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Uncertainty Analysis

Uncertainty Analysis. P M V Subbarao Professor Mechanical Engineering Department. A Measure of Confidence Level…. The Uncertainty.

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Uncertainty Analysis

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  1. Uncertainty Analysis P M V Subbarao Professor Mechanical Engineering Department A Measure of Confidence Level…..

  2. The Uncertainty • The uncertainty of the measurement result y arises from the uncertainties u (xi) (or ui for brevity) of the input estimates xi that enter equation. • Types of uncertainty may be categorized according to the method used to evaluate them.

  3. Components of Uncertainty • “Component of uncertainty arising from a random effect” : Type A • These are evaluated by statistical methods. • “Component of uncertainty arising from a systematic effect,”: Type B • These are evaluated by other means.

  4. Representation of uncertainty components • Standard UncertaintyEach component of uncertainty, is represented by an estimated standard deviation, termed standard uncertaintyui, and equal to the positive square root of the estimated variance • Standard uncertainty: Type AAn uncertainty component obtained by a Type A evaluation is represented by a statistically estimated standard deviation si, • Equal to the positive square root of the statistically estimated variance si2. • For such a component the standard uncertainty is ui = si.

  5. Standard uncertainty: Type B • This uncertainty is represented by a quantity uj , • May be considered as an approximation to the corresponding standard deviation; which is it is equal to the positive square root of uj2. • uj2 may be considered an approximation to the corresponding variance si2 and which is obtained from an assumed probability distribution based on all the available information. • Since the quantity uj2 is treated like a variance and uj like a standard deviation, • for such a component the standard uncertainty is simply uj.

  6. Evaluating uncertainty components: Type B • A Type B evaluation of standard uncertainty is usually based on scientific judgment using all of the relevant information available, which may include: • previous measurement data, • experience with, or general knowledge of, the behavior and property of relevant materials and instruments, • manufacturer's specifications, • data provided in calibration and other reports, and • uncertainties assigned to reference data taken from handbooks. • Broadly speaking, the uncertainty is either obtained from an outside source, or obtained from an assumed distribution.

  7. Uncertainty obtained from an outside source • Procedure:  Convert an uncertainty quoted in a handbook, manufacturer's specification, calibration certificate, etc., • Multiple of a standard deviation • A stated multiple of an estimated standard deviation to a standard uncertainty. • Confidence interval • This defines a "confidence interval" having a stated level of confidence, such as 95 % or 99 %, to a standard uncertainty.

  8. Uncertainty obtained from an assumed distribution • Normal distribution: "1 out of 2" • Procedure:  Model the input quantity in question by a normal probability distribution. • Estimate lower and upper limits a- and a+ such that the best estimated value of the input quantity is (a+ + a-)/2 • There is 1 chance out of 2 (i.e., a 50 % probability) that the value of the quantity lies in the interval a- to a+. • Then ujis approximately 1.48 a, where a = (a+ - a-)/2 is the half-width of the interval.

  9. Uncertainty obtained from an assumed distribution • Normal distribution: "2 out of 3" • Procedure:  Model the input quantity in question by a normal probability distribution. • Estimate lower and upper limits a- and a+ such that the best estimated value of the input quantity is (a+ + a-)/2 • and there are 2 chances out of 3 (i.e., a 67 % probability) that the value of the quantity lies in the interval a- to a+. • Then ujis approximately a, where a = (a+ - a-)/2 is the half-width of the interval

  10. Normal distribution: "99.73 %" • Procedure:  If the quantity in question is modeled by a normal probability distribution, there are no finite limits that will contain 100 % of its possible values. • Plus and minus 3 standard deviations about the mean of a normal distribution corresponds to 99.73 % limits. • Thus, if the limits a- and a+ of a normally distributed quantity with mean (a+ + a-)/2 are considered to contain "almost all" of the possible values of the quantity, • that is, approximately 99.73 % of them, then ujis approximately a/3, • where a = (a+ - a-)/2 is the half-width of the interval.

  11. For a normal distribution, ± u encompases about 68 % of the distribution; for a uniform distribution, ± u encompasses about 58 % of the distribution; and for a triangular distribution, ± u encompasses about 65 % of the distribution.

  12. Propagation of Uncertainty in Independent Measurements

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