March 2026 | Technical Memorandum NTIA TM-26-581
Analysis of Multiple Noisy Measurements and their Resamplings via Expected Values of Correlation and Mean-Squared-Error
Cite This Publication
Jaden Pieper and Stephen D. Voran, “Analysis of Multiple Noisy Measurements and their Resamplings via Expected Values of Correlation and Mean-Squared-Error,” Technical Memorandum NTIA TM-26-581, U.S. Department of Commerce, National Telecommunications and Information Administration, Institute for Telecommunication Sciences, March 2026. https://doi.org/10.70220/hqqs0vzx
Jaden Pieper and Stephen D. Voran 
Abstract:
This memorandum provides mathematical relationships for agreement between true measurand values, means of multiple imperfect measurements, and resampled measurement means. Agreement is quantified here in terms of Pearson correlation coefficient (PCC) and mean-squared error (MSE). The derivations of the key relationships range from simple to rather intricate, but the resulting equations are all simple, intuitive, and easy to use. More specifically, given a number of measurands, a number of measurements per measurand, and some basic variance information, the equations provide the expected PCC (or MSE) between measurand values, means of measurements, and means of resampled measurements. Depending on the measurement context and the specific goals for PCC or MSE, these results can guide readers in deciding on an appropriate number of measurements to gather.
Keywords: mean opinion score; MOS; subjective testing; correlation; bootstrapping; mean-squared error; measurement error; resampling
For technical information concerning this report, contact:
Jaden Pieper
Institute for Telecommunication Sciences
(202) 236-7516
jpieper@ntia.gov
For funding information concerning this report, click this link.
Funding Information
Performing Agency
U.S. Department of Commerce
National Telecommunications and Information Administration
Institute for Telecommunication Sciences
325 Broadway
Boulder, CO 80305
https://ror.org/00mj5bc69
Funding Agency
U.S. Department of Commerce
National Telecommunications and Information Administration
Herbert C. Hoover Building
14th and Constitution Ave., N.W.
Washington, D.C. 20230
https://ror.org/032241511
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