  A simple formula is proved to be a tight estimate for the condition number of
the full rank linear least squares residual with respect to the matrix of least
squares coefficients and scaled 2-norms. The tight estimate reveals that the
condition number depends on three quantities, two of which can cause
ill-conditioning. The numerical linear algebra literature presents several
estimates of various instances of these condition numbers. All the prior values
exceed the formula introduced here, sometimes by large factors.
