March 11, 2020

Some tourists in the Museum of Natural History are marveling at some dinosaur bones. One of them asks the guard, “Can you tell me how old the dinosaur bones are?”

The guard replies, “They are 3 million, four years, and six months old.”

“That’s an awfully exact number,” says the tourist. “How do you know their age so precisely?”

The guard answers, “Well, the dinosaur bones were three million years old when I started working here, and that was four and a half years ago.”

Lets be honest about what we know and what we do not know

We write the values that have real meaning

3 million years means 3±0.5 ⨉ 10^{6}

Adding 4.5 years is meaningless

- All non-zero digits are significant
- In 1234 all digits are significant
- Same in 12.34 and 1.234

- Zeros surrounded by non-zero are significant
- Same in 1204. Four significant digits

- Zeros to the left are not significant
- Four significant digits in 0.0001234

- Zeros to the right are not significant
**unless**there is a decimal point- 12340 has four significant digits
- 123.40 has five significant digits
- 1234.0 has five significant digits
- 12340. has five significant digits

Notice that 1234.0 is not the same as 1234

Also, 12340. is not the same as 12340

This is a *convention* but not everybody uses it

It is much safer to use scientific notation

It is safer to represent numbers in *scientific notation* \[
\begin{align}
1234.0 & = 1.2340\cdot 10^3\\
1234 & = 1.234 \cdot 10^3\\
12340. & = 1.2340 \cdot 10^4\\
12340 & = 1.234 \cdot 10^4
\end{align}
\]

All digits in the *mantissa* are significant

(*mantissa* is the number being multiplied by 10 to the power of the *exponent*)

- Last week we measured a volume
- Round the uncertainty to a single figure
- Instead of 2013.765 ± 78.345 write 2013.765 ± 80

- Then round the main value to the last well known place
- Instead of 2013.765 ± 80 write 2010 ± 80

- The digits “3.765” were a
*white lie*. Let’s not fool ourselves

Last class we saw the easy rule for error propagation

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

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.

\[\begin{align} (x ±𝚫x)^2& = x^2 ± 2x\cdot𝚫x\\ & = x^2 ± 2x^2\cdot\frac{𝚫x}{x} \\ & = x^2 ± 2𝚫x\% \end{align}\] and \[\begin{align} \sqrt{x ±𝚫x}& = \sqrt x ± \frac{1}{2\sqrt x}\cdot 𝚫x\\ & = \sqrt x ± \frac{1}{2}\sqrt x\cdot \frac{𝚫x}{x}\\ & = \sqrt x ± \frac{1}{2}𝚫x\% \end{align}\]

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

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)

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

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

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)

Standard deviation of *rectangular* distribution is \[u=\frac{a}{\sqrt{3}}\] when the width of the rectangle is \(2a\)

Standard deviation of *noise* can be estimated from the data: \[s=\sqrt{\frac{1}{n-1}\sum_i (x_i - \bar x)^2}\]

If the measures are random, their average is also random

It has the same *mean* but less *variance*

Standard error of the *average* of samples is \[\frac{s}{\sqrt{n}}\]