# I wanted to know how it is determined uncertainty

## Main Question or Discussion Point

Hi everyone! I wanted to know how it is determined uncertainty ? i watched some lesson but i didn't understand if it is a variable chosen from who is making the measure of that depends on the tool that i use for the measure ?

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mfb
Mentor

What?
It depends on your setup how the uncertainty is determined.

As an example: If you measure the length of some object with a regular ruler, you will not get an exact value. You can get the millimeters right, but certainly not a precision of 0.01 mm. So you have an uncertainty somewhere in between - a common choice is 1/2 of the ticks of the ruler (=> 0.5mm).

Let me see if I've understood, for example if i have an object that measures the mass like a balance and it has a sensibility (precision) of 100 grams and i measure myself and it shows me a value like 75 kg i can say that my mass is 75 kg ±(100/2)grams... is that right?

mfb
Mentor

±100g or ±50g or something like this would depend on the specific setup (and the way to express the uncertainty), but that is the general idea.

sorry i didn't understand, you before said that i have to take the precision of the tool/2 but now in the last answer you wrote that i can take even the value of the precision of my tool to be the uncertainty of my measure.
Can you please be more precise ?

mfb
Mentor

I specified which tool I meant - this can be important.
If you have a digital scale, for example, it might be more like ± the least significant digit, while on an analog scale, you might be able to see where (between the scale ticks) the value is.

Let me see if i've understood the topic:

Let's divide the measuments in two cases (analog and digital)

1) ANALOG: If i have an analog tool, for example an analog balance and it has a precision of 100 grams, and i find that the mass of an object is 9,5kg is a good choice to add to my measure ±(100/2) grams so my mass would be: 9.5kg ± 50 grams

2)DIGITAL: If i have a digital tool, for example a digital voltmeter and it has a precision of two digits after the dot, and i have a measure for example of 5,54V than i should add ± 0.01 so my measure would be 5.54V ± 0.01V.

IS THAT RIGHT ? Can you evidence my mistakes if there are any?

Thank you for the help and sorry for my bad english

mfb
Mentor

Should be fine. If those uncertainties dominate the final uncertainty and the result is something interesting (especially: not a lab course), it could be interesting to investigate them further.

Sorry what do you mean ? Can you explain me better ?

Thank you for the help! You are very kind!

mfb
Mentor

Hmm, maybe it gets clearer with an example:

Imagine you want to measure the gravitational acceleration with a pendulum. You measure its length (~1m) with a ruler (1mm ticks, uncertainty ±0.5mm) and one oscillation period (~2s) manually with a stopwatch (can display time in 0.01s-intervals, uncertainty ±0.01s). You repeat the experiment several times and note that your measured times vary by ±0.05s, simply by the fact that you do not start/stop the stopwatch at the same time in every repetition.

Using those values and the measurement of a single oscillation, you can calculate $g=\frac{4\pi^2l}{T^2}$ - I do not care about the values, just the uncertainties here: The relative uncertainty of l is small (1/2000), while the relative uncertainty of the time is quite large (0.05s/2s = 1/40). The final uncertainty of g will be dominated by the imperfect operation of the stopwatch. The result would stay the same if you used ±1mm for the length uncertainty.

However, that is a bad experiment design. It would be better to measure the time of 100 periods. Now the relative uncertainty on the time is 1/4000, and the relative uncertainty for the length is 1/2000 (with ±0.5mm). Both are important, and you should think about both. Did you really measure the length with that precision? If you had to read the scale at both the pendulum mass and the fixed point of the pendulum, you should assign an uncertainty to both readings.
And so on.

Ahh ok ok i understand, you are saying that to be more precise for certain measures i can reduce the uncertainty by doing it a number of time and then use the new uncertainty.
Thank you very much!! You are great! Now i understand!

Another doubt about physics calculation to be more precise:

If i have a long formula like for example x*y+ 5y +cos(z) +x*y*z

and x has 3 significant figures, y has 2 significant figures and z has 5 significant figures, i heard that i can keep in mind rounding only at the final result and this is ok but what about significant figures rule ? how i apply these rules to my calculations? Should i use those rules every time i do a simple calculation? And if so what about rounding after apllied significant figures rule ?

Thank you, i would be really grateful for this because on the web i couldn't find anything...

mfb
Mentor

Ahh ok ok i understand, you are saying that to be more precise for certain measures i can reduce the uncertainty by doing it a number of time and then use the new uncertainty.
I did not say that, but it is true.

Round the final value and nothing in between - and if you want to do it properly, use gaussian error propagation. Without knowing the values for x,y,z, it is impossible to tell which part will be relevant for the total uncertainty.
In addition, you cannot simply give the result with 2 significant figures.

Be sure you don't understate the uncertainty of your measures by assuming you can set one end of a ruler perfectly at one end of a length to be measured, or start a time period measure perfectly at the beginning of the period.

Length and time periods must each have two measures and two estimates of uncertainty.

When you measure a length with a ruler, you don't set one end of the ruler at one end of the length and read the opposite end of the length against the ruler scale.
You lay the ruler randomly along the length to be measured so the ruler overhangs the length on both ends... this ensures that you don't assume a perfect measurement at either end, and entails that you will make two readings, one for each end, and both of those readings will be a true measurement with a proper estimate of uncertainty.

Likewise with a measurement of time, there must be two uncertainties, the beginning of the period and the end. You can't just assume the start time was perfect and end up with only one uncertainty at the end representing the uncertainty for the period.

I did not say that, but it is true.

Round the final value and nothing in between - and if you want to do it properly, use gaussian error propagation. Without knowing the values for x,y,z, it is impossible to tell which part will be relevant for the total uncertainty.
In addition, you cannot simply give the result with 2 significant figures.
Ok i know about rounding up! But what about significant figures ? i should keep in mind of them in every calculation ??

Can you make me an example with the formula i gave you or with a formula you choose and use random numbers just to make me understand how the calculations work with significant figures and if i have to count them after every calculation or only in certain situation?

anyway thank you for the significant help!!

mfb
Mentor

Ok i know about rounding up! But what about significant figures ? i should keep in mind of them in every calculation ??
Significant figues (relative uncertainties) can be a handy tool to evaluate the uncertainty of products, but they are usually useless for sums. Gaussian error propagation is more the general tool, it works for both (and many more operations).

Can you make me an example with the formula i gave you or with a formula you choose and use random numbers just to make me understand how the calculations work with significant figures and if i have to count them after every calculation or only in certain situation?
I am sure you can find many websites with examples. It is a standard questions.

So everytime i'm making physics exercises i should use gaussian propagation ?

mfb
Mentor

Unless you have a good reason to use something else: Yes.

Sorry in my physics exercises i never heard about gaussian propagation, (talking about mechanics or electrical physics exercises) only about sig.figures...
Anyway on the web can'0t find anything.

mfb
Mentor

That should give a good idea, with the assumption that the uncertainties are not correlated. Usually that is a good approximation - there is no reason that the length measurement and the time measurement influence each other in the pendulum, for example.

is uncertainty applicable with the speed of light? like 3x10^8m/s≥c

mfb
Mentor

The speed of light in vacuum is exact (by definition) in SI units (and most other scientific systems I think), it does not have any uncertainty.