02 Errors Propagation

Why propagation?

With the example of measuring the diameter of the plastic donut, we know that there are errors in measurements. We display the measurement like this:

$$\text{value}=\text{mean}\pm \text{standard error unit}$$

With the example and measurement of the previous lecture, we know that the value is:

$$\text{diameter}=3.02\pm 0.02in$$

Now with that measurement, we want to make a box for the donut. We will calculate with that measurement, but how much is the error for the box? That’s why we propagate the errors.

Errors notation

To write value $x$ with its error, we write $x\pm \delta x$. In these examples, we assume that the errors are all random and uncorrelated.

Arbitrary functions of one variable

With one variable to propagate, we calculate the errors using this formula:

$$\delta q=\left|\frac{dq}{dx}\right|\delta x$$

This formula can be applied to any function based on one independent variable like $q=f(x)$.

Uncertainty in a function of several variables

Let $q=f(x,y,z,...)$, then:

$$\delta q=\sqrt{(\frac{dq}{dx}\delta x{)}^2+(\frac{dq}{dy}\delta y{)}^2+(\frac{dq}{dz}\delta z{)}^2+\dots }$$

You might ask why not adding them up? Because of their porbabilistic nature of errors and their uncorrelation.

Partial derivatives

When taking the derivatives of a function with multiple variables, we focus on on variable at a time. The other variable will be treated as constants. For example, taking the derivative of $g(x,y,z)=x^{2}y{z}^3+x{y}^3{z}^2$:

$$\frac{dg}{dx}=2xy{z}^3+y^3{z}^2$$ $$\frac{dg}{dy}=x^2{z}^3+x3y^2{z}^2$$ $$\frac{dg}{dz}=x^{2}y3z^2+x{y}^{3}2z$$