# Talk:Propagation of uncertainty

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## The Resistance example is wrong, I think.

The errors should add in quadrature. So square each element and take the square root. Take the example Y=A+B. Then the error is \sqrt(dA^2+dB^2), not dA+dB. That is what you get when you use the equation right above the example problem. —Preceding unsigned comment added by 132.163.47.52 (talk) 23:21, 4 July 2010 (UTC)

I fixed the error. And I corrected the Absolute Error Box. I am fairly certain they were incorrect. —Preceding unsigned comment added by 132.163.47.52 (talk) 23:59, 4 July 2010 (UTC)

I genuinely think that this article is important for the whole of science. Too many scientists (including myself) graduate without becoming confident with error calculations, and I suspect that many scientists never do so. I see here an opportunity to improve the quality of a whole body of scientific work.

Currently, the approach is too mathematical (and not even rigorous), and does not deal with the subtleties of error calculations, like the difference between using errors and standard deviations, and which situations it is ok to use approximations.

When the equations are approximations, why not use the "is approximately equal to" instead of the equals sign? The fact that it is an approximation should be clearly marked, in any case.

I suggest the article should clearly delineate the different possible approaches - perhaps this could even become the basis for standardisation of error calculations in future publications. Thanks, MichaelR 83.72.194.27 11:12, 7 October 2007 (UTC)

## Contradiction between formula and example?

It seems there is a conflict between the general formula and the very first example - perhaps I'm missing something here, but it strikes me as confusing that the error is given as $\Delta X=\Delta A+\Delta B$ when the equation suggests it should be $\Delta X=(\Delta A^{2}+\Delta B^{2})^{\frac{1}{2}}$.

Thanks gabe_rosser 00:20, 23 May 2007 (BST)

Yes, Gabe's formula above is much closer to what I use. I'm no statististian, but the sources that showed me how to do it that way are. I think maybe some of the formulas presented in the article are approximations to make things simpler for hand calculation. If that's what they are, it should be clearly stated, because it is obviously causing confusion. ike9898 18:23, 8 August 2007 (UTC)
I agree with Gabe, the formula given for ΔR does not follow from the described method of finding ΔR. I am a 4th year Mechanical Engineering student, and I am certain that Gabe is correct; thus, I will be correcting the error.

## i or j

Is it necessary to use both i and j as indices for the summation of the general formulae? it appears to me that i only appears in the maths whilst j only appears in the english. True? If not, it could be more clearly explained as to the reasons for the change / use of both.

Thanks Roggg 09:35, 20 June 2006 (UTC)

## Geometric mean

Example application: the geometric mean? Charles Matthews 16:33, 12 May 2004 (UTC)

From the article (since May 2004!): "the relative error ... is simply the geometric mean of the two relative errors of the measured variables" -- It's not the geometric mean. If it were, it would be the product of the two relative errors in the radical, not the sum of the squares. I'll fix this section. --Spiffy sperry 21:47, 5 January 2006 (UTC)

## Delta?

In my experience, the lower-case delta is used for error, while the upper-case delta (the one currently used in the article) is used for the change in a variable. Is there a reason the upper-case delta is used in the article? --LostLeviathan 02:01, 20 Oct 2004 (UTC)

## Missing Definition of Δxj

A link exists under the word "error" before the first expression of Δxj in the article, but this link doesn't take one to a definition of this expression. The article can be improved if this expression is properly defined. —Preceding unsigned comment added by 65.93.221.131 (talkcontribs) 4 October 2005

## Formulas

I think that the formula given in this article should be credited to Kline-Mcklintock. —lindejos

First, I'd like to comment that this article looks like Klingonese to the average user, and it should be translated into English.

Anyway, I was looking at the formulas, and I saw this allegation: X = A ± B (ΔX)² = (ΔA)² + (ΔB)², which I believe is false.

As I see it, if A has error ΔA then it means A's value could be anywhere between A-ΔA and A+ΔA. It follows that A±B's value could be anywhere between A±B-ΔA-ΔB and A±B+ΔA+ΔB; in other words, ΔX=ΔA+ΔB.

If I am wrong, please explain why. Am I referring to a different kind of error, by any chance?

aditsu 21:41, 22 February 2006 (UTC)

As the document I added to External links ([1]) explain it, we are looking at ΔX as a vector with the variables as axes, so the error is the length of the vector (the distance from the point where there is no error).
It still seems odd to me, because this gives the distance in the "variable plane" and not in the "function plane". But the equation is correct. —Yoshigev 22:14, 23 March 2006 (UTC)
Now I found another explanation: We assume that the variables has Gaussian distribution. The addition of two Gaussians gives a new Gaussians with a width equals the quadrature of the width of the originals. (see [2]) —Yoshigev 22:27, 23 March 2006 (UTC)

## Article title

The current title "Propagation of errors resulting from algebraic manipulations" seems to me not so accurate. First, the errors don't result from the algebraic manipulations, they "propagate" by them. Second, I think that the article describe the propagation of uncertainties. And, third, the title is too long.

Seems okay. A problem with the article is that the notation x + Δx is never explained. From your remarks, it seems to mean that the true value is normally distributed with mean x and variance Δx. This is one popular error model, leading to the formula (Δ(x+y))² = (Δx)² + (Δy)².
Another one is that x + Δx means that the true value of x is in the interval $[x-\Delta x, x+\Delta x]$. This interpretation leads to the formula $\Delta(x+y) = \Delta x + \Delta y$, which aditsu mentions above.
I think the article should make clear which model is used. Could you please confirm that you have the first one in mind? -- Jitse Niesen (talk) 00:58, 24 March 2006 (UTC)
Not exactly. I have in mind that for the measured value x, the true value might be in $[x-\Delta x, x+\Delta x]$, like your second interpretation. But for that true value, it is more probable that it will be near x. So we get a normal distribution of the probable true value around the measured value x. Then, 2Δx is the width of that distribution (I'm not sure, but I think the width is defined by the standard deviation), and when we add two of them we use (Δx)² + (Δy)², as explained in Sum of normally distributed random variables.
I will try to make it clearer in the article. —Yoshigev 17:45, 26 March 2006 (UTC)

As you can see, I rewrote the header and renamed the article. —Yoshigev 17:44, 27 March 2006 (UTC)

## This artical was a disgrace to humanity

First the article defined $\Delta A$ as the absolute error of $A$ but then the example formulas section went ahead and defined error propagation with respect to $\Delta A$ as the standard deviation of $A$. Even then the examples had constants that weren't even considered in the error propagation. Then to add insult to injury I found a journal article which shows that at least two of the given definitions were only approximations so I had to redo the products formula and added notes to the remaining formulas explaining that they were only approximations with an example of how they are only approximations. (doesn't it seem just a little bit crazy to only find approximations to the errors when you can have an exact analysis of the error propagation, to me it just seems like approximating an approximation.) So now there are TWO columns: ONE for ABSOLUTE ERRORS and ANOTHER for STANDARD DEVIATIONS! sheash! it's not that hard to comprehend that absolute errors and standard deviations are NOT EQUIVILANT! $\sigma_A$ is the standard deviation of $A$, and $\Delta A$ is the absolute error NOT the standard deviation of $A$. --ANONYMOUS COWARD0xC0DE 04:02, 21 April 2007 (UTC)

The problem is now the formula given in the table don't match the stated general formula for absolute error in terms of partial derivatives. I think this makes the article rather confusing, particularly for those readers who don't have access to the journal article you've linked. One is left without a clear idea (1) where the general formula comes from, (2) why it is inexact for the case of a product of two or more quantities, and (3) where the exact forms come from.
The answer is relatively simple. The general form given is basically a Taylor series, truncated after the first order term. It is valid only if the errors are small enough that we can neglect all higher order terms. Of course, if we have a product of n quantities we would need an n-th order Taylor series to produce the exact result. The article needs to be edited to make this clear.
This is the person who wrote the absolute error box, that is actually incorrect. —Preceding unsigned comment added by 132.163.47.52 (talk) 15:33, 5 July 2010 (UTC)
Specifically, the general form should look something like this:
$\Delta f(x_1,\cdots,x_k) = \sum_{n_1=0}^{\infin} \cdots \sum_{n_k=0}^{\infin} \left | \frac{\partial^{n_1} \cdots \partial^{n_k} f}{{\partial x_1^{n_1}} \cdots \partial x_k^{n_k}} \right | \frac{(\Delta x_1)^{n_1}\cdots (\Delta x_1)^{n_k}}{n_1!\cdots n_k!}$
However the term with $n_1 \cdots n_k$ all equal to zero is excluded. Note that this expression simplifies to the approximate form if we include only the lowest order terms (i.e., terms with $n_1 + \cdots + n_k = 1$)
--Tim314 05:15, 30 May 2007 (UTC)

## Caveats?

Can we get some caveats placed in here? i.e. a statement that these formula are only valid for small sigma/mean and that the formula for the quotient particularly is problematic given Hodgson's paradox and the discussion in ratio distribution. The Goodman (1960) article has some exact formula that modify the approximate forms here as well.169.154.204.2 01:17, 23 August 2007 (UTC)

## Re-wrote example table

I searched for this page to remind myself of one of the formulae, which I use on a regular basis. (When I'm at work, I flip to a tabbed page in my copy of [Bevington].) I could see that the table contained many errors, mostly due to confusions between $\Delta A\!\,$ and $\left(\Delta A\right)^2 \!\,$, and some of the table entries included extra terms, presumably for correlations between $A\text{, }B\text{, }C\!\,$, though they are stated in the caption to be uncorrelated. I didn't notice it right away, but the table had a third column with $\Delta A\!\,$ replaced with an undefined (but commonly used) $\sigma_A\!\,$.

In its current form, the table is (exactly, not approximately) correct for uncorrelated, normally-distributed real variables. Hopefully, it also gives an indication of how these rules can be combined. (Some are special cases of others, but they all derive from the general rule in the previous section, anyway.)

So what about formulae for correlated variables? I don't know much about that because I always use the general derivative method for problems with correlations. I only use the example formulae for simple cases. Jim Pivarski 23:25, 7 October 2007 (UTC)

## Pitfall of linear error propagation: eg dividing by zero

There is an important exception to the usual treatment of error propagation. That is when a power series expansion of the function F(x) to be calculated breaks down within the uncertainty of the variable x. The usual treatment assumes that the function is smooth and well-behaved over a sufficiently large domain of x near the measured value x_0. Then one expands in a power series about x_0 and takes only the first linear term. However, if F is not well-behaved (for example if F(x) goes to infinity at some point near x_0) the correct uncertainty on F(x) may be completely different from the usual formula Delta F = F'(x_0) Delta x.

The simplest example is the uncertainty in 1/x. If we measure x=100 with an uncertainty of 1%, then 1/x has an uncertainty of 1% also. If we measure x=100 with an uncertainty of 100%, then 1/x has an *infinitely* large uncertainty, because x may take the value 0!

Is this effect treated in any known texts? --Tdent 15:09, 24 October 2007 (UTC)

## caveats added, but more details needed

I put in a couple of references to problems with ratios alluded to above, but what is required is a better quantification for when these issues are important. The Geary-Hinkel transformation has limits of validity and they can be used to show when that formula (which reduces to the standard case in favorable circumstances) is useful. Hopefully someone can add to that? 74.64.100.223 (talk) 16:07, 29 November 2007 (UTC)

## Incorrect Taylor Series Expansion

I think the Taylor series expansion in the section in Non-linear combinations is wrong

$f_k \approx f^0_k+ \sum_i^n \frac{\partial f_k}{\partial {x_i}} x_i +\sum_i^n\sum_{j (j \ne i)}^n \frac{\partial f_k}{\partial {x_i}}\frac{\partial f_k}{\partial {x_j}}x_ix_j$

$f_k \approx f^0_k+ \sum_i^n \frac{\partial f_k}{\partial {x_i}} x_i +\sum_i^n\sum_{j}^n \frac{\partial^2 f_k}{\partial {x_i}\partial {x_j}}x_ix_j$

This is a major difference. Using only the first-order approximation,

$f_k \approx f^0_k+ \sum_i^n \frac{\partial f_k}{\partial {x_i}} x_i$

which is what I think the author of that section is after. This can be used to estimate error under a linear approximation (a hyperplane tangent to the non-linear function) using the functions in the previous section. Perhaps I'm missing something here, but if not, the section and examples are seriously flawed. Schomerus (talk) 22:38, 22 February 2008 (UTC)

Technically you are right. However the missing term $+\sum_i^n \left(\frac{\partial f_k}{\partial {x_i}}\right)^2 x_i^2$ does not contribute to the error propagation formula. In that sense the approximation is acceptable. Rest assured that the error propagation formulas are correct and can be found in numerous text-books. Petergans (talk) 09:20, 23 February 2008 (UTC)
My concern over the Taylor series expansion is that the correct coefficient for the second order term should be given by the second derivative of $f$ not the product of first derivatives. It is definitely not the case in general that
$\frac{\partial f_k}{\partial {x_i}}\frac{\partial f_k}{\partial {x_j}} = \frac{\partial^2 f_k}{\partial {x_i}\partial {x_j}}$
in any sense, approximate or otherwise. For example, in the linear case
$f = x_1 + x_2$
expanding around $(0,0)'$ according to the the first formula
$f \approx x_1 + x_2 +x_1x_2$
which makes little sense given that a first order approximation to a linear equation should be exact, not an approximation at all, and in any case shouldn't contain any second order terms. If I've misunderstood something here could you provide a reference so I can check the derivation for myself?Schomerus (talk) 17:50, 23 February 2008 (UTC)

Now I see it. I think that the mistake is that it's not a Taylor series expansion, but a total derivative sort of thing. total derivative#The total derivative via differentials appears to be similar to the expression I'm looking for, though not exactly. I'm sure that the expression

$\sigma^2_f=\left(\frac{\partial f}{\partial a}\right)^2\sigma^2_a+\left(\frac{\partial f}{\partial b}\right)^2\sigma^2_b+2\frac{\partial f}{\partial a}\frac{\partial f}{\partial b}COV_{ab}$

is correct, but clearly it is not derived in the way that I thought. I don't have any references to hand. Can you help me out here? Petergans (talk) 22:38, 23 February 2008 (UTC)

I finally figured it out - the second term is not needed at all. The function is linearized by Taylor series expansion to include first derivatives only so thay the error propagation formula for linear combinations can be applied. Thanks for spotting the error. Petergans (talk) 08:05, 24 February 2008 (UTC)

That's right. The expression is a direct consequence of applying the formula for linear combinations to the linear approximation (hyperplane tangent at $f(\bar{x})$ ) obtained by first order Taylor series expansion. Thanks for making the corrections in the article. You could add that the approximation expands around $\bar{x}$ making $f_0= f(\bar{x})$, which is another potential source of inaccuracy when the linear approximation fails since for non-linear functions in general $\overline{f(x)} \ne f(\bar{x})$Schomerus (talk) 17:34, 24 February 2008 (UTC)

The equation that is currently in place with absolute values being taken can not generate the example equation 4 ( A/B ) -> -cov(ab) the absolute value bars would make A*B error propagation the same as A/B. —Preceding unsigned comment added by 129.97.120.140 (talk) 18:31, 28 January 2011 (UTC)

## Useful Reference?

Been trying to understand this for a while (and how the formula are derived) and not that easy to understand from this page. Finally found a useful reference which then makes the rest of this page make sense so thougt I'd share. The reference is

Taylor, J. R., 1997: An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. 2nd ed. University Science Books, 327 pp.

and section 9.2 helps make things clear.

Added a comment here as I don't have time to edit main page. —Preceding unsigned comment added by 139.166.242.47 (talk) 11:44, 29 February 2008 (UTC)

Better books that covers these issues is "Propagation of Errors" by Mike Peralta. This entry April 2013. — Preceding unsigned comment added by 144.15.255.227 (talk) 17:45, 29 April 2013 (UTC)

## Undefined variable in caveats section

In the caveats section, the formula

$y=mz+c: \sigma^2_y=z^2\sigma^2_m+\sigma^2_c+2z\rho \sigma_m\sigma_c$

is written, but the variable $\rho$ is not defined. Please fill in its definition. Zylorian (talk) —Preceding undated comment was added at 17:04, 6 October 2008 (UTC).

Additionally, this formula is wrong. It's not the correct formula for uncertainty in a linear regression; that uncertainty can't be arrived at through error propagation. I'm not sure that the idea of error propagation could ever be correctly applied to least squares parameters. Perhaps that paragraph should be removed from the caveats section. DrPronghorn (talk) 04:04, 9 February 2012 (UTC)

OK, I went ahead and removed that section. It's not clear to me that error propagation is the correct place to discuss the errors in fitted parameters. In the case of fitted models, the errors are governed by the fitting process and original fit dataset, not by error propagation. DrPronghorn (talk) 04:19, 22 February 2012 (UTC)

Better book that covers these issues is "Propagation of Errors" by Mike Peralta. — Preceding unsigned comment added by 144.15.255.227 (talk) 17:43, 29 April 2013 (UTC)

## Errors in section 1 and confusions in section "Partial derivatives"

The third equation for $M^f_{ij}$ is wrong or inconsistent with itself. The term $A_{ik}$ should have its subscripts reversed, right?

Likewise, the same error in the next equation down.

This is my first post on wikipedia, I leave this to the author to correct it if it needs to be done.

In agreement with the comments above, I found the section "Partial derivatives" a bit contradictory or at least lacking explanation. It is contradictory because in the third paragraph $\Delta x$ is said to be commonly given to be the standard deviation $\sigma$, which is the square root of the variance $\sigma^2$, yet in the section "Partial derivatives" the square root of the variance is not equal to the absolute error in the table. It was precisely this issue for which I had consulted wikipedia. However there is enough information here to have helped me solve my problem.

Dylan.jayatilaka (talk) 05:23, 7 January 2009 (UTC)

## Created new article on this subject

...rather than try to edit this one. Take a look: Experimental uncertainty analysis. Right or wrong, I believe it to be far more "accessible" than this one is. How to reconcile my new article with this one and with Uncertainty analysis remains to be seen. Please give me a week or so to produce the table at the bottom of "my" article- it will go beyond the table in this article, and I think it will be a very useful thing for many readers. Also I have a mention and a reference related to the 1/x thing I see above here- the solution of course is to use a higher-order expansion, and the necessary eqns are in that book. Regards... Rb88guy (talk) 18:30, 12 March 2009 (UTC)

## Derivative is not partial in Example calculation: Inverse tangent function

If $f(x) = \arctan(x)$ then you cannot take $\frac{\partial f}{\partial x}$, because $f$ only depends on one variable. It should be $\frac{df}{dx}$ — Preceding unsigned comment added by Finkeltje (talkcontribs) 15:31, 1 February 2013 (UTC)

## Linear Combinations section had following formula:

$\sigma^2_f= \sum_i^n \sum_j^n a_i \Sigma^x_{ij} a_j= \mathbf{a \Sigma^x a^t}$

I changed this to

$\sigma^2_f= \sum_i^n \sum_j^n a_i \Sigma^f_{ij} a_j= \mathbf{a \Sigma^f a^t}$