Distribution Theory for Glass's Estimator of Effect Size and Related Estimators
Abstract
Glass's estimator of effect size, the sample mean difference divided by the sample standard deviation, is studied in the context of an explicit statistical model. The exact distribution of Glass's estimator is obtained and the estimator is shown to have a small sample bias. The minimum variance unbiased estimator is obtained and shown to have uniformly smaller variance than Glass's (biased) estimator. Measurement error is shown to attenuate estimates of effect size and a correction is given. The effects of measurement invalidity are discussed. Expressions for weights that yield the most precise weighted estimate of effect size are also derived.
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Raw effect sizes were corrected for small-sample bias, and most correction factors were close to 1.0 because sample sizes were large.
“Raw effect sizes were corrected for bias (i.e., the upward-bias of effect sizes among small samples; Hedges, 1981); however, most correction factors were close to 1.0 given the large sample sizes.”
From Gender Differences in Depression in Representative National Samples: Meta-Analyses of Diagnoses and Symptoms (2017) · cites this paper as Hedges, 1981
The effect size estimates were bias-corrected (Hedges' g).
“we applied a correction to bias-correct estimates (resulting in what is often referred to as Hedge ' s g ).”
From The Impact of Peer Assessment on Academic Performance: A Meta-analysis of Control Group Studies (2020) · cites this paper as Hedges 1981
Standardized mean difference (d) has a slight positive bias.
“As standardized mean difference ( d ) is known to have a slight positive bias (Hedges 1981), we applied a correction to bias-correct estimates (resulting in what is often referred to as Hedge ' s g ).”
From The Impact of Peer Assessment on Academic Performance: A Meta-analysis of Control Group Studies (2020) · cites this paper as Hedges 1981
To correct for possible bias caused by small samples, Cohen's d was used to calculate Hedges' g.
“To correct for possible bias caused by small samples, Cohen's d was then used to calculate Hedges' g (Hedges 1981).”
From The gamification of learning: a meta-analysis (2020) · cites this paper as Hedges 1981
Effect sizes were computed as Hedges' g standardised mean differences to correct for small-sample bias and improve comparability across tasks.
“Effect sizes were computed as Hedges’ g standardised mean differences (SMDs) to correct for small-sample bias and improve comparability across tasks88.”
From Meta-analysis of computerised working memory training: behavioural gains, training parameters, transfer mechanisms, and neural correlates (2026) · cites this paper as [88]
Cholinesterase inhibitors and memantine appear ineffective for agitation in dementia.
“Cholinesterase inhibitors and memantine appear ineffective. 13 , 14”
From Non-pharmacological interventions for agitation in dementia: systematic review of randomised controlled trials (2014) · cites this paper as [13]
Hedge's G a helps to control for common overestimations of effect sizes in samples below 20.
“Unlike Cohen's d, Hedge's G a helps to control for common overestimations of effect sizes in samples below 20 (Hedges 1981).”
Preliminary searching indicates that most samples are over 100 and none are under 50, so the small-sample correction is not likely to be needed.
“Preliminary searching indicates that most samples are over 100 and none are under 50, so the small‐sample correction (to use Hedges' g) is not likely to be needed (Hedges, 1981).”
From PROTOCOL: Single-track year-round education for improving academic achievement in us k-12 schools: results of a meta-analysis (2018) · cites this paper as Hedges, 1981
Hedges' g is unitless, allowing comparison across metrics, and corrects for upward bias in Cohen's d.
“We chose Hedges’ g as our measure of effect size because it is unitless, so it allows us to compare human–AI performance across different metrics, and it corrects for upward bias in the raw standardized mean difference (Cohen’s d)61.”
From When combinations of humans and AI are useful: A systematic review and meta-analysis (2024) · cites this paper as [61]
Risk of bias of included RCTs was assessed using Cochrane Collaboration's Tool, independently by a minimum of two researchers.
“Risk of bias of the included RCTs was assessed using the Cochrane Collaboration’s Tool for Assessing Risk of Bias 120 , which were independently conducted by a minimum of two researchers (KJM, PA, and/or DH).”
The analysis used Hedges' g as a conservative adjustment for small samples.
“Next, an estimation for Hedges' g (Hedges, 1981) was applied as a conservative measure to correct for sample size.”
Hedges's g was used to adjust effect sizes to correct for small sample size biases.
“Since differences in sample sizes between studies might introduce bias into the calculated effect sizes, Hedges's g was used to adjust the effect sizes to correct for small sample size biases (Hedges, 1981).”
From RETRACTED ARTICLE: The effect of ChatGPT on students’ learning performance, learning perception, and higher-order thinking: insights from a meta-analysis (2025) · cites this paper as Hedges, 1981
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New trends in gender and mathematics performance: A meta-analysis.
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- Authors
- Hedges, L. V.
- Affiliations
- University of Chicago (US)
- Corresponding author
- Hedges, Larry V.
- ORCID
- Hedges, Larry V.
- Venue
- Journal of Educational Statistics
- Publisher
- JSTOR
- Published
- 1981
- Volume
- 6
- Issue
- 2
- Pages
- 107
- ISSN
- 0362-9791 (ISSN-L)
- OpenAlex
- W2059527510
- Cited by
- 5,695
- References
- 9
- Citations by year
- 2026: 623 · 2025: 486 · 2024: 543 · 2023: 414 · 2022: 463 · 2021: 405 · 2020: 359 · 2019: 306 · 2018: 264 · 2017: 222 · 2016: 198 · 2015: 175 · 2014: 169 · 2013: 132 · 2012: 118
- FWCI
- 61.80
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Educational Statistics (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Funders
- Spencer Foundation
- Topics
- Social Sciences › Decision Sciences › Statistics, Probability and Uncertainty › Meta-analysis and systematic reviews
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Epidemiology
- Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
- Keywords
- effect size, exact distribution, small sample bias, minimum variance unbiased estimator, measurement error, measurement error correction
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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