Calculating uncertainties, using calculus. What do these letters represent?

[jen78]

Calculating uncertainties, using calculus. What do these letters represent??

Homework Statement

I think I've forgotten some basic calculus. Please help, this is reeally easy and quick answer for someone who know this, but its taking me forever to figure out.

I attached document that I am going through, --Specifically, I am stuck on equation "U2" on page 3. --I just need to know what exact numbers they are plugging into get 1.43mm?? Or if you think this is explained poorly, any further explanation is REALLY appreciated

Thanks for any help

Homework Equations

The Attempt at a Solution

 

[LowlyPion]

Welcome to PF.

It may be a while before the pdf is approved.

Can you describe what it is more specifically?

 

[jen78]

yeah thanks for responding.
heres what I am doing so far, --this is WITHOUT using calculus, (im just having trouble learning how to use calculus to make this process go a lot faster (thats the whole benefit of using calc for uncertainty right?))
For example, let's say i have 5 measurements of a length of a string. They are
1. 50mm
2. 48mm
3. 51mm
4. 54mm
5. 52mm

So what I am doing so far is adding themn up, then dividing by 5 to get the mean.

= 51mm

Then i take the difference from this mean in each measurement
1. -1
2. -3
3. 0
4. 3
5. 1

Next i square each of these differences and get the mean
= 2.8
Finally i take sqare root of this to get
= 1.67

So, 1.67 is the uncertainty in my 5 measurements right?
Like i said i want to use calculus to do this. It would get totally tredious to do this way for every measuremtn.
THANKS ALOT!

 

[LowlyPion]

Personally I wouldn't be comfortable with this approach.

I would be more interested in determining the uncertainty from the process itself, such as determining inexactness because of viewing angle, or the minimum scale of my measuring device, etc.

Since your string is presumably the same piece of string measured 5 times, then I would want to understand the outliers like the ones that were 3mm off. What was the source of error that introduced a result like 3mm, as opposed to the 1mm you would expect from reading the device alone.

As to your coursework however, I guess I don't understand where they want to take you. Perhaps your attachment will have more?

This link might be useful to you:
http://spiff.rit.edu/classes/phys273/uncert/uncert.html

 

[jen78]

ok thganks, how long does it ussually take before the pdf is viewable? Maybe my example wasnt good, but manual is a lot better obviously

 

[LowlyPion]

ok thganks, how long does it ussually take before the pdf is viewable? Maybe my example wasnt good, but manual is a lot better obviously

Think Caribbean Islands. It gets here when it gets here.

I'll check back later.

 

[RoryP]

Ive never heard of calculating uncertaincies using calculus but it sounds interesting so when the pdf comes through i will be intreagued to find out how.
but until then have you considered using absolute and percentage uncertaincies?? that's what we had to use for our A-level physics practical exam =]

 

[LowlyPion]

--Specifically, I am stuck on equation "U2" on page 3. --I just need to know what exact numbers they are plugging into get 1.43mm?? Or if you think this is explained poorly, any further explanation is REALLY appreciated

If you want to reproduce them then you need to add up the squares of all the deviations (as you did in your simpler example), i.e. the Δ's from the mean that they have squared and divided by 9. (Not 10, but 9 as per the discussion.)

This isn't exactly calculus. It is more tedious statistics.

Note the alternative method of excluding the 1/3, which has the practical effect of ignoring the outliers and then your ± extremes determine the mean for your distribution curve. Personally, I'd choose that method, as the outliers are likely involving other imprecisions in your measurement technique that may be all the harder to quantify.

 

[maxsthekat]

--Specifically, I am stuck on equation "U2" on page 3. --I just need to know what exact numbers they are plugging into get 1.43mm?? Or if you think this is explained poorly, any further explanation is REALLY appreciated

Having just taken a physics course about error, I feel your pain ;)
That's the equation for standard deviation. You can find charts for this (and there's a whole branch of mathematics, statistics, that uses this equation extensively), but one standard deviation, in both directions (plus and minus from the average) is approximately 68% probability.

In other words, let's say you take 40 measurements, and you get an average of 12 seconds (numbers/units all pulled out of my butt), with a standard deviation of 2 seconds. That means on your next measurement (#41), you have a 68% probability of that measurement falling between 10 seconds (average - 1 standard dev) and 14 seconds (average + standard dev).

So, how do you find the standard deviation?

1. Find the average, that's the x with the bar across it in the equation
2. For each x value, subtract the average from it, and square the answer
3. Add all of those results together
4. Divide by N-1 (where N is your number of trials/measurements)
5. Take the square root of that whole-shebang.

Viola! Standard deviation!

Now, the standard deviation isn't an uncertainty-- it just tells you the likelihood of a measurment falling within a certain range. For uncertainty of your data, take your standard deviation, and divide it by the square root of N (your number of trials).

Hope that helps :)

 

[maxsthekat]

Oh! I should add, the "general" calculus formula for finding uncertainty (when you're using equations, instead of sets of data, like your problem) is

assume f is a function, eg: F(a,b,c) = a^4 + b^3 + abc^5
uncertainty in f = Square root [ (dF/da * uncertainty in a)^2 + (dF/fb * uncertainty in b)^2 + (dF/dc * uncertainty in c)^2]

In other words, you take the partial derivative with respect to each variable, multiply that by the uncertainty, square it, repeat for each term, add all together, and take the square root...

So, for our example:
F(a,b,c) = a^4 + b^3 + abc^5
dF/da = 4a^3 + bc^5
dF/db = 3b^2 + ac^5
dF/dc = 5ab^4

uncertainty = Sqrt [ ((a^4+b^3+abc^5)*uncertainty_a)^2 + ((3b^2+ac^5)*uncertainty_b)^2 + ((5ab^4)*uncertainty_c)^2]

This equation assumes you have the uncertainties for a, b, and c. It also assumes you have values for a, b, and c.