ENES100-0702
Prof. R. J. Phaneuf
Fall 2002
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Uncertainty of Measured Quantities
Throughout your project you’ll be dealing with measured quantities. The certainty of any measurement is limited by both accuracy and precision.
Accuracy is a measure of the overall fidelity of an operation or measurement. It is affected by the quality of the instruments used. For example, if a meter stick which is improperly calibrated is used in determining the length of parts of an apparatus, those lengths will be consistently inaccurate due to this systematic error. The best way to judge accuracy is to perform measurements in more than one way, using independently calibrated instruments.
Precision is a measure of the reproducibility of an operation or measurement in the presence of random errors. For example, if the meter stick mentioned above is accurately calibrated, but laid out in mm, then attempts to estimate the length of an object with it might be limited to a precision of 0.2 mm. The best way to judge precision is to make multiple independent measurements, and see how much spread there is in the measured values.
Combining Uncertainties
In performing calculations using measured quantities the uncertainties associated with limited accuracy and precision should be combined to determine the uncertainty in the result. In adding or subtracting two quantities, the uncertainties add:
In multiplying or dividing two quantities the fractional uncertainties add:
,
where and are the measured uncertainties in A, B, respectively, and is the resulting uncertainty in the product.
In more complex operations you can estimate the resulting uncertainty by varying the individual quantities over the range given by the uncertainty, and seeing how much the result varies.
Significant Figures
A shorter approach to dealing with limited reliability in a calculation is to keep track of significant figures, only reporting the result to within the uncertainty associated with the individual quantities.
Each or the quantities should be written in scientific notation, with the number of digits after the decimal point indicating the reliability of the quantity. In a sum or difference, each of the terms should be represented by the same power of 10. The result will have the number of significant figures associated with the minimum number of digits after the decimal point in any term plus 1.
Example:
The result is only quoted to 3 significant figures.
In a product, simple scientific notation should be used, and the result should be quoted only to the least number of significant figures of any of the factors.
Example:
Again, the result is quoted to only 3 significant figures.
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