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Uncertainty in Measurement: Reading Instruments02:46

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Counting is the type of measurement that is free from uncertainty, provided the number of objects being counted does not change during the process. Such measurements result in exact numbers. By counting the eggs in a carton, for instance, one can determine exactly how many eggs are there in the carton. Similarly, the numbers of defined quantities are also exact. For example, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilograms. Quantities...
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The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this...
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Propagation of Uncertainty from Random Error00:59

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An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
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Uncertainty: Confidence Intervals00:54

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The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor...
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Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy of their results. Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or the accepted value. Precise values agree with each other; accurate values agree with a true value. 
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In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.
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Measurement of Quantum Interference in a Silicon Ring Resonator Photon Source
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Bayesian uncertainty evaluation applied to the tilted-wave interferometer.

Manuel Marschall, Ines Fortmeier, Manuel Stavridis

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    A Bayesian approach enhances uncertainty quantification for tilted-wave interferometry, a key technique for precise asphere and freeform surface measurement. This method provides accurate form estimates and uncertainties for complex optical surfaces.

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    Area of Science:

    • Optical metrology
    • Computational modeling
    • Statistical inference

    Background:

    • Tilted-wave interferometry (TWI) is crucial for high-accuracy form measurement of aspheres and freeform optics.
    • Accurate surface determination requires sophisticated computational models mirroring the physical measurement process.
    • Quantifying measurement uncertainty in TWI is challenging due to system complexity.

    Purpose of the Study:

    • To develop a robust uncertainty quantification method for TWI.
    • To address the inverse problem in TWI using a Bayesian statistical framework.
    • To provide pixel-by-pixel form estimates with associated uncertainties.

    Main Methods:

    • A Bayesian approach was formulated based on a statistical model of the TWI computational model.
    • An approximate inference scheme using Monte Carlo sampling was employed for posterior distribution analysis.
    • Experimental design was used to identify key influencing factors on measurement uncertainty.

    Main Results:

    • The proposed Bayesian method successfully quantifies uncertainty in TWI measurements.
    • Pixel-by-pixel form estimates and their uncertainties were obtained for two test surfaces.
    • Key influencing factors affecting measurement accuracy were identified and analyzed.

    Conclusions:

    • The Bayesian approach offers a statistically sound method for uncertainty quantification in TWI.
    • This technique significantly improves the reliability of form measurements for complex optical surfaces.
    • The methodology is validated and demonstrates efficacy for practical metrology applications.