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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...

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.

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.

Random Error

Random Error

Random or indeterminate errors originate from various uncontrollable variables, such as variations in environmental conditions, instrument imperfections, or the inherent variability of the phenomena being measured. Usually, these errors cannot be predicted, estimated, or characterized because their direction and magnitude often vary in magnitude and direction even during consecutive measurements. As a result, they are difficult to eliminate. However, the aggregate effect of these errors can be...

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...

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Related Experiment Video

Updated: Aug 6, 2025

Measuring Statistical Learning Across Modalities and Domains in School-Aged Children Via an Online Platform and Neuroimaging Techniques

Measuring Statistical Learning Across Modalities and Domains in School-Aged Children Via an Online Platform and Neuroimaging Techniques

Published on: June 30, 2020

Order of statistical learning depends on perceptive uncertainty.

Tatsuya Daikoku^1,2, Masato Yumoto³

¹International Research Center for Neurointelligence, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan.

Current Research in Neurobiology

|March 17, 2023

Summary

The human brain flexibly adjusts statistical learning (SL) strategies based on uncertainty. Higher uncertainty prompts the brain to use more complex, higher-order SL to reduce unpredictability.

Keywords:

ANOVA, analysis of variance ECDs, equivalent current dipoles ERF, event-related magnetic fields ERP, event-related potentials Information theory MEG, magnetoencephalography MMN, mismatch-negativity Markovian N-gram Predictive coding SL, statistical learning TP, transition probability entropy

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Area of Science:

Cognitive Neuroscience
Neuroscience
Psychology

Background:

Statistical learning (SL) enables the brain to encode sequence transition probabilities (TPs) and their uncertainty.
Uncertainty is known to modulate prediction in the human brain.
How the brain adapts SL strategies based on uncertainty remains unclear.

Purpose of the Study:

To investigate how uncertainty modulates neural effects of SL.
To determine if varying uncertainty levels alter the order of SL strategies employed by the brain.

Main Methods:

Auditory sequences with manipulated uncertainty (low, intermediate, high) were created using different TP ratios (90:10, 80:20, 67:33).
Neural responses were recorded while participants listened to these sequences.
Conditional entropy was used to quantify uncertainty levels (0.47, 0.72, 0.92 bit).

Main Results:

Lower TPs elicited stronger neural responses, consistent with prior research.
Participants utilized higher-order SL strategies when exposed to high-uncertainty sequences.
Neural responses indicated a stronger reaction to less predictable elements within sequences.

Conclusions:

The human brain demonstrates flexibility in adapting SL strategy order according to uncertainty levels.
Uncertainty is a key factor influencing the selection of SL strategies.
Higher-order SL may be employed by the brain to mitigate uncertainty in information processing.