Propagation of error
From Psychology Wiki
Community portal · Tasks to do · News · Help
Clinical · Educational · Ind&Org · Other fields · Professional · Transpersonal · World
Assessment |
Biopsychology |
Comparative |
Cognitive |
Developmental |
Language
Personality |
Philosophy |
Research Methods |
Social |
Statistics
Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory
In statistics, propagation of uncertainty (or propagation of error) is the affect of variables' uncertainties (or errors) on the uncertainty of a function based on them. Mainly, the variables are measured in an experiment, and have uncertainties due to measurement limitations (e.g. instrument precision) which propagate to the result.
The uncertainty is usually defined by the absolute error — a variable that is probable to get the values x±Δx is said to have an uncertainty (or margin of error) of Δx. In other words, for a measured value x, the true value is probable to be in [x−Δx, x+Δx]. Uncertainties can also be defined by the relative error, Δx/x, and then it is usually written as percentage. It is assumed that the probability of the true value to be in distinct distances from the measured value is normally distributed, with the uncertainty being the standard deviation.
This article explains how to calculate the uncertainty of a function, if the variables' uncertainties are known.
Contents |
[edit] General formula
Let 
be a function which depends on 
variables 
. The uncertainty of each variable is given by 
:
If the variables are uncorrelated, we can calculate the uncertainty Δf of f that results from the uncertainties of the variables:
where 
designates the partial derivative of 
for the 
-th variable.
If the variables are correlated, the covariance between variable pairs, Ci,k := cov(xi,xk), enters the formula with a double sum over all pairs (i,k):
where Ci,i = var(xi) = Δxi².
After calculating 
, we can say that the value of the function with it's uncertainty is:
[edit] Example formulas
This table shows the uncertainty of simple functions, resulting from uncorrelated variables A, B, C with uncertainties ΔA, ΔB, ΔC, and a precisely-known constant c.
| function | function uncertainty |
|---|---|
| X = A ± B | (ΔX)² = (ΔA)² + (ΔB)² |
| X = cA | (ΔX) = c(ΔA) |
| X = c(A×B) or X = c(A/B) | (ΔX/X)² = (ΔA/A)² + (ΔB/B)² |
| X = c(A×B×C) or X = c(A/B)×C | (ΔX/X)² = (ΔA/A)² + (ΔB/B)² + (ΔC/C)² |
| X = cAn | (ΔX/X) = |n| (ΔA/A) |
| X = ln cA | ΔX = (ΔA/A) |
| X = exp A</td> | (ΔX/X) = ΔA |
[edit] Example application: Resistance measurement
A practical application is an experiment in which one measures current, I, and voltage, V, on a resistor in order to determine the resistance, R, using Ohm's law, 
Given the measured variables with uncertainties, I±ΔI and V±ΔV, the uncertainty in the computed quantity, ΔR is
Thus, in this simple case, the relative error ΔR/R is simply the square root of the sum of the squares of the two relative errors of the measured variables.
[edit] External links
- An Informal Guide to Error Analysis from Union College.
- Uncertainties and Error Propagation, Appendix V from the Mechanics Lab Manual, Case Western Reserve University.
[edit] See also
| This page uses content from the English-language version of Wikipedia. The original article was at Propagation of error. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License. |
