March 11, 2020
Significant figures
Dinosaur Bones
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.”
Being honest
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 ⨉ 106
Adding 4.5 years is meaningless
Significant Figures
Which figures are significant?
- 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
Which figures are significant?
- 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
Be aware of notation
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
Using 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)
Writing our result
- 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
Sources of uncertainty
The easy rule
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 ÷
General formula
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.
Examples
\[\begin{align} (x ±𝚫x)^2& = x^2 ± 2x\cdot𝚫x\\ & = x^2 ± 2x^2\cdot\frac{𝚫x}{x} \\ & = x^2 ± 2𝚫x\% \end{align}\] and \[\begin{align} \sqrt{x ±𝚫x}& = \sqrt x ± \frac{1}{2\sqrt x}\cdot 𝚫x\\ & = \sqrt x ± \frac{1}{2}\sqrt x\cdot \frac{𝚫x}{x}\\ & = \sqrt x ± \frac{1}{2}𝚫x\% \end{align}\]
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
Not all uncertainties are alike
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
Visually
Visually
Noise
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
Example
Distribution of thermal noise
Real v/s measured data
Discretization uncertainty
Combined
How to combine
The exact distribution is hard to calculate
International standards suggest using computer simulation
They recommend Montecarlo methods
(what we did here)
Standard deviation of discretization
Standard deviation of rectangular distribution is \[u=\frac{a}{\sqrt{3}}\] when the width of the rectangle is \(2a\)
Standard deviation of thermal noise
Standard deviation of noise can be estimated from the data: \[s=\sqrt{\frac{1}{n-1}\sum_i (x_i - \bar x)^2}\]
Standard deviation of average
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 \[\frac{s}{\sqrt{n}}\]