Uncertainties

It was shown that the standard deviation of the angle of the fitted line was proportional to the standard deviation of the distance of the data points from the line. A small angle approximation was used to derive this result and the relation was also true for the tangent of the angle of the line which is its slope. If we take the Δξ to be the maximum distance along the line from the center of the data then we can rewrite the relation as shown in the image below. We get a relation between the uncertainty in the slope of the line and the spread of the data along the line. This relation can be used to show how the uncertainty in the measurement of velocity is related to the interval of time over which the measurements are taken.


One might notice the similarity of this "uncertainty relation" with that of the Heisenberg Uncertainty Principle of Physics. The momentum, p = mv, is defined as mass times velocity. The connection between the two relations is that if there is an intrinsic source of error in the motion of a particle then there will be a minimum value of the product of the two intervals. This minimum occurs because not all the deviations involved are a result of the process of measurement but some are related to the motion of the particle itself.