Where the Slope's Standard Error Comes From

The slope is a statistic with a sampling distribution

Every new sample from the same population gives a slightly different least-squares slope $b$. Across all possible samples, $b$ has a sampling distribution centred at the true slope $\beta$ (unbiased), approximately normal when the LINER conditions hold, with standard deviation estimated by

Here $s$ is the standard deviation of the residuals (the typical prediction error, labelled "S" in output) and $s_x$ is the ordinary standard deviation of the $x$-values.

Three levers on $SE_b$

• Smaller $s$ — points hugging the line give a more precise slope.
• Larger $s_x$ — spreading the $x$-values out makes the line harder to tilt; this is the one you can control when designing a study.
• Larger $n$ — more points, through $\sqrt{n-1}$.

Example: $s=2{,}5$, $s_x=4$, $n=26$ give $SE_b=\dfrac{2{,}5}{4\sqrt{25}}=0{,}125$. Doubling the spread of $x$ halves it; quadrupling $n$ also halves it.

Three different "s" in one output

Output tables list $s$ (residual SD), the standard error of the slope $SE_b$ (in the coefficient row, "SE Coef"), and sometimes the SE of the intercept. Questions that ask to "interpret $s$" want: "the actual $y$-values typically differ from the predicted values by about $s$ units". Questions about the precision of the slope want $SE_b$. Confusing them is the single most common Unit 9 error.