Comparing Theory and Measurement

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Presentation transcript:

Comparing Theory and Measurement Agreement between theory and experiment does NOT imply correctness. Counter-examples include: bad theory agreeing with bad data bad theory agreeing with good data by coincidence good theory agreeing with bad data because a variable was not considered or controlled in the experiment Scientific information can be misused selectively. Comparisons must be made within the context of uncertainty.

How Sure Are We ? When a physical process is quantified, uncertainties associated with describing the process occur. Uncertainties result from Experiments Modeling

Systematic and Random Uncertainties An error is the difference between the measured and the true value. An uncertainty is an estimate of the error. Uncertainties are categorized as either systematic (bias) or random (precision). An uncertainty is assumed to be systematic if no statistical information is provided.

Systematic and Random Uncertainties Systematic, Bi: arises from comparisons with standards (calibration) involves no statistics; the number is given alone related to the accuracy (‘to within ±Bi units’) Random, Pi: based upon repeated measurements involves statistics ( ) related to the precision (scatter) for one more measurement for multiple measurements

Systematic and Random Uncertainties Figure 9.2

Precision and Accuracy Precision Accuracy good poor good good poor poor

Measurement Uncertainty Analysis Overall goal: Obtain an estimate of Ux, where x' = x ± Ux (%C) x can be either a single value or an average value. The magnitude of Ux depends upon the percent confidence (%C), the contributing uncertainties, and how the contributing uncertainties are combined.

Measurement Uncertainty Analysis The overall uncertainty, Ux, is related to the combined standard uncertainty, uc, through a coverage factor, where For most experiments, N ≥ 10  tn,95 ≈ 2 (<10 % error). This implies Ux,95 ≈ 2uc = . This is called the large scale approximation.

Student’s t Table Gives the value of t for a given n and P % confidence . What is t for N = 12 ? Table 8.4

Quadrature Combination of Uncertainties The combined standard uncertainty, uc, comes from the combined estimated variance, uc2, which is expressed as: assuming each xi is independent of the other J-1 variables. r denotes a result (a variable that is a function of one or more measurands) qi is the absolute sensitivity coefficient, which weights the uncertainty contribution of xi to the result. qi = 1 when the result is simply a measurand.