Antilog Error Propagation
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Gable's calendar Explanation In many instances, the quantity of interest is calculated from a error propagation natural log combination of direct measurements. Two questions face us: Given the
Error Propagation Ln
experimental uncertainty in the directly measured quantities, what is the uncertainty in the final result?
Logarithmic Error Calculation
In designing our experiment, where is effort best spent in improving the precision of the measurements? The approach is called propagation of error. The theoretical
Compound Error Definition
background may be found in Garland, Nibler & Shoemaker, ???, or the Wikipedia page (particularly the "simplification"). We will present the simplest cases you are likely to see; these must be adapted (obviously) to the specific form of the equations from which you derive your reported values from direct log uncertainty measurements. Addition and subtraction Note--$$S=√{S^2}$$ Formula for the result: $$x=a+b-c$$ x is the target value to report, a, b and c are measured values, each with some variance S2a, S2b, S2c. $$S_x=√{S^2_a+S^2_b+S^2_c}$$ (Sx can now be translated to a confidence interval by means previously discussed. Multiplication/division Formula for the result: $$x={ab}/c$$ As above, x is the target value to report, a, b and c are measured values, each with some variance S2a, S2b, S2c. $$S_x=x√{{(S_a/a)}^2+{(S_b/b)}^2+{(S_c/c)}^2}$$ Exponentials (no uncertainty in b) Formula for the result: $$x=a^b$$ $$S_x=xb(S_a/a)$$ Special cases: Antilog, base 10: $$x=10^a$$ $$S_x=2.303xS_a$$ Antilog, base e: $$x=e^a$$ $$S_x=xS_a$$ Logarithms Base 10: $$x=log{a}$$ $$S_x=0.434(S_a/a)$$ Base e: $$x=ln{a}$$ $$S_x={S_a/a}$$ Navigation CH361 Home Equations for Statistics Q-Test Table t-test Tables Linear Regression Propagation of Error Contact Info Do you notice something missing, broken, or out of whack? Maybe you just need a little extra help using the Brand. Either wa
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the measuring instrument (e.g. to the nearest mm). Errors in measurements have a carry-through effect in calculations, and this can be analysed to determine the overall error in the final solution. SymbolsWe will use the http://www.learneasy.info/MDME/MEMmods/MEM30012A/error_analysis.html following symbols:S = absolute error of a measurement. x = the measurement itself (the measurand)S/x = relative errorThe error (S) is never known exactly. If we knew exactly what the error was we could subtract it and get a perfect measurement. Errors are statistical, the measurement is most probably within a certain range. The symbol S is used because it stands for Standard Deviation. (See Statistics) This error can also be called error propagation the uncertainty of a measurement.It is important to maintain the same method of describing the error throughout the calculations. We usually use +- tolerancing to describe the error.Sources ofError Errors can come from various sources. Resolution error is easy to estimate, but the others are usually quite approximate and may have to be estimated by the person takingthe measurement. Random ErrorsLimitation of precision: Resolution of the instrument.Error = Smallest Resolution/2 Misalignment. antilog error propagation Parallax error of needle/scale and eye, misaligned instrument (Eg.The dial gauge is not vertical, the tape measure is at an angle, the caliper is not perpendicular etc).Round off or inaccuracy in formula or constant (e.g. Gravity = 9.81)Systematic Errors: (Inherent in the measurement).Errors in the calibration of the measuring instruments.Examples: Stretch of a tape, inaccurate graduations, worn or incorrectly adjusted instruments. The only way to check this is by calibration against a known standard or correct method.Incorrect measuring technique:Examples: Incorrect method: pushing too hard on a caliper, parallax error due to viewing at an angle.Bias of the experimenter. Examples: The experimenter might consistently read an instrument incorrectly, or might let knowledge of the expected value of a result influence the measurements.Systematic Errors do not improve by taking many readings, because the average is not zero. Regular calibration is all about minimising systematic error.These errors should be added together to give the absolute measurement error;Absolute error = (Resolution / 2) +(misalignment error) + (systematic error) + (inherent error)Absolute and RelativeErrorAbsolute Error is the tolerance of the measurement, or theapproximate error of a single measurement. The best estimate is the standard deviation of the measurement, which can only be determined with many measurements taken. Failing this,an estimation can be made using the error sources above
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