__Combining Uncertainties in
Experimental Results__

__H. Slade__

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 steps results, until the final
result is obtained.

Let L_{v} = [m_{w}C_{w}(T_{f}
T_{0}) + m_{c}C_{c}(T_{f } T_{0})
m_{s}C_{w}(T_{bp} T_{f})]/m_{s}

_{ }

(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, lets define three terms, which added together make
up L_{v}:

Term1 = m_{w}C_{w}(T_{f} T_{0})/m_{s}

Term2 = m_{c}C_{c}(T_{f} T_{0})/m_{s}

Term3 = C_{w}(T_{bp} T_{f})

To get the uncertainty for the experimental result, L_{v},
we find the *absolute* uncertainties
for each of the three Terms. Starting
with Term1, note that there are four factors multiplied together: m_{w},
C_{w}, (T_{f} T_{0}), and 1/m_{s}. Because they represent a product (Even the
divisor, m_{s}, is equivalent to multiplying by a reciprocal, 1/m_{s}.),
we will need to add their *fractional*
uncertainties together. Before that, we
need to go to a simpler level, the combination (T_{f} T_{0}). That is a sum (subtraction included). Starting with *absolute* uncertainties for T_{f} and T_{0}, ΔT_{f} and ΔT_{0}
respectively, these are added together to give the *absolute* uncertainty for (T_{f}
T_{0}),

Δ(T_{f} T_{0})
= ΔT_{f} + ΔT_{0}

Continuing with Term1, its uncertainty is the sum of the *fractional* uncertainties for the four
factors identified above. We assumed *absolute* uncertainties ΔT_{f}
and ΔT_{0} (which may be your estimates of measuring limits), and
we now assume *absolute* uncertainties
Δm_{w}, ΔC_{w}, and Δm_{s} for the
remaining factors. The next step is to convert
each of the latter three and the derived *absolute*
uncertainty, Δ(T_{f} T_{0}), into the *fractional*
uncertainties Δm_{w}/m_{w}, ΔC_{w}/C_{w},
Δm_{s}/m_{s}, and [Δ(T_{f} T_{0})]/(T_{f}
T_{0}). As stated above, we
add these together to get the *fractional *uncertainty,

[ΔTerm1]/Term1
= Δm_{w}/m_{w} + ΔC_{w}/C_{w} + Δm_{s}/m_{s}
+ [Δ(T_{f} T_{0})]/(T_{f} T_{0})

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 L_{v},
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, ΔL_{v} = ΔTerm1 + ΔTerm2 + ΔTerm3

Of course, ΔL_{v} 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.