What's Missing?

You may have noticed that there is no term for the deviation along the direction of the line. The reason is that any error is included in the position along the line. And there are no error estimates for the position of the center of the distribution and the angle of the eigenvector. The error estimate for determining the position of a point is a function of the standard deviations and the number of data points. One would also expect that the error in the direction of the line would be of the order of angle of a right triangle whose height is the standard deviation normal to the line and base is the maximum length of the fitted line from the center as the base. The results would depend on the error distribution and could be checked by numerical computations. By assuming a center and direction for the line and an error model and then comparing the estimates of the fit with the initial assumptions one gets feedback on the method used to fit the data.