If I take a series of measurements for the length and width of arecatangular boa

Question

If I take a series of measurements for the length and width of arecatangular board and find the mean and the Standard deviation forboth the length and the width. How do I answer thisquestion?

Suppose you were to calculate the area of the rectangle youmeasured in the experiment. How might you estimate the errorof the area measurement? (Remember that you have alreadycalculated the standard deviation for each side.)

The mean for the length is 0.28m
The Standard Deviation for the length is .02582

The mean for the width is .206m
The Standard Deviation for the width is .009661

Explanation / Answer

Ahh, yes, you will be encountering this many a time in yourphysics lab. The mean is simple enough. You just use the formula L*W =A,and viola, you have your mean value!
The Standard Deviation is much more complicated, and involvespartial derivatives. If you dont' know them yet, a Partialderiviative is when you take the derivative with respect to avariable. You are used to dy/dx. This means you take the derivativeof an equation with repsect to X. Here, the formula you want isthis:
A = (L*W)/ L *L + (L*W)/ W *W
Confused? I'll break it down. Please note, the fanciful dright there just means partial derivative. It's like the normal din dy/dx, but it indicates a partial derivative over an ordinaryone.
(L*W)/ L means to treat L as the variable, andW as a constant. So, the derivative of L*W is W.

L is the standard deviation that you are given.
Repeat this process, but with W now.
So, the general formula is this: Take the derivative of the formulayou are using with respect to one of the variables, then multiplyby the standard deviation of the variable you are using.
Once you have done this for each variable, add them all up (Pleaenote, sign does NOT matter with "uncertainty", as this is called.Take the absolute value of everything), and viola, you haveyour uncertainty. The units should be the same as theaverage.

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