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Uncertainty in Measurement: Reading Instruments

Uncertainty in Measurement: Reading Instruments

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...

Uncertainty in Measurement: Accuracy and Precision

Uncertainty in Measurement: Accuracy and Precision

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.

Propagation of Uncertainty from Systematic Error

Propagation of Uncertainty from Systematic Error

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 particular...

Propagation of Uncertainty from Random Error

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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...

Uncertainty: Confidence Intervals

Uncertainty: Confidence Intervals

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 't,' or...

Uncertainty: Overview

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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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Related Experiment Video

Updated: May 19, 2026

Split Point Analysis and Uncertainty Quantification of Thermal-Optical Organic/Elemental Carbon Measurements

Split Point Analysis and Uncertainty Quantification of Thermal-Optical Organic/Elemental Carbon Measurements

Published on: September 7, 2019

Improving optical measurement uncertainty with combined multitool metrology using a Bayesian approach.

Nien Fan Zhang¹, Richard M Silver, Hui Zhou

¹Statistical Engineering Division, National Institute of Standards and Technology, 100 Bureau Drive MS 8980, Gaithersburg, Maryland 20899, USA. zhang@nist.gov

|September 5, 2012

Summary

This study introduces a hybrid metrology approach combining optical measurements with techniques like atomic force microscopy. This Bayesian method reduces uncertainty in critical dimension measurements for nanometer-scale features.

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Area of Science:

Materials Science
Metrology
Nanotechnology

Background:

Advanced optical technologies are crucial for dimensional metrology of features 22 nm and smaller.
Current optical measurement modeling involves curve libraries and nonlinear regression, but faces challenges like parametric correlation and noise.
Measurement uncertainty is a significant issue in optical critical dimension (OCD) measurements.

Purpose of the Study:

To propose a Bayesian statistical approach for hybrid metrology.
To combine data from different physical measurement techniques.
To reduce uncertainties in parameter estimation for nanometer-scale metrology.

Main Methods:

Assembling a library of curves through multidimensional parameter space simulation.
Employing a nonlinear regression routine for experiment-to-theory agreement.
Utilizing a Bayesian approach to integrate data from multiple metrology techniques (e.g., OCD, AFM, SEM).

Main Results:

Demonstrated reduction in parameter estimator uncertainties through hybrid metrology.
Showcased the effectiveness of combining diverse measurement techniques.
Improved accuracy and reliability in dimensional metrology for sub-22 nm features.

Conclusions:

The proposed Bayesian hybrid metrology approach effectively reduces measurement uncertainty.
Integrating multiple physical measurement techniques offers a robust solution for advanced dimensional metrology.
This method enhances the precision required for characterizing nanoscale features in semiconductor manufacturing.