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On many occasions, physicists, other scientists, and engineers need to make estimates of a particular quantity. These are sometimes referred to as guesstimates, order-of-magnitude approximations, back-of-the-envelope calculations, or Fermi calculations. The physicist Enrico Fermi was famous for his ability to estimate various kinds of data with surprising precision. Estimating does not mean guessing a number or a formula at random. Instead, estimation means using prior experience and sound...
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Estimating Population Mean with Unknown Standard Deviation01:22

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In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gosset (1876–1937) of the...
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Estimating Population Standard Deviation01:26

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When the population standard deviation is unknown and the sample size is large, the sample standard deviation s is commonly used as a point estimate of σ. However, it can sometimes under or overestimate the population standard deviation. To overcome this drawback, confidence intervals are determined to estimate population parameters and eliminate any calculation bias accurately. However, this only applies to random samples from normally distributed populations. Knowing the sample mean and...
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To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean is the point estimate of the unknown population mean μ.
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The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...
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It isn't easy to measure a parameter such as the mean height or the mean weight of a population. So, we draw samples from the population and calculate the mean height or mean weight of the individuals in the sample. This sample data acts as a representative measure of the population parameter. These sample statistics are known as estimates. 
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A Bayesian perspective on magnitude estimation.

Frederike H Petzschner1, Stefan Glasauer2, Klaas E Stephan3

  • 1Translational Neuromodeling Unit (TNU), Institute for Biomedical Engineering, University of Zürich & ETH Zürich, Switzerland.

Trends in Cognitive Sciences
|April 7, 2015
PubMed
Summary
This summary is machine-generated.

Magnitude estimation biases across senses suggest shared brain mechanisms. A unifying Bayesian framework explains these biases, guiding future research in health and psychiatric conditions like schizophrenia.

Keywords:
Stevens’ power lawWeber-Fechner lawgenerative modelperceptual inferencepsychophysicsschizophrenia

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Area of Science:

  • Neuroscience
  • Psychophysics
  • Computational modeling

Background:

  • Human perception relies on estimating physical world magnitudes (e.g., loudness, distance, time).
  • Magnitude estimates are frequently biased, deviating from veridical perception.
  • Similar biases across sensory modalities suggest shared underlying neural processing.

Observation:

  • Characteristic biases in magnitude estimation are observed across different sensory systems.
  • These cross-modal similarities hint at universal neurobiological principles governing magnitude judgments.

Findings:

  • A unifying Bayesian framework is proposed to understand magnitude estimation biases.
  • This framework reinterprets existing psychophysical data and reconciles classical theories.
  • It offers a novel perspective on the neurobiological mechanisms of magnitude perception.

Implications:

  • The Bayesian approach provides a formal theory to guide research on magnitude estimation.
  • It can illuminate neurobiological underpinnings in both healthy individuals and those with psychiatric disorders, such as schizophrenia.
  • This framework facilitates a deeper understanding of sensory processing and cognitive biases.