Residual vs Standard Error

A residual measures how far off the line is for a single point, so a positive residual sits above the line and a negative one below. It equals $\text{residual} = y - \hat{y}$, the observed response minus the predicted response $\hat{y}$ (read y-hat). For example, if the line predicts 72 but a student actually scored 80, the residual is $80 - 72 = 8$. Least-squares regression chooses the line that makes the sum of the squared residuals as small as possible.

The standard error plays the same role for a statistic that the standard deviation plays for raw data, and it gets smaller as the sample size grows. For example, the standard error of a sample proportion is $\sqrt{\frac{p(1-p)}{n}}$ (the square root of p times one minus p, divided by n). For a sample mean it is $\frac{\sigma}{\sqrt{n}}$ (sigma, the population standard deviation, over the square root of n). A smaller standard error means the estimate is more precise, which narrows a confidence interval.