Reading summary(lm) as description

Coefficients, the residual standard error on n − 2 df, Multiple R-squared — and the columns to leave for Module 13.

The printout the paper hands you

A regression question on the paper prints summary(lm(score ~ income)) and asks you to read it. Three lines of it are the description this module has built by hand; the rest is inference, and Module 13 will reopen this exact output to read them. Learn now which is which.

Worked example — the eight clients in R

> fit <- lm(score ~ income)
> coef(fit)
(Intercept)      income 
 544.176665    2.065922

lm(y ~ x) fits the least-squares line — the response on the left of ~, the explanatory variable on the right. coef() prints the two estimates at full precision: 544.18 and 2.066, the L4.3 values.

Now the block summary(fit) prints:

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 544.1767    10.5049   51.80 3.47e-09 ***
income        2.0659     0.1432   14.42 6.96e-06 ***

Residual standard error: 9.642 on 6 degrees of freedom
Multiple R-squared:  0.972,	Adjusted R-squared:  0.9673 
F-statistic: 208.1 on 1 and 6 DF,  p-value: 6.956e-06

Read as description — three things.

Estimate column: $\hat{β}_{0} = 544.1767$ , $\hat{β}_{1} = 2.0659$ . The line.
Residual standard error: 9.642 on 6 degrees of freedom — this is $s = S S E / (n - 2) = 557.8/6 = 9.64$ from L4.6, and the 6 is $n - 2 = 8 - 2$ : two parameters estimated from eight points.
Multiple R-squared: 0.972 — $R^{2} = S S R / S S T$ from L4.5.

Leave for Module 13. Std. Error, t value and Pr(>|t|) — the standard error of each estimate, the test statistic for "is this coefficient zero?", and its p-value. Adjusted R-squared (a penalised version for multiple predictors) and the F-statistic line (the overall test, equivalent to the slope's $t$ -test when there is one predictor). The stars are significance codes for those p-values.

The third-decimal question

A hand answer that squares a rounded $r$ will not match the printout: $0.98 6^{2} = 0.9722$ , while R prints 0.972 from the unrounded 0.98589 ( $0.9720$ ). The difference is rounding $r$ before squaring, not a different formula — R computes $R^{2} = S S R / S S T$ from the sums, which is what $r^{2}$ equals when $r$ is carried in full. On the paper, quote the printed value; when you compute by hand, carry four decimals.

Reading habits

The row is the variable (income), never the response; the response is named only in the Call. (Intercept) is $\hat{β}_{0}$ . Everything after the Estimate column on each row belongs to inference; when a question says "interpret the fitted line", it wants the two estimates, the residual standard error and $R^{2}$ — and nothing from the other columns.