# How to compute R-squared for a simple linear model (in R)

## Task

Let’s say we have fit a linear model to two columns of data, one for a single independent variable $x$ and the other for a single dependent variable $y$. How can we compute $R^2$ for that model, to measure its goodness of fit?

Related tasks:

## Solution

We assume you have already fit a linear model to the data, as in the code below, which is explained fully in a separate task, how to fit a linear model to two columns of data.

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xs <- c( 393, 453, 553, 679, 729, 748, 817 )
ys <- c( 24, 25, 27, 36, 55, 68, 84 )
model <- lm( ys ~ xs )

You can get a lot of information about your model from its summary.

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summary( model )

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Call:
lm(formula = ys ~ xs)
Residuals:
1 2 3 4 5 6 7
9.163 2.199 -9.072 -16.795 -4.431 6.047 12.890
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -37.32142 18.99544 -1.965 0.10664
xs 0.13272 0.02959 4.485 0.00649 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 11.62 on 5 degrees of freedom
Multiple R-squared: 0.8009, Adjusted R-squared: 0.7611
F-statistic: 20.12 on 1 and 5 DF, p-value: 0.006486

In particular, it contains the $R^2$ value.

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summary( model )$r.squared

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[1] 0.8009488

Content last modified on 24 July 2023.

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Contributed by Nathan Carter (ncarter@bentley.edu)