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Estimating Population Mean with Unknown Standard Deviation

Estimating Population Mean with Unknown Standard Deviation

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...

Estimating Population Standard Deviation

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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...

Estimating Population Mean with Known Standard Deviation

Estimating Population Mean with Known Standard Deviation

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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Related Experiment Video

Updated: Sep 9, 2025

Predicting the Effectiveness of Population Replacement Strategy Using Mathematical Modeling

Predicting the Effectiveness of Population Replacement Strategy Using Mathematical Modeling

Published on: July 4, 2007

Time-varying reproductive number estimation for practical application in structured populations.

Erin Clancey¹, Eric T Lofgren¹

¹Paul G. Allen School for Global Health, Washington State University, Pullman, WA, USA.

Epidemiologic Methods

|August 29, 2025

Summary

EpiEstim's real-time estimates of the time-varying reproductive number ( $ℛ_{t}$ ) can be temporally biased in structured populations. Adjusting methods can recover accuracy for epidemic response.

Keywords:

EpiEstim generation interval distribution population structure simulation time-varying reproductive number

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Last Updated: Sep 9, 2025

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Following the Dynamics of Structural Variants in Experimentally Evolved Populations

Following the Dynamics of Structural Variants in Experimentally Evolved Populations

Published on: February 3, 2023

Area of Science:

Epidemiology
Statistical modeling
Computational biology

Background:

The time-varying reproductive number ( $ℛ_{t}$ ) is crucial for real-time epidemic assessment.
The EpiEstim framework provides estimates for $ℛ_{t}$ but its performance in structured populations is under-evaluated.
Population structure can introduce temporal biases in $ℛ_{t}$ estimates.

Purpose of the Study:

To evaluate the temporal performance of EpiEstim in small, non-randomly mixing populations.
To assess the impact of population structure on $ℛ_{t}$ estimates.
To demonstrate a method for correcting temporal biases in EpiEstim $ℛ_{t}$ estimates.

Main Methods:

Simulated COVID-19 outbreak data from a two-population mechanistic model.
Comparison of true $ℛ_{t}$ with EpiEstim-derived ${\hat{ℛ}}_{t}$ estimates.
Analysis of temporal bias by comparing time points when $ℛ_{t}$ crosses the critical threshold of 1.

Main Results:

EpiEstim ${\hat{ℛ}}_{t}$ estimates prematurely fell below 1 in structured populations.
Weekly aggregated data showed later ${\hat{ℛ}}_{t}$ threshold crossings than daily data, with population structure having no additional effect.
Recovering temporal accuracy was achieved by using lagging subpopulation data for total population ${\hat{ℛ}}_{t}$ estimation.

Conclusions:

Population structure can bias EpiEstim $ℛ_{t}$ estimates near the critical threshold.
Prudent application of EpiEstim to structured population data is necessary for accurate epidemic response.
Methods exist to mitigate temporal biases in $ℛ_{t}$ estimation within structured populations.