Propagation of Errors

In many cases our final results from an experiment will not be directly measured, but will be some function of one or more other measured quantities. For example, if we want to measure the density of a rectangular block, we might measure the length, height, width, and mass of the block, and then calculate density according to the equation

Each of the measured quantities has an error associated with it ---- and these errors will be carried through in some way to the error in our answer, .

Writing the equation above in a more general form, we have:

The change in for a small error in (e.g.) M is approximated by

where is the partial derivative of with respect to . In the worst-case scenario, all of the individual errors would act together to maximize the error in . In this case, the total error would be given by

If the individual errors are independent of each other (i.e., if the size of one error is not related in any way to the size of the others), some of the errors in will cancel each other, and the error in will be smaller than shown above. For independent errors, statistical analysis shows that a good estimate for the error in is given by

Differentiating the density formula, we obtain the following partial derivatives:

Substituting these into the formula for ,

Dividing by to obtain the fractional or relative error,

This gives us quite a simple relationship between the fractional error in the density and the fractional errors in . It may be useful to note that, in the equation above, a large error in one quantity will drown out the errors in the other quantities, and they may safely be ignored. For example, if the error in the height is 10% and the error in the other measurements is 1%, the error in the density is 10.15%, only 0.15% higher than the error in the height alone.

 Introduction Main Body •Experimental Error •Minimizing Systematic Error •Minimizing Random Error •Propagation of Error •Significant Figures Questions
The passage discusses the propagation of errors in experimental measurements, specifically how errors in measured quantities affect the error in a calculated result, such as density. It includes a detailed derivation of the formula for the propagation of errors and an example of how to apply it.
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the text does not contain any specific mathematical questions or problems for the reader to solve. it is primarily an explanation and derivation of the propagation of errors formula.
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Propagation of Errors

In many cases our final results from an experiment will not be directly measured, but will be some function of one or more other measured quantities. For example, if we want to measure the density of a rectangular block, we might measure the length, height, width, and mass of the block, and then calculate density according to the equation

Each of the measured quantities has an error associated with it ---- and these errors will be carried through in some way to the error in our answer, .

Writing the equation above in a more general form, we have:

The change in for a small error in (e.g.) M is approximated by

where is the partial derivative of with respect to . In the worst-case scenario, all of the individual errors would act together to maximize the error in . In this case, the total error would be given by

If the individual errors are independent of each other (i.e., if the size of one error is not related in any way to the size of the others), some of the errors in will cancel each other, and the error in will be smaller than shown above. For independent errors, statistical analysis shows that a good estimate for the error in is given by

Differentiating the density formula, we obtain the following partial derivatives:

Substituting these into the formula for ,

Dividing by to obtain the fractional or relative error,

This gives us quite a simple relationship between the fractional error in the density and the fractional errors in . It may be useful to note that, in the equation above, a large error in one quantity will drown out the errors in the other quantities, and they may safely be ignored. For example, if the error in the height is 10% and the error in the other measurements is 1%, the error in the density is 10.15%, only 0.15% higher than the error in the height alone.

 Introduction Main Body •Experimental Error •Minimizing Systematic Error •Minimizing Random Error •Propagation of Error •Significant Figures Questions
user
The passage discusses the propagation of errors in experimental measurements, specifically how errors in measured quantities affect the error in a calculated result, such as density. It includes a detailed derivation of the formula for the propagation of errors and an example of how to apply it.

TYPE: NO_QUESTION

The text does not contain any specific mathematical questions or problems for the reader to solve. It is primarily an explanation and derivation of the propagation of errors formula.