OLS with randon noise in one variable

The classical, correct use of least squares linear regression where one variable has negligible errors. The other has random errors with ‘normal’ or gaussian distribution.

Under these conditions it can be shown that the least squares fit is the best estimation of the underlying linear relationship that is possible from the given data.

In this example, the regression line and the true slope are indistinguishable on the graph.

This is in contrast with the case where there is significant error in both variables: