# How to compute the standard error of the estimate for a model (in R)

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One measure of the goodness of fit of a model is the standard error of its estimates. If the actual values are $y_i$ and the estimates are $\hat y_i$, the definition of this quantity is as follows, for $n$ data points.

$\sigma_{\text{est}} = \sqrt{ \frac{ \sum (y_i-\hat y_i)^2 }{ n } }$

If we’ve fit a linear model, how do we compute the standard error of its estimates?

## Solution

Let’s assume that you already fit the linear model, as shown in the code below. This one uses a small amount of fake data, but it’s just an example. See also how to fit a linear model to two columns of data.

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x <- c(34, 9, 78, 60, 22, 45, 83, 59, 25)
y <- c(126, 347, 298, 309, 450, 187, 266, 385, 400)
model <- lm(y ~ x)


The standard error for each estimate is shown as part of the model summary, reported by R’s built-in summary function. See the column entitled “Std. Error” in the output below.

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summary(model)

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Call:
lm(formula = y ~ x)

Residuals:
Min       1Q   Median       3Q      Max
-193.776   -4.334   15.459   71.143  118.116

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  354.082     76.733   4.614  0.00244 **
x             -1.009      1.473  -0.685  0.51536
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 107.1 on 7 degrees of freedom
Multiple R-squared:  0.06283,	Adjusted R-squared:  -0.07106
F-statistic: 0.4693 on 1 and 7 DF,  p-value: 0.5154


If we need to extract just the model coefficients table, or even just the “Std. Error” column of it, we can use code like the following.

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coef(summary(model))
coef(summary(model))[,2]

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Estimate   Std. Error t value    Pr(>|t|)
(Intercept) 354.082248 76.732772   4.6144853 0.002441995
x            -1.009013  1.472939  -0.6850334 0.515358250

(Intercept)           x
76.732772    1.472939


The standard error of the estimate for the intercept is is 76.733 and the standard error of the estimate for the slope is 1.473.