# Questions with uncertainties for lab reports

## Homework Statement

I am unsure how to do uncertainties for my lab experiment. For example, we are suppose to estimate or guess the uncertainty for like the time and measurements.. Is there any general rule for proper guessing? Then I am unsure on what to do with the uncertainties in math operations. I get confused between the uncertainty and standard deviation, as in... which i should use?

## Homework Equations

For example, I have t=0.123 and the uncertainty of the instrument is probably 0.001. and i have a distance of .5m and i dont know the uncertainty(this part we guess on?).. probably 0.00005m. when what does the uncertainty become when you divide the distance over time? I think you are suppose to do sqrt( .001^2 + .00005^2) but that gives such a small number it doesnt make much sense. and if i had velocity at this time. would my best estimate value be the calculated velocity? or the average of the calculated velocities? Because I am suppose to find the velocity at this time with its uncertainty. the template shows the uncertainty with delta v, and i thought delta meant the standard deviation.. so would that mean the delta v for all the trials i have done be the same if it were the standard deviation?

Any help would be greatly appreciated.

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LowlyPion
Homework Helper
Welcome to PF.

As to your measurement uncertainties for distance there are a number of potential opportunities for error. Usually the smallest gradation of your measuring instrument is a good place to start. If you have a meter stick that only shows 1mm ticks then ± 1mm would be a good place to start. Also don't forget your reaction time, or any systematic error as a result of say your angle in viewing etc.

As to the equation you are using to propagate your uncertainty - the RSS of the errors for a product or division operation, you need to temper that with understanding that you should be using the relative uncertainty - i.e the percentage uncertainty - rather than the absolute uncertainty.

Hence if you measure something travel 30 cm ± 1mm then the relative uncertainty is .0033. If your time was say 1.4s ± .1s then that relative uncertainty is 1/14 = .071. Applying the RSS yields .071 because the one relative error was quite a bit smaller than the other. So if you expressed x/t as V then you would want to say v = 21.43 cm/s ± (.071)*21.43 or 21.4 ± 1.5 cm/s

So for adding and subtracting values with uncertainties, I should just add the uncertainties right?

And what should i do if I have 10 trials and I am suppose to show the uncertainty for all the trials? Would I take the calculated value and divide by the standard deviation for each so I would get different uncertainties for all of them?

And what do I do with trig functions like when i have to arc sin something with uncertainties?

Thanks again a bunch!

LowlyPion
Homework Helper

So for adding and subtracting values with uncertainties, I should just add the uncertainties right?

And what should i do if I have 10 trials and I am suppose to show the uncertainty for all the trials? Would I take the calculated value and divide by the standard deviation for each so I would get different uncertainties for all of them?

And what do I do with trig functions like when i have to arc sin something with uncertainties?

Thanks again a bunch!
For addition and subtraction operations, since they are like quantities then adding absolute uncertainties would be the way to go.

As for taking 10 measurements, presumably your error if its a result of say the finest scale element of your instrument would still be the same for the average as for each individually.

With functions like ln or Sin, cos etc, then express the range. I would take the largest value and the smallest value evaluated through the function. For instance if you have 37° ± 2° then for say sin37° I would evaluate sin 35° and sin 39°, take the difference, halve it and express it ± that half difference.

ah i mean like 2 * (3 +-1 )
and for multiplication and exponents, if the exponent is 2, just do it like multiplication?

and again thanks.

LowlyPion
Homework Helper
Yes, exponents are like multiplication, albeit multiplication by itself.

ah i mean like 2 * (7 +-3 )

and again thanks.
so what about that? would the uncertainty be +-6? or just +-3 ?

LowlyPion
Homework Helper
so what about that? would the uncertainty be +-6? or just +-3 ?
For your example 2*7±3 You have 2 with a relative uncertainty of 0, and 7 with a ± .429 relative uncertainty. Square them and take the root and it's still .429. Apply that to 14 and you get 14 ± 42.9% or 14 ± 6.

when i am presenting the final results, i am suppose to show the uncertainty and fractional uncertainty. so for example, my best estimate would be my average right and the uncertainty would NOT be the standard deviation, but would be the average of the uncertainties I had calculated for the 10 trials i have done? So then the fractional uncertainty would be the average uncertainty/best estimate?

cause i was suppose to calculate g and i get.. g= 9.834 ± 1.543 m/s2; 0.157
where the 1.543 is the average uncertainty of the uncertainties of the calculated g in each trial. and 0.157 is just 1.543/9.834

Now if this is not right.. does that mean the uncertainty should have been the standard deviation of the calculated g? or the calculated uncertainties of g?

LowlyPion
Homework Helper
but would be the average of the uncertainties I had calculated for the 10 trials i have done?
Your uncertainties don't arise from the readings per se. They arise from limitations in the measuring device or the observer or other conditions. You can average the results, but the uncertainties should be used to adjust the expression of the final average.

If you think about it, you do not assign a different uncertainty for each observation. Now you may want to average the differences from theoretical and compare them with your imputed uncertainties, as a test that your uncertainties were adequately accounting for systematic and observational errors.

well i mean the uncertainties i get are not from measurements, but after calculations, those become my uncertainties..
as same questions as before and should i be rounding my answer as i do each computation? or should i leave it and round at the end - because i am suppose to show the value after every single little computation so i suppose i should round it?