## Combinations of Errors

This section of the website is being **actively developed** between September 2011 and April 2012, and **new content **will be added on a day-to-day basis during that period. Please send any comments/suggestions to graham.currell@uwe.ac.uk

**Study Text**: "Essential Mathematics and Statistics for Science", 2nd ed, G Currell and A A Dowman (Wiley-Blackwell)

**QVA (questions and video answers) Tutorials:**

Combining uncertainties / propagation of errors Study Text: Section 8.3.3 (p226)

Errors and uncertainties in concentrations and dilutions (video feedback in preparation) Study Text: Section 8.3.3 (p226)

**Combining and propagating random errors/uncertainties**

Assuming **random **uncertainties where:

* u _{a} *and

*u*

_{b}are the

**absolute**uncertainties in variables,

*a*and

*b*.

*Ru*

_{a}and

*Ru*

_{b}are the

**relative percentage**uncertainties in variables,

*a*and

*b*.

To convert between absolute and relative percentage uncertainties:

*Ru _{a} *= 100 ×

*u*/

_{a}*a*and

*u*

_{a}=

*a*×

*Ru*

_{a}/ 100 etc

To calculate combined uncertainties use absolute or relative uncertainties depending on the combination of variables which give a final value *x*:

If

• *x* = *a* + *b* or* x* = *a* - *b* :- then use *u*_{x} = √{(*u*_{a})^{2} + (*u*_{b})^{2}}

• *x* = *a*×*b* or *x* = *a*/*b* :- then use *Ru*_{x} = √{(*Ru*_{a})^{2} + (*Ru*_{b})^{2}}

• *x* = *k*×*a* (where *k* is a constant) :- then use *Ru*_{x} = *Ru*_{a} or *u*_{x} = *k*×*u*_{a
} •* x* = *a*^{n} (where *n *is a constant) :- then use *Ru*_{x} = *n*×(*Ru*_{a})

Note that it is possible to use the simple (not percentage) relative uncertainties (i.e. without introducing the ‘100’ into the calculations), but it is necessary to be consistent throughout the calculation. We use percentage uncertainty here because many scientists are more familiar with expressing relative uncertainty as a percentage.

See Study Text: Section 8.3.3

**Absolute and relative uncertainty**

The uncertainty of ±0.03 cm^{3} in a 10 cm^{3} class A graduated pipette would be considered as the **absolute **uncertainty.

The **relative **uncertainty in the same situation is given by:

** Relative uncertainty **= **Absolute uncertainty **/ **Value**

e.g. relative uncertainty in the above example = 0.03 cm^{3 }/ 10 cm^{3} = 0.003

**Percentage **uncertainty is the relative uncertainty expressed as a percentage

e.g. percentage uncertainty in the above example = 100 × 0.03 cm^{3} / 10 cm^{3} = 0.3%

Absolute uncertainties have the same **units **as the value itself (e.g. cm^{3 }in the above example), relative uncertainties are a simple ratio with no units, and percentage uncertainties are ratios expressed as percentages.