I dont understand how to calculate percentage uncertainty?

[mutineer123]

I don't understand how to calculate percentage uncertainty!?

I know in chem we calculate uncertainties by smallest division/the reading X 100. But in physics i am clueless. For instance here's an example

The power loss P in a resistor is calculated using the formula P = V^2/R.
The uncertainty in the potential difference V is 3% and the uncertainty in the resistance R is 2%.
What is the uncertainty in P?

(this is not homework) I just want to know how to get sums like these.

 

[sambristol]

mutineer123,
if your quantity is a function of other variables then you first calculate the absolute error and then divide by the measured value to get the % error. At all points in the calculation you must use the absolute error not the %. The method is to find the error relating to each variable (R and V in your example) these are squared and the squares added, finialy take the square root. This will give you the absolute error in the power.

To find the contribution from each of the variables you differentiate the function with respect to the variable and multiply by the absolute error of that variable. This is the quantity you square as mentioned above.

See section 4 of http://www.phas.ubc.ca/~phys209/error_analysis/errors.pdf

Regards
sam

 

[Emilyjoint]

On my course we are taught that when quantities are multiplied or divided the % errors are added together.
It was shown like this...imagine V=100 +or-3%. This means that V could be as large as 103 or4 as small as 97. V^2 could be as large as (103x103) = 10609 which is pretty much 6% larger than (100x100)
Also V^2 could be as small as (97x97)= 9409 which is also pretty much 6% smaller than (100x100).
So error in V^2 is +or-6%.
You can do this exercise to include the 2% error in R and you will find that the overall % error is pretty much 8% (3 + 3 + 2)
Our teacher did several examples like this and it was convincing.

 

[rcgldr]

On my course we are taught that when quantities are multiplied or divided the % errors are added together.

That's a first order approximation. You could also calculate values using the minimum, non-error, and maximum values, then express the uncertainty as percentage of minimum or maximum versus non-error value. This is what is normally done for "worst case" situations in a design process. In this case

P_min = (.97 V)² / (1.02 R) ~= 0.922451 ~= 1 - 0.077549

P_max = (1.03 V)² / (.98 R) ~= 1.082551

P_nonerror = (1.00 V)² / (1.00 R) = 1.000000

P_avg_with_error ~= 1.002501 ± 8.005 %

This is excessive precision since the orginal values only show percentages (assume 3 digits of precision), but generally high precision calculations are done for worst case until a final answer is produced. On a side note, if you're looking for distribution for the uncertainty, it's complicated by the fact that a 2% resistor could have been selected from a large batch of resistors where all the resistors with 1% or less error were removed and labeled as 1% resistors, so you have a bell curve with an empty center. If the 2% resistors were removed with the rest of the resistors to be labeled as greater than 2% resistors, then you'd have a bell curve only with the region betwen +/- 1% to 2%, empty center, empty edges.

 

[Emilyjoint]

Our method of simply adding %s when you have quantities multiplied or divided gave the same value...8%

 

[rcgldr]

Our method of simply adding %s when you have quantities multiplied or divided gave the same value...8%

Yes, but that was using an average value that was .25% higher (1.0025). Using 1.0 for the expected power, then it's - 7.75% or +8.25%. With just two components, one of them squared, the first order is close enough, but with a larger number of components, the first order term will be farther off from a full expression.

 

[mfb]

In general, the error depends on the correlation of the individual errors from V and R (and the distribution of the errors, but that is often close to a Gaussian distribution).
Without correlation, you shouldn't add the percentages, but the squares of them.

The 6% error for V^2 are fine, but the total error is then given by $$\sqrt{6^2+2^2}\%=\sqrt{40}\%=6.3\%$$

The difference to the "worst case" scenario of rcgldr comes from the fact that a larger voltage can come together with a larger resistance at the same time, too: (1.03U)^2/(1.02R) =~ 1.04 U^2/R

 

[rcgldr]

The difference to the "worst case" scenario of rcgldr comes from the fact that a larger voltage can come together with a larger resistance at the same time, too: (1.03U)^2/(1.02R) =~ 1.04 U^2/R

Worst case scenarios are those which produce max or min power. In this case max voltage, min resistance, or vice versa.

Gaussian distribution

As mentioned in my previous post, depends if the parts are selected from pool of parts where "better" or "worse" parts are removed from the pool to end up with the "specified" parts, in this case the 2% resistors. Why the voltage varies would depend on what is causing that variation.

 

[mutineer123]

Okay Thanks guys, Il apply this knowledge to the questions Il solve, and if I have any difficulty Il get back to the thread. Again Thanks alot

 

[HallsofIvy]

On my course we are taught that when quantities are multiplied or divided the % errors are added together.
It was shown like this...imagine V=100 +or-3%. This means that V could be as large as 103 or4 as small as 97. V^2 could be as large as (103x103) = 10609 which is pretty much 6% larger than (100x100)
Also V^2 could be as small as (97x97)= 9409 which is also pretty much 6% smaller than (100x100).
So error in V^2 is +or-6%.
You can do this exercise to include the 2% error in R and you will find that the overall % error is pretty much 8% (3 + 3 + 2)
Our teacher did several examples like this and it was convincing.

That's an engineering rule of thumb- when quantities are added (or subtracted), their errors add; when quantities are multiplied (or divided) their relative errors add.

Suppose X has error dX, Y has error dY. If Z= X+ Y then dZ= dX+ dY while if Z= XY, dZ= XdY+ YdX so, dividing by Z= XY,
$$\frac{dZ}{Z}= \frac{XdY}{XY}+ \frac{YdX}{XY}= \frac{dY}{Y}+ \frac{dZ}{Z}$$

 

[Emilyjoint]

I know that when quantities are added the absolute errors must be added.
The total error is then converted to a % (relative?) error if the quantity is part of a multiplication such as working out an area or volume from lengths.
Then adding the %s seems to give the overall error for the number of significant figures involved

 

[mfb]

Worst case scenarios are those which produce max or min power. In this case max voltage, min resistance, or vice versa.

They are worst case scenarios only if your given uncertainty means "100% of the values are within this range". Which might be a good approximation for resistors compared to their color code - but in that case, you can simple measure this resistance, and get a better value.

 

[Emilyjoint]

I am getting confused by so many techniques for dealing with errors, the different techniques do not seem to make much difference.
If adding % errors give 8% total error and doing the square root calculatio give 6.4% total error how will this show in the significant figures of an actual answer?...if at all

 

[mfb]

With just two errors, the difference is not so significant. But imagine 4 different uncertainties of 5% each. Adding the uncertainties linearly gives 20% uncertainty, adding them in quadrature gives 10% total uncertainty.
100+-10 and 100+-5 (as an example) are really different measurements in a scientific context.