Uncertainty and measurement

2024-06-1617:39 Status:Complete

Fundamental SI and Derived Units

Derived units are combinations of fundamental units. Some examples are:

m/s (Unit for velocity)
N (kg*m/s^ (Unit for force)
J (kgm^2/s^2) (Unit for energy) However the fundamental units are as follows:

Quantity	SI unit	Symbol
Mass	Kilogram	kg
Distance	Meter	m
Time	Second	s
Electric current	Ampere	A
Amount of substance	Mole	mol
Temperature	Kelvin	K

Scientific Notation and Metric Multipliers

In scientific notation, values are written in the form $a 1 0^{n}$ , where $a$ is a number between 1 and 10 and $n$ is any integer. Some examples are:

The speed of light is $300000000 m / s$ . In scientific notation, this is expressed as $3 \times 1 0^{8}$
A centimeter (cm) is 1/100 of a meter (m). In scientific notation, one cm is expressed as $1 \times 1 0^{-} 2 m$ .

Prefix	Abbreviation	Value
peta	P	10^15
tera	T	10^12
giga	G	10^9
mega	M	10^6
kilo	k	10^3
hecto	h	10^2
deca	da	10^1
deci	d	10^-1
centi	c	10^-2
milli	m	10^-3
micro	μ	10^-6
nano	n	10^-9
pico	p	10^-12
femto	f	10^-15

Significant Figures

For a certain value, all figures are significant, except:

Leading zeros
Trailing zeros if this value does not have a decimal point, for example:

12300 has 3 significant figures. The two trailing zeros are not significant.
012300 has 5 significant figures. The two leading zeros are not significant. The two trailing zeros are significant.

When multiplying or dividing numbers, the number of significant figures of the result value should not exceed the least precise value of the calculation.

The number of significant figures in any answer should be consistent with the number of significant figures of the given data in the question.

Orders of Magnitude

Orders of magnitude are given in powers of 10, likewise those given in the scientific notation section previously.

Orders of magnitude are used to compare the size of physical data.

Distance	Magnitude (m)	Order of magnitude
Diameter of the observable universe	10^26	26
Diameter of the Milky Way galaxy	10^21	21
Diameter of the Solar System	10^13	13
Distance to the Sun	10^11	11
Radius of the Earth	10^7	7
Diameter of a hydrogen atom	10^-10	10
Diameter of a nucleus	10^-15	15
Diameter of a proton	10^-15	15
Mass	Magnitude (kg)	Order of magnitude
The universe	10^53	53
The Milky Way galaxy	10^41	41
The Sun	10^30	30
The Earth	10^24	24
A hydrogen atom	10^-27	-27
An electron	10^-30	-30

Time	Magnitude (s)	Order of magnitude
Age of the universe	10^17	17
One year	10^7	7
One day	10^5	5
An hour	10^3	3
Period of heartheart	10^0	0

Random and Systematic Errors

Random error	Systematic error
- Caused by fluctuations in measurements centered around the true value (spread). - Can be reduced by averaging over repeated measurements. -Not caused by bias.	- Cannot be reduced by averaging over repeated measurements. - Caused by fixed shifts in measurements away from the true value. - Caused by bias.
Examples: - Fluctuations in room temperature - The noise in circuits - Human error	Examples: - Equipment calibration error such as the zero offset error - Incorrect method of measurement

Absolute, Fractional and Percentage Uncertainties

Physical measurements are sometimes expressed in the form x±Δx. For example, 10±1 would mean a range from 9 to 11 for the measurement.

Absolute uncertainty	Δx
Fractional uncertainty	Δx /x
Percentage uncertainty	Δx/x*100%
Calculating with uncertainties

Addition/Subtraction	y=a±b	Δy=Δa+Δb (sum of absolute uncertainties)
Multiplication/Division	y=a*b or y=a/b	Δy/y=Δa/a+Δb/b (sum of fractional uncertainties)
Power	y=a^n	Δy/y=

Instrumental (Rectangular) Uncertainties

Keep in mind that there is always a combination of both Gaussian and Instrumental uncertainty. Use the appropriate one depending on which one dominates. For instance when stopping a stopwatch the error comes primarily from the time needed for a person to register the measurement and mechanically record it. There is some instrumental uncertainty (even if the press was precise), however, the Gaussian error made from the delay in stopping the stopwatch would dominate the rectangular error of the instrument.

Instrumental Uncertainty

Take a balance of some chemical powder in a scale, dependent on the calibrated masses. Consider that even the smallest difference of mass will cause the arms to swing to the other side.

For this reason, a rectangular probability density is used to describe uncertainty (As opposed to Gaussian). Similar to a digital reading of “0.78”, it is unknown how likely the value displayed is the actual value. For this reason, when using an instrument, read the displayed/specified/implied (e.g. if a scale has a precision of 0.01, the uncertainty is 0.005…) uncertainty on the instrument.

To find the confidence interval of a rectangular distribution perform the following as done in this example: $δ_{rec t an gu l a r} = \frac{a}{3}$

Gaussian Uncertainties

For “random error” or other error produced by human fault. To visualize this, take measurements and plot them on a curve. If they look similar to a Gaussian curve (normal distribution), then the dominant error is from “random error.”

$\overset{x}{ˉ} = \frac{1}{N} \sum_{i = 1}^{N} x_{i}$ Where $N$ is the number of times the measurement is taken, and $x_{i}$ is the value of the $i^{t h}$ measurement. These can be used to find the standard deviation (describing how narrow/wide the distribution is). $σ (x) = \frac{1}{N - 1} \sum_{i = 1}^{N} (x_{i} - \overset{x}{ˉ})^{2}$ Where $σ (x)$ is the standard deviation. $x_{i} - \overset{x}{ˉ}$ is used to find how far the data point at $i$ is. It is the square to make it positive (later undone by the square root). The sum of all the distances from the mean, $\overset{x}{ˉ}$ is then divided by the trials, $N$ (excluding itself $\to N - 1$ ).

To simplify calculations, a confidence interval (CI) is taken at $95% \approx 4 σ$ $δ_{g a u ss ian} = \frac{95% C I}{4}$

Summary

Error Bars

Data represented that have error bars are bars on graphs which indicate uncertainties. They can be horizontal or vertical, with the total length of two absolute uncertainties.

Correlation

NOTE: a strong linear correlation with a slope or r = 0 is still considered to have no correlation.

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