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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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A point estimate of the population mean is obtained from a single sample. Such a point estimate does not represent a population well because it needs to account for variability in the population. Single point estimate can also be biased despite the sample being selected randomly. Thus, a point estimate is often unreliable. A confidence interval is needed to reduce this unreliability.
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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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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.
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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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Precise Estimates of Persistence Time for SIS Infections in Heterogeneous Populations.

Damian Clancy1

  • 1Department of Actuarial Mathematics and Statistics, Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK. d.clancy@hw.ac.uk.

Bulletin of Mathematical Biology
|September 13, 2018
PubMed
Summary

This study provides simple estimates for infection persistence times in heterogeneous populations. It helps understand how variations in susceptibility and infectiousness affect disease spread and extinction.

Keywords:
Endemic fade-outLarge deviationsStochastic epidemic modelsStochastic networksSuperspreaders

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Network Science

Background:

  • Understanding disease dynamics in heterogeneous populations is crucial for effective public health interventions.
  • Previous models often simplified population heterogeneity, limiting their applicability.
  • The susceptible-infectious-susceptible (SIS) model provides a basic framework for infectious disease transmission.

Purpose of the Study:

  • To derive simple and precise estimates of mean infection persistence time in heterogeneous SIS models.
  • To approximate the quasi-stationary distribution of the infection process.
  • To investigate the impact of various forms of heterogeneity on disease dynamics.

Main Methods:

  • Development of analytical methods for estimating mean persistence time.
  • Derivation of a novel approximation for the quasi-stationary distribution.
  • Application of the model to analyze different heterogeneity scenarios.

Main Results:

  • Simple and precise estimates for mean persistence time were obtained.
  • A new, accurate approximation for the quasi-stationary distribution was derived.
  • The influence of heterogeneity in infectiousness, susceptibility, and infectious period distributions was quantified.

Conclusions:

  • The derived estimates offer valuable tools for analyzing infection persistence in complex populations.
  • The findings enhance our understanding of how heterogeneity influences disease extinction and endemic states.
  • The model's applicability extends to heterogeneous directed network models under the annealed network approximation.