Class 11: Statistical uncertainty

Methodology of Scientific Research

Andrés Aravena, PhD

March 22, 2023

Preparing for tomorrow

Do yo know how to solve this equation?

\[ax^2+ bx + c =0\]

Your answer will help me prepare tomorrow’s class

The easy uncertainty propagation rule

Last class we saw the easy rule for error propagation

\[ \begin{aligned} (x ± Δx) + (y ± Δy) & = (x+y) ± (Δx+Δy)\\ (x ± Δx) - (y ± Δy) & = (x-y) ± (Δx+Δy)\\ (x ± Δx\%) \times (y ± Δy\%)& =xy ± (Δx\% + Δy\%)\\ (x ± Δx\%) ÷ (y ± Δy\%)& =x/y ± (Δx\% + Δy\%) \end{aligned} \]

Here \(Δx\%\) represents the relative uncertainty, that is \(Δx/x\)

We use absolute uncertainty for + and -, and relative uncertainty for ⨉ and ÷

Be careful whit absolute and relative

It is easy to get confused with relative errors

Instead of \((x ± Δx\%)\) it is better to write \[x(1± Δx/x)\]

Mathematical notation was invented to make things clear, not confusing

Exercise: Let’s verify the formulas

Remember that we assume that \(Δx/x\) is small

Errors may compensate

Probabilistic uncertainty propagation

These rules are “pessimistic”. They give the worst case

In general the “errors” can be positive or negative, and they tend to compensate

(This is valid only if the errors are independent)

In this case we can analyze uncertainty using the rules of probabilities

Variance quantifies uncertainty

In this case, the value \(Δx\) will represent the standard deviation of the measurement

The standard deviation is the square root of the variance

Then, we combine variances using the rule

“The variance of a sum is the sum of the variances”

(Again, this is valid only if the errors are independent)

The probabilistic rule

\[ \begin{align} (x ± Δx) + (y ± Δy) & = (x+y) ± \sqrt{Δx^2+Δy^2}\\ (x ± Δx) - (y ± Δy) & = (x-y) ± \sqrt{Δx^2+Δy^2}\\ (x ± Δx\%) \times (y ± Δy\%)& =x y ± \sqrt{Δx\%^2+Δy\%^2}\\ \frac{x ± Δx\%}{y ± Δy\%} & =\frac{x}{y} ± \sqrt{Δx\%^2+Δy\%^2} \end{align} \]

Confidence interval for the measurement

When using probabilistic rules we need to multiply the standard deviation by a constant k, associated with the confidence level

In most cases (but not all), the uncertainty follows a Normal distribution. In that case

  • \(k=1.96\) corresponds to 95% confidence
  • \(k=2.00\) corresponds to 98.9% confidence
  • \(k=2.57\) corresponds to 99% confidence
  • \(k=3.00\) corresponds to 99.9% confidence


In movies, people carries suitcases with cash

How much money is there?

(you can make any justified hypothesis)

What is the weight of this ferry?

More pictures of the ferry

And a classic one