Why is propagation of uncertainties quadratic rather than linear? [duplicate]

Question

1) Up until now, during practical work sessions, I always used these formulas for uncertainty propagation:

if $C = A+B$ or $C = A-B$ $$\Delta C = \Delta A + \Delta B$$ if $C = AB$ or $C = \frac{A}{B}$ $$\frac{\Delta C}{C} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$$ if $C = A^m$ $$\frac{\Delta C}{C} = |m|\frac{\Delta A}{A}$$

These formulas are derived from the expression for the differential of a function of multiple variables:

if $C = f(A,B)$ $$ dC = \frac{\partial C}{\partial A}dA + \frac{\partial C}{\partial B}dB \Rightarrow \Delta C = \frac{\partial C}{\partial A}\Delta A + \frac{\partial C}{\partial B}\Delta B$$

and I think that makes sense because the value C is just a function of two variables that happen to be measurements.

2) But this morning, I was told that this is wrong and I should actually use these instead:

if $C = A+B$ or $C = A-B$ $$(\Delta C)^2 = (\Delta A)^2 + (\Delta B)^2$$ if $C = AB$ or $C = \frac{A}{B}$ $$\left(\frac{\Delta C}{C}\right)^2 = \left(\frac{\Delta A}{A}\right)^2 + \left(\frac{\Delta B}{B}\right)^2$$

which are apparently derived from a general formula:

$$(\Delta C)^2 = \left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 + \hspace{0.3cm}...$$

or

$$\Delta C = \sqrt{\left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 + \hspace{0.3cm}...}$$

3) So:

• Why is this formula better?
• Where does it come from?
• What does it actually represent? (Do I recognize the shape of a norm in that last formula?)
• What is wrong with the formula of the differential?

I'm asking a lot, but the propagation of uncertainty always confused the ship out of me.

Answer 1

I think that the easiest way to understand this is in the formula for addition. If you consider your quantity $C$ which depends on $A$ and $B$ such that $C=A+B$, then the formula $\Delta C = \Delta A + \Delta B$ overestimates the error values. You can visualize this as a rectangle with $A$ on the x-axis and $B$ on the y-axis with the area enclosed being $\Delta C$. However, the likelihood of having errors in the corners, i.e. the extreme values for both $A$ and $B$ are very small. Therefore, instead of using a rectangle to represent the error, we instead use an ellipse with $\Delta C^2 = \Delta A^2 + \Delta B^2$. This gets rid of the extremely unlikely error values.

Answer 2

These new formulae are a form of "adding in quadrature." In my understanding it's a way of combining errors whilst acknowledging that the 'worst case' of adding two errors is not likely to happen, if those errors are independent. For example, if you measure the area of a rectangle by measuring its length with a ruler, and its width with some kind of laser/light measuring device, then the two errors in these measurements are independent, and it will be fairly unlikely that overall you would get two errors in the same direction (e.g. both overmeasuring). The quadrature tries to take this into account by playing them off against each other in some sense, and you are right, there is some notion of the norm involved.