6.1 Examples

As a first example let us derive the formulas for the sum, difference, product and ratio of two quantities x and y with errors x and y.

1. Error of a Sum: u = x + y

(65)

2. Error of a Difference: u = x - y

(66)

If the covariance is 0, the errors on both a sum and difference then reduce to the same sum of squares. The relative error, u/u, however, is much larger for the case of a difference since u is smaller. This illustrates the disadvantage of taking differences between two numbers with errors. If possible, therefore, a difference should always be directly measured rather than calculated from two measurements!

1. Error of a Product: u = xy

Dividing the left side by u2 and the right side by x2 y2,

(67)

1. Error of a Ratio: u = x/y

Dividing both sides by u2 as in (iii), we find

(68)

which, with the exception of the sign of the covariance term is identical to the formula for a product. Equation (68) is generally valid when the relative errors are not too large. For ratios of small numbers, however, (68) is inapplicable and some additional considerations are required. This is treated in detail by James and Roos [Ref. 1].

Example 5. The classical method for measuring the polarization of a particle such as a proton or neutron is to scatter it from a suitable analyzing target and to measure the asymmetry in the scattered particle distribution. One can, for example, count the number of particles scattered to the left of the beam at certain angle and to the right of the beam at the same corresponding angle. If R is the number scattered to the right and L the number to the left, the asymmetry is then given by

Calculate the error on as a function of the counts R and L.

This is a straight forward application of (64). Taking the derivatives of , we thus find

where the total number of counts Ntot = R + L. The error is thus

The covariance is obviously 0 here since the measurements are independent. The errors on R and L are now given by the Poisson distribution, so that R2 = R and L2 = L. Substituting into the above, then yields

If the asymmetry is small such that R L Ntot / 2, we have the result that