Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

One-Way ANOVA: Equal Sample Sizes01:15

One-Way ANOVA: Equal Sample Sizes

3.2K
One-Way ANOVA can be performed on three or more samples with equal or unequal sample sizes. When one-way ANOVA is performed on two datasets with samples of equal sizes, it can be easily observed that the computed F statistic is highly sensitive to the sample mean.
Different sample means can result in different values for the variance estimate: variance between samples. This is because the variance between samples is calculated as the product of the sample size and the variance between the...
3.2K
One-Way ANOVA: Unequal Sample Sizes01:15

One-Way ANOVA: Unequal Sample Sizes

5.7K
One-way ANOVA can be performed on three or more samples of unequal sizes. However, calculations get complicated when sample sizes are not always the same. So, while performing ANOVA with unequal samples size, the following equation is used:
5.7K
Variance01:15

Variance

9.3K
 The deviations show how spread out the data are about the mean. A positive deviation occurs when the data value exceeds the mean, whereas a negative deviation occurs when the data value is less than the mean. If the deviations are added, the sum is always zero. So one cannot simply add the deviations to get the data spread. By squaring the deviations, the numbers are made positive; thus, their sum will also be positive.
The standard deviation measures the spread in the same units as the...
9.3K
Sampling Distribution01:12

Sampling Distribution

12.3K
Given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example...
12.3K
Testing a Claim about Standard Deviation01:19

Testing a Claim about Standard Deviation

2.4K
A complete procedure to test a claim about population standard deviation or population variance is explained here.
The hypothesis testing for the claim of population standard deviation (or variance) requires the data and samples to be random and unbiased. The population distribution also must be normal. There is no specific requirement on the sample size as the estimation is based on the chi-square distribution.
As a first step, the hypothesis (null and alternative) concerning the claim about...
2.4K
Estimating Population Mean with Unknown Standard Deviation01:22

Estimating Population Mean with Unknown Standard Deviation

7.6K
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...
7.6K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

Elastic functional Cox regression model with shape predictors.

Journal of applied statistics·2026
Same journal

An improved two-stage binary relevance method for multilabel classification.

Journal of applied statistics·2026
Same journal

Classification of multivariate functional data with an application to ADHD fMRI data.

Journal of applied statistics·2026
Same journal

Assessing the performance of longitudinal T-lymphocytes as biomarkers of immune recovery in HIV-infected children with or without TB co-infection.

Journal of applied statistics·2026
Same journal

Sparse long-only Markowitz portfolio optimization.

Journal of applied statistics·2026
Same journal

Homogeneity of multinomial populations when data are classified into a large number of groups.

Journal of applied statistics·2026
See all related articles

Related Experiment Video

Updated: Jun 11, 2025

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
04:35

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Published on: July 3, 2020

3.3K

A Simple Variance Estimator for Unequal Probability Sampling without Replacement.

Yves G Berger1

  • 1Social Statistics, University of Southampton, UK.

Journal of Applied Statistics
|October 7, 2024
PubMed
Summary
This summary is machine-generated.

The Hájek variance estimator offers a more accurate alternative to the Hansen-Hurwitz estimator for unequal probability sampling. It simplifies variance estimation in business surveys by relying only on first-order inclusion probabilities.

Keywords:
Design-based inferenceHansen–Hurwitz variance estimatorSen–Yates–Grundy variance estimatorinclusion probabilitiesπ-estimator

More Related Videos

Decomposing the Variance in Reading Comprehension to Reveal the Unique and Common Effects of Language and Decoding
06:33

Decomposing the Variance in Reading Comprehension to Reveal the Unique and Common Effects of Language and Decoding

Published on: October 11, 2018

6.8K
Observation and Analysis of Blinking Surface-enhanced Raman Scattering
05:52

Observation and Analysis of Blinking Surface-enhanced Raman Scattering

Published on: January 11, 2018

7.4K

Related Experiment Videos

Last Updated: Jun 11, 2025

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
04:35

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Published on: July 3, 2020

3.3K
Decomposing the Variance in Reading Comprehension to Reveal the Unique and Common Effects of Language and Decoding
06:33

Decomposing the Variance in Reading Comprehension to Reveal the Unique and Common Effects of Language and Decoding

Published on: October 11, 2018

6.8K
Observation and Analysis of Blinking Surface-enhanced Raman Scattering
05:52

Observation and Analysis of Blinking Surface-enhanced Raman Scattering

Published on: January 11, 2018

7.4K

Area of Science:

  • Statistics
  • Survey Methodology

Background:

  • Traditional survey sampling often uses the Sen-Yates-Grundy variance estimator, which is complex to implement due to joint inclusion probabilities.
  • The Hansen-Hurwitz (with replacement) variance estimator is commonly used in practice but can overestimate variance with large sampling fractions in business surveys.

Purpose of the Study:

  • To review the Hájek variance estimator as a more accurate and practical alternative for unequal probability sampling without replacement.
  • To propose a simplified expression for the Hájek estimator, making it as easy to implement as the Hansen-Hurwitz estimator.
  • To demonstrate the ease of implementing the Hájek estimator using standard statistical packages.

Main Methods:

  • Review of the Hájek (1964) variance estimator, focusing on its reliance on first-order inclusion probabilities.
  • Development of a simplified alternative expression for the Hájek variance estimator.
  • Demonstration of implementation using standard statistical software.

Main Results:

  • The Hájek variance estimator is generally more accurate than the Hansen-Hurwitz estimator, especially with large sampling fractions.
  • A new, simplified expression for the Hájek estimator is proposed, comparable in complexity to the Hansen-Hurwitz estimator.
  • The Hájek estimator can be readily implemented in common statistical packages.

Conclusions:

  • The Hájek variance estimator provides a more accurate and practically feasible approach for estimating variance in unequal probability sampling designs.
  • Its simplified form and ease of implementation make it a valuable tool for business surveys experiencing large sampling fractions.