# 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

## 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