In calculating the concentration of lead in Taconite Lake, the only uncertainty considered was the uncertainty in the concentration of lead. Quite often results
are calculated by combining a number of quantities that each have their own associated uncertainties. These uncertainties must also be taken into account when
calculating the final uncertainty. In the case of Taconite Lake, if the samples had been diluted before analysis, the random uncertainties of glassware volume
would need to be propagated (combined with) the uncertainty found in the concentration of lead.
Typically, one aims to find some value \(Z\) mathematically, and its associated uncertainty \(\Delta Z\) via uncertainty propagation formulae.
Table 4 below summarizes the rules for the propagation of uncertainty for random uncertainty. For systematic uncertainty, there are different rules that must be applied to calculate the combined total uncertainty. Since systematic uncertainty can be corrected for, it is better to find the source of uncertainty and correct it rather than propagate its uncertainty. Consult a textbook or your lab manual for the equations on how to do propagation of systematic uncertainty, or for cases involving both random and systematic uncertainty.
| Mathematical Operation | Equation to propagate uncertainty | Comment |
|---|---|---|
| Addition or Subtraction: $$Z = A + B - C $$ | $$ \Delta Z = \sqrt{\Delta A^2 + \Delta B^2 + \Delta C^2}$$ | Root of summed absolute uncertainties squared Uncertainty adds even when math operation is subtraction |
| Multiplication or Division: $$Z = \frac{A \cdot B}{C}$$ | $$ \Big( \frac {\Delta Z}{Z} \Big) = \sqrt{\Big( \frac {\Delta A}{A} \Big)^2 + \Big( \frac {\Delta B}{B} \Big)^2 + \Big( \frac {\Delta C}{C} \Big)^2} $$ | Root of summed relative uncertainties squared |
To learn how to use the formulas, we will work through two examples.
First example: Determine the final volume and uncertainty of HCl delivered by a buret during a titration. The initial volume of HCl was 11.23±0.03 mL and the final volume was 19.36±0.03 mL.
A volume of liquid delivered by a buret is calculated by taking the difference between the final volume and the initial volume. Uncertainty is present in both measurements, so to determine the uncertainty in the volume delivered, we must take both uncertainties into account. We will be using the following two formulas to determine the volume and the uncertainty. Since we're taking the difference between two values, we're doing subtraction, and thus we follow the top formula in the table above when calculating the uncertainty in the difference.
$$Volume \, HCl \, delivered = 19.36 mL - 11.23 mL = 8.13 mL $$
$$Uncertainty \, (one \, sig. \, fig.) = \sqrt{(0.03)^2+(0.03)^2 } = 0.04 mL$$
You would report the volume of HCl delivered as 8.13 \(\pm\) 0.04 mL. This has an associated confidence level which depends on the confidence level of the input uncertainties. Ensure the input uncertainties all have the same confidence level so you know the confidence level of the output value!
Second example: Determine the volume (and its uncertainty) of a glove box with dimensions 195.0 \(\pm\) 2.0 cm by 78.0 \(\pm\) 1.0 cm by 92.0 \(\pm\) 1.0 cm.
Assume these uncertainties are standard deviations, meaning they are at the 68% confidence level. Nevermind why these uncertainties are so large; perhaps the
quality of the measuring tapes available is extremely poor.
The volume is calculated by multiplying the length, width and height. Since uncertainty is present in the measurement of each dimension, we must take all into account to determine the final uncertainty. Since we're using multiplication, we use the the bottom formula in the table above when calculating the uncertainty in the product.
$$Volume \, of \, Glove \, Box = 195.0 cm \, \cdot \, 78.0 cm \, \cdot \, 92.0 cm = 1.4 \, \times \, 10^5 cm^3 $$
$$Uncertainty \, in \, volume = 1.4 \, \times \, 10^5 cm^3 \sqrt{{\Big(\frac{2.0cm}{195.0cm}\Big)}^2 + {\Big(\frac{1.0cm}{78.0cm}\Big)}^2 + {\Big(\frac{1.0cm}{92.0cm}\Big)}^2} = 0.3 \times 10^5 cm^3 $$
You would report the volume of the glove box to be \((1.4 \pm 0.3) \times \, 10^5 cm^3\) at the 68% confidence level. Recalling the rule of thumb that t=2 for 95% confidence level, we could equally well report \((1.4 \pm 0.6) \times \, 10^5 cm^3\) at 95% C.L. Note how we report the uncertainty using only one significant digit.
Notes:
1) If using more than three variables in your math operations, generalize the above formulae by adding D, E, F ...
2) There are some cases when other mathematical operations such as logarithmic or exponential functions are present. A general formula exists to determine how to calculate the uncertainty for these operations, but it is faster and simpler to look up the formulas for uncertainty propagation when using these operations.