Bertolt Brecht, Life of Galileo (1939)

Uncertainty of measurement is the doubt about the result of a measurement, due to

- resolution
- random errors
- systematic errors

How big is the margin? How bad is the doubt?

We declare an interval: [*x*_{min},
*x*_{max}]

Most of the time we write x ± 𝚫x

Example: 20cm ± 1cm

Do not to confuse *error* and *uncertainty*

*Error* is the difference between the measured and the “true”
value

*Uncertainty* is a quantification of the doubt about the
result

Whenever possible we try to correct for any known errors

But any error whose value we do not know is a source of uncertainty

For a single read, the uncertainty depends *at least* on the
instrument resolution

For example, my old water heater showed temperature with 5°C resolution: 50, 55, 60,…

If it shows 55°C, the real temperature is somewhere between 53°C and 57°C

We write 55°C ± 2.5°C

For a single read, 𝚫x = half of the resolution

Sum of two measurements:

\[(x ± 𝚫x) + (y ± 𝚫y) = (x+y) ±
(𝚫x+𝚫y)\]

Difference between measurements:

\[(x ± 𝚫x) - (y ± 𝚫y) = (x-y) ±
(𝚫x+𝚫y)\]

To calculate \((x ± 𝚫x) × (y ± 𝚫y)\) we first write the uncertainty as percentage

\[(x ± 𝚫x/x\%) × (y ± 𝚫y/y\%)\]

Then we sum the percentages:

\[xy ± (𝚫x/x + 𝚫y/y)\%\]

Finally we convert back to the original units:

\[xy ± xy(𝚫x/x + 𝚫y/y)\]

\[ \begin{aligned} (x ± 𝚫x) \times (y ± 𝚫y) & = x(1 ± 𝚫x/x) \times y(1 ± 𝚫y/y)\\ & = xy(1 ± 𝚫x/x)(1 ± 𝚫y/y) \\ & = xy(1 ± 𝚫x/x ± 𝚫y/y ± (𝚫x/x)(𝚫y/y)) \\ & = xy(1 ± 𝚫x/x + 𝚫y/y) \\ & = xy ± xy(𝚫x/x + 𝚫y/y)\\ \end{aligned} \]

We discard \((𝚫x/x)(𝚫y/y)\) because it is small

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

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

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

Assuming that the errors are small compared to the main value, we can find the error for any “reasonable” function

Taylor’s Theorem says that, for any derivable function \(f,\) we have \[f(x±𝚫x) = f(x) ± \frac{df}{dx}(x)\cdot 𝚫x + \frac{d^2f}{dx^2}(x+\varepsilon)\cdot \frac{𝚫x^2}{2}\] When \(𝚫x\) is small, we can ignore the last part.

\[\begin{align} (x ±𝚫x)^2& = x^2 ± 2x\cdot𝚫x\\ & = x^2 ± 2x^2\cdot\frac{𝚫x}{x} \\ & = x^2 ± 2𝚫x\% \end{align}\]

\[\begin{align} \sqrt{x ±𝚫x}& = \sqrt x ± \frac{1}{2\sqrt x}\cdot 𝚫x\\ & = \sqrt x ± \frac{1}{2}\sqrt x\cdot \frac{𝚫x}{x}\\ & = \sqrt x ± \frac{1}{2}𝚫x\% \end{align}\]

Calculate the uncertainty in

The density of the stone ball

the Drake’s formula

the number of piano tuners in your city

Last class we only considered one kind of uncertainty: the instrument resolution

This is a “one time” error

- We notice it immediately
- It does not change if we measure again

There are other sources of uncertainty: **noise**

When the instrument resolution is good, we observe that the measured values change on every read

In many cases this is due to thermal effects, or other sources of noise

Usually the variability follows a Normal distribution

The exact distribution is hard to calculate

International standards suggest using computer simulation

They recommend Montecarlo methods

(what we did here)

Aristotle (384–322 BC), Nicomachean Ethics