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## Presenting Precision of Zero Result

The question is how to present the result of an experimental measurement. Standard practice includes various relative error measures in which the standard deviation or the maximum deviation is divided by the mean of the result, or by the theoretical value or other standard.

The problem arose in a simple Conservation of Momentum experiment using gliders on an air track. When the final velocity of the projectile and the target are non zero, relative percent error is useful. When the two gliders have the same mass with the target initially at rest, the final velocity of the projectile is zero. = 0 m/sec +/- 0.01 m/sec defines the result adequately. There are obvious problems with dividing by either the experimental or theoretical result to achieve a relative error presentation.

Does anyone have any suggestions as to a better way to express the quality of such a measurement?

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Physics: Common Sense made obscure by Mathematics
3 REPLIES 3

## Presenting Precision of Zero Result

I don't have the suggestion that you ask for, but I would like to make one anyway.

Just present the actual variation in measured vs. theoretical final velocities. There are two reasons that I would suggest that a proportional error is not called for here.

One reason is that velocity is a quantity that has an arbitrary zero point. It is never a good idea to use proportional error with this type of quantity. For example, one does not use percent error with the Celsius temperature scale. [Proportional error is meaningful with the Kelvin scale, which has an absolute zero point.] One also does not use proportional error with angles, which have an arbitary line for a zero point.

The other reason I would suggest not over-using proportional error is that it really makes no sense to use it when the size of the error has nothing to do with the size of the measurement. This applies in two ways: (1) Is there any likelihood of greater error when the measurement is greater? (2) Do you, because of the size of the quantity being measured, have a need for increased precision?

Michael Thackston

## Presenting Precision of Zero Result

Thanks Michael,

That is the best advice I have received so far. Usually the precision of the result is independent of the magnitude of what is being measured. The meter stick is good to =+/- 1 mm no matter if you are measuring 10 cm or 50.

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Physics: Common Sense made obscure by Mathematics

## Presenting Precision of Zero Result

The error you are trying to quantify is, as you say , about zero - so a division by zero is not helpful, but my way of thinking is this:
If the glider comes in at 1m/sec, and you expect it to stop, a residual speed of .01m/sec would be a 1% error. If it still had 0.2m/sec that would be 20%.
I suspect you can do something even more useful by including the residual speed of the target in the calculation - that will give you not only the errors, but also the system losses.

Don
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