The Sampling Distribution of the Slope

Every sample gives you one least-squares slope, but that slope is just an estimate of the true slope in the population. Take a different random sample of the same size from the same population, and you get a different slope. Do this over and over, in theory, and the slopes form their own distribution: the sampling distribution of the slope. Its center lands near the true population slope, and its spread tells you how much the estimate can be trusted. The standard error of the slope is the standard deviation of that distribution, estimated from your one sample, and computer regression output reports it right next to the slope itself.

Sample size drives that spread. A slope fit through 20 points can swing wildly if you rerun the study, because each point has outsized leverage on where the line falls. A slope fit through 200 points barely moves between samples, because individual points get outvoted by the crowd. As n grows, the standard error of the slope shrinks, roughly in proportion to 1 over the square root of n, the same pattern you have already seen for the standard error of a sample mean or proportion.

This connects directly to inference for regression. Dividing the sample slope by its standard error gives a t statistic, used to test whether the true slope is actually zero, meaning no linear relationship at all. A sample slope that is not exactly zero is expected from sampling variability alone, even when the true slope is zero. Only a slope that is many standard errors away from zero counts as real evidence of a relationship, which is exactly what that t statistic and its p-value measure.