Combining Uncertainties in Experimental Results

The lab book Appendix for Phys 140L and Phys 150L has a discussion of combining the uncertainties in measured parameters to arrive at an uncertainty value for an experimental result that depends on these parameters.[1]  In particular, it talks about absolute uncertainties and relative uncertainties[2].  The discussion gets rather deep, and some of the examples are hard to follow, so here is a somewhat simpler approach that yields a larger uncertainty for the experimental result than the statistical method of the appendix, but still reasonable.  Application of the methods depends on the mode of combination of the individual parameters in calculation of the experimental result.

Addition and Subtraction of Measured Parameters

When two or more measured parameters are added or subtracted to get an experimental result, their absolute uncertainties (the absolute value of each uncertainty) must be added together.  I.e., the combined absolute uncertainty for a result of adding or subtracting parameters A, B, and C with individual absolute uncertainties ΔA,  ΔB, and ΔC is

ΔR = ΔA + ΔB + ΔC

In the next paragraph we discuss fractional uncertainties.  To obtain a fractional uncertainty from the absolute uncertainty, ΔR, divide ΔR/R.

Multiplication and Division of Measured Parameters

When two or more measured parameters are multiplied or divided to get an experimental result, their fractional uncertainties are added together.  These are ratios of the absolute uncertainty to the measured parameter value.*    I.e., for a result of multiplying or dividing parameters A, B, and C, the combined fractional uncertainty for individual fractional uncertainties (ΔA)/A,  (ΔB)/B, and (ΔC)/C is

ΔR/R = (ΔA)/A + (ΔB)/B + (ΔC)/C ,

where R is the product/quotient function of A, B, and C.

If the absolute uncertainty ΔR is desired, multiply this sum by R.  (As was true of absolute uncertainties, do not keep any negative signs when adding fractional uncertainties.

* Sometimes people prefer to use percentage uncertainties instead of fractional.  Both are referred to as relative uncertainties.  However, a percentage ratio requires multiplying the fractional ratio by 100, and remembering to divide the percentage ratio by 100 when converting back to an absolute uncertainty.  To avoid this complication, this discussion will assume fractional uncertainties are in use whenever relative uncertainties are mentioned.

Powers of Measured Parameters

When a measured parameter appears in a formula for an experimental result as a power different from 1, multiply its fractional uncertainty by n, where n is the exponent of the measured parameter in the result relationship (This power may be a fraction, representing a radical operation like square root.  Do not keep any negative signs in the resulting uncertainty.)  Then combine this fractional uncertainty appropriately with any others as will be discussed next.

Complex Relationships of  Measured Parameters

When measured parameters enter into a formula for an experimental result in different manners (e.g., some by addition and subtraction, and some by multiplication and division, or powers, including roots), first group together factors, quotients, and terms at the simplest levels.  An example will make the procedure clear.  Intermediate results come from very direct, simple calculations based on the previous steps results, until the final result is obtained.

Let Lv = [mwCw(Tf  T0) + mcCc(Tf  T0)  msCw(Tbp  Tf)]/ms

(This is the formula for latent heat of vaporization of water, calculated for an experimental setup described for the Phys 140L lab on heat.)

First, lets define three terms, which added together make up Lv:

Term1 = mwCw(Tf  T0)/ms

Term2 = mcCc(Tf  T0)/ms

Term3 =  Cw(Tbp  Tf)

To get the uncertainty for the experimental result, Lv, we find the absolute uncertainties for each of the three Terms.  Starting with Term1, note that there are four factors multiplied together: mw, Cw, (Tf  T0), and 1/ms.   Because they represent a product (Even the divisor, ms, is equivalent to multiplying by a reciprocal, 1/ms.), we will need to add their fractional uncertainties together.  Before that, we need to go to a simpler level, the combination (Tf  T0).  That is a sum (subtraction included).  Starting with absolute uncertainties for Tf  and T0, ΔTf and ΔT0 respectively, these are added together to give the absolute uncertainty for (Tf  T0),

Δ(Tf  T0) = ΔTf + ΔT0

Continuing with Term1, its uncertainty is the sum of the fractional uncertainties for the four factors identified above.  We assumed absolute uncertainties ΔTf and ΔT0 (which may be your estimates of measuring limits), and we now assume absolute uncertainties Δmw, ΔCw, and Δms for the remaining factors.  The next step is to convert each of the latter three and the derived absolute uncertainty, Δ(Tf  T0),  into the fractional uncertainties Δmw/mw, ΔCw/Cw, Δms/ms, and [Δ(Tf  T0)]/(Tf  T0).  As stated above, we add these together to get the fractional uncertainty,

[ΔTerm1]/Term1 = Δmw/mw + ΔCw/Cw + Δms/ms + [Δ(Tf  T0)]/(Tf  T0)

Notice that the Term1 uncertainty is fractional, like its components.  In like manner, calculate [ΔTerm2]/Term2 and [ΔTerm3]/Term3, which are also fractional uncertainties.

The last step is to get the uncertainty for Lv, for which we will need to add absolute uncertainties for Term1, Term2, and Term3.  Before that, each of the fractional uncertainties [ΔTerm1]/Term1, [ΔTerm2]/Term2 and [ΔTerm3]/Term3 need to be converted to absolute.  How? Calculate the value of each term from your parameters (measured or given) and multiply by the fractional uncertainty for that term.

E.g.,  ΔTerm1 = Term1 x [ΔTerm1]/Term1

Then, ΔLv = ΔTerm1 + ΔTerm2  + ΔTerm3

Of course, ΔLv is also an absolute uncertainty.

Since there is frequent need for conversion back and forth between absolute and fractional uncertainties to get a final derived uncertainty, it is convenient to tabulate both values for these uncertainties, along with the value of the related parameter, factor, or term. An Excel spreadsheet is a good way to do this, referencing intermediate results in the computational steps that follow.  What appears extremely complex when combined algebraically becomes a series of rather simple steps.[3]

In summary, the uncertainty of a result derived from the mathematical combination of measured parameters can be conservatively estimated by a series of conversions of absolute and fractional uncertainties.  These intermediate uncertainties are always added together without respect to negative signs, absolute with absolute and fractional with fractional.  Absolute uncertainties are added for parameter sums and differences.  Fractional uncertainties are added for parameter products, quotients, and powers other than one.

[1] Statistical Treatment of Experimental Data, p. 107 & ff.

[2] Ibid, pp. 111-116

[3] The method of combination of uncertainties described here gives a more conservative estimate of calculated uncertainty than is statistically required for two or more independent parameters.  I.e., the overall uncertainty calculated this way is larger than  need be.  The statistically justified method described in the lab book appendix adds up the squares of the individual uncertainties (either absolute or fractional) and takes as its result the square root of the sum.  For the purpose of introducing the concept of combining uncertainties, however, simply adding the individual uncertainties (either absolute or fractional) is sufficient.