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Uncertainty: Overview

Uncertainty: Overview

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.

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...

Propagation of Uncertainty from Random Error

Propagation of Uncertainty from Random Error

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...

The Uncertainty Principle

The Uncertainty Principle

Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He...

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...

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.

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Updated: Dec 25, 2025

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

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Published on: September 7, 2019

Some aspects on the uncertainty calculation in Mueller ellipsometry.

Matthias Wurm, Tobias Grunewald, Sven Teichert

|April 1, 2020

Summary

Accurate uncertainty estimation in spectroscopic Mueller ellipsometry is crucial. This study introduces a novel merit function that correctly accounts for depolarization, improving measurement uncertainty calculations for thin films like SiO2 on Si.

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Scattering And Absorption of Light in Planetary Regoliths

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Last Updated: Dec 25, 2025

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Scattering And Absorption of Light in Planetary Regoliths

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Published on: July 1, 2019

Area of Science:

Materials Science
Optical Metrology
Spectroscopy

Background:

Spectroscopic Mueller ellipsometry (SME) is a powerful technique for characterizing thin films.
Accurate uncertainty estimation is vital for reliable metrological results in SME.
Current methods often fail to correctly estimate uncertainties due to improper handling of depolarization.

Purpose of the Study:

To investigate the metrological aspects of SME, specifically uncertainty estimation.
To identify the reasons for discrepancies in uncertainty calculations using conventional merit functions.
To propose a new, improved merit function for more accurate uncertainty estimation in SME.

Main Methods:

Utilized simulated Mueller matrices to analyze uncertainty estimation.
Investigated the impact of sample-induced and measured depolarization on parameter fitting.
Developed and proposed a novel merit function that incorporates measured depolarization as a weighting parameter.
Introduced an extension to include measurement noise in the merit function.

Main Results:

Demonstrated that commonly used merit functions yield incorrect uncertainties for parameters like SiO2 layer thickness on Si.
Identified non-optimal treatment of depolarization as the primary cause of these discrepancies.
The proposed merit function, which weights depolarization, provides more accurate uncertainty estimations.
The extended merit function enables reliable statistical uncertainty calculations.

Conclusions:

The novel merit function offers a computationally inexpensive and effective solution for accurate uncertainty estimation in SME.
Properly accounting for depolarization is essential for reliable metrological results in spectroscopic Mueller ellipsometry.
The proposed method enhances the reliability of thin film characterization using SME without significant computational overhead.