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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 Guinness...

Confidence Interval for Estimating Population Mean

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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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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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(point estimate - error bound, point estimate + error bound)
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A confidence...

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...

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Another key consideration is determining the appropriate number of samples required to...

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Related Experiment Video

Updated: Jul 18, 2026

Problem-Solving Before Instruction (PS-I): A Protocol for Assessment and Intervention in Students with Different Abilities

Problem-Solving Before Instruction (PS-I): A Protocol for Assessment and Intervention in Students with Different Abilities

Published on: September 11, 2021

Sample size planning for the standardized mean difference: accuracy in parameter estimation via narrow confidence

Ken Kelley¹, Joseph R Rausch

¹Inquiry Methodology Program, Indiana University, Bloomington, IN 47405, USA. kkiii@indiana.edu

Psychological Methods

|December 13, 2006

Summary

Researchers developed sample size planning methods for narrow confidence intervals using the accuracy in parameter estimation approach. This ensures precise estimates for the standardized mean difference in studies.

Related Experiment Videos

Last Updated: Jul 18, 2026

Problem-Solving Before Instruction (PS-I): A Protocol for Assessment and Intervention in Students with Different Abilities

Problem-Solving Before Instruction (PS-I): A Protocol for Assessment and Intervention in Students with Different Abilities

Published on: September 11, 2021

Area of Science:

Statistics
Psychometrics
Social Sciences

Background:

Accurate statistical inference relies on appropriate sample size (SS) planning.
The accuracy in parameter estimation (AIPE) approach focuses on achieving precise estimates by controlling confidence interval (CI) width.

Purpose of the Study:

To develop and present methods for planning sample size (SS) for the standardized mean difference using the accuracy in parameter estimation (AIPE) approach.
To ensure that the resulting confidence interval (CI) for the standardized mean difference is sufficiently narrow.

Main Methods:

Developed methods for SS planning based on achieving a narrow CI width via the AIPE approach.
Proposed a modification to adjust SS for a desired CI width with a specified degree of certainty.
Discussed the analytic approach to CI formation for the population standardized mean difference.

Main Results:

Provided tables with necessary SS values for various scenarios.
The proposed methods allow researchers to plan SS to obtain narrow confidence intervals for the standardized mean difference.
A modification allows for planning SS to ensure a desired CI width with high certainty (e.g., 99%).

Conclusions:

The AIPE approach offers a robust framework for sample size planning focused on precision.
The developed methods and provided tables facilitate the application of AIPE for standardized mean difference estimation.
The Methods for the Behavioral, Educational, and Social Sciences R software package implements these SS planning methods.