model. - The remaining variation ( or in this example) is explained by other factors not included in the model (could be other variables, or _random error_). The higher the percentage of $$r^2$$, the better our linear regression model is at explaining the relationship between $$X$$ and $$Y$$. Visible text: The higher the percentage of , the better our linear regression model is at explaining the relationship between and .