Combining errors in independent measurements.

Many experiments depend on several measured quantities, each of which has an

associated error. The rules for combining the errors are given below. These would not be

examined, but they can be used in your everyday work and will give you a feel for the

magnitudes involved and more importantly will train you in the quantitative assessment of error.

In all the following cases, the two independent measurements A and B with errors ∆A and

∆B, are combined to give the quantity X which has the associated error ∆X.

Errors in sums or differences:

The straightforward combination of individual errors in sums or differences gives an overall

error bound that is too pessimistic. It is unlikely that the individual errors will both be in the

same direction, though not of course impossible.

If X = A + B or X = A – B , then: ∆X = √{(∆A)2 + (∆B)2}

The expression is the same whether the quantities are added or subtracted. If more than two

quantities are involved the expression is simply extended in a similar fashion, that is if X = A +

B – C, then:

∆X = √{(∆A)2 + (∆B)2 + (∆C)2}.

The resulting error is always larger than any of the individual errors, but not as large as their

sum.

If two values that are very similar are subtracted, the resulting error can be very large. Such

procedures are best avoided if at all possible – see example 2.

Example 1: In an experiment to determine the enthalpy of neutralisation of sodium hydroxide by

hydrochloric acid, the initial temperature was (19.2 ± 0.2) oC, and the final temperature (26.4 ±

0.2) oC. What is the temperature rise?

∆T = (T2 – T1) ± ∆T

= (26.4 – 19.2) oC ± ∆T

= 7.2 oC ± ∆T.

www.notesolution.com