Uncertainties in Quantities and Error Propagation
- Page ID
- 175486
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Table of Contents:
Relative Standard Deviation
As with accuracy, precision is usually expressed in a relative sense. In this case it is relative to the average and typically expressed in terms of ppt and is referred to as the RSD (relative standard deviation). For this example the RSD is
\[\begin{align}\text{RSD (in ppt)} &= \left(\frac{\text{st dev}}{|\text{average}|}\right) \times 1000 \text{ ppt}\\&= \left(\frac{0.00067 \text{ g/L}}{0.1095 \text{ g/L}}\right) \times 1000 \text{ ppt}\\&=6.1 \text{ ppt}\end{align}\nonumber\]
Note that that RSD does not have any real units because it is a ratio of two values with the same units.
Based on the particular experimental procedure you will be using, recommendations on the expected RSD will be made. By comparing your RSD to the recommendation you can evaluate how well you are performing the experimental procedures in terms of reproducibility.
Range of the Experimental Data
Sometimes you may want to quickly estimate the precision of a data set without computing the standard deviation. One way to do this is to calculate the range of the data. Using the examples for density of the nitrogen gas sample given above, the range is
\[\begin{align}\text{RSD (in ppt)} &= \left(\frac{\text{largest}}{\text{smallest}}-1\right) \times 1000 \text{ ppt}\\&= \left(\frac{0.1102}{0.1089}-1\right) \times 1000 \text{ ppt}\\&=12 \text{ ppt}\end{align}\nonumber\]
Note that the range is different than the RSD, and is very simple to calculate. This makes the range a useful tool for estimating precision as soon as some data is collected. You will be provided recommendations regarding an acceptable range to serve as a guide during your data collection. However, in writing reports you will use standard deviation and RSD to report the precision of your results. As with the standard deviation, always express the RSD with two significant figures.
Uncertainty in Calculated Values
In most experiments the final value of interest will be based on calculations using experimental data that has a measured degree of uncertainty. When this is the case, the uncertainty in the experimental measurements must be used to estimate the uncertainty in the final value. These uncertainty calculations are straightforward when using a spreadsheet. The approach to the process depends on if the calculated value is the result of addition and subtraction, or the result of multiplication and division.
Addition or Subtraction
For the case where a calculated value is based on the addition or subtraction of data that each has an associated standard deviation, the uncertainty in the calculated value is determined as follows. To obtain the sum or difference (n) of these average values (call them \(\overline{x}\) and \(\overline{y}\)):
\[n = \overline{x} + \overline{y} \text{ or } n = \overline{x} - \overline{y}\]
Calculate the standard deviation for each of the average values, call these (std x) and (std y). The uncertainty in the value n is obtained by squaring each of the standard deviations and taking the square root of their sum. This would appear as follows in a single cell in the spreadsheet
\[\text{std } n = \sqrt{(\text{std }x)^2 + (\text{std }y)^2}\]
or
\[\text{std } n = \text{SQRT((std x)^2 + (std y)^2}\]
This value is then the uncertainty for n and will have the same units as x and y.
Multiplication or Division
For the case where a calculated value is based on the multiplication or division of data that each has an associated standard deviation or uncertainty, the uncertainty in the calculated value is determined as follows. For each value that will be used in the calculation, determine the relative standard deviation (RSD) as described above. All of the RSD values must be determined with the same multiplier (% or ppt). As an example, assume that three values x, y and z are used to calculate the final value n using the expression
\[n-(x \times y)/z\]
Thus your spreadsheet should have values for RSD x, RSD y and RSD z, each in their own cell in the spreadsheet, each expressed as ppt. The RSD in the value n is obtained by squaring each of the RSD values and taking the square root of their sum. This would appear as follows in a single cell in the spreadsheet
\[\text{RSD} = \sqrt{(\text{RSD }x)^2 + (\text{RSD }y)^2 + (\text{RSD }z)^2}\]
or
\[\text{RSD} = \text{SQRT((RSD x)^2 + (RSD y)^2 + (RSD z)^2)}\]
The addition of the RSD^2 terms is the same regardless of the order of multiplication or division for any of the values. Since all the values are relative standard deviations, the calculated value is the relative standard deviation or relative uncertainty for n, which has no units. To determine the standard deviation or the uncertainty with the same units as n, it is only necessary to look at the earlier expression for RSD and rearrange to solve for the standard deviation.
\[\text{uncertainty or st dev} = \frac{\text{RSD (in ppt)}}{1000 \text{ ppt}} \times \text{average}\]