Welch's t-Test: When and How to Use It

Welch's t-test is a two-sample statistical test that compares the means of two independent groups without requiring the assumption that the two populations have equal variances. It is the preferred default choice for comparing two group means in most research settings because it maintains reliable error rates when variances differ between groups, and it loses little statistical power compared to Student's t-test when variances are equal. This article explains the purpose of Welch's t-test, its assumptions, how to calculate it, when to choose it over Student's t-test, and how to interpret results correctly in research and applied settings.

The Problem with Student's t-Test

Student's t-test has been the standard method for comparing two independent group means for decades. The test assumes that the two samples are drawn from populations with equal variances, an assumption called homogeneity of variance. When this assumption is violated, Student's t-test can produce misleading results.

Simulation research has demonstrated that Student's t-test can have an inflated Type I error rate when the equal variance assumption is violated. Type I error is the probability of incorrectly rejecting a true null hypothesis, meaning the test claims a difference exists when it does not. When sample sizes are unequal and variances differ between groups, the inflation can be substantial, leading researchers to report significant differences that are statistical artifacts instead of real effects 20.

The equal variance assumption rarely holds in practice. In fields such as psychology, the assumption of equal variances will seldom be satisfied in real data 20. Similar concerns apply across the life sciences, agriculture, and environmental monitoring, where treatment groups often show different levels of variability. For example, a new feed additive might change also the average weight gain of animals but also the spread of weight gains within the group.

Choosing between Student's t-test and Welch's t-test based on the outcome of a preliminary test of variance equality is a common but flawed strategy. This two-step approach often fails to provide an appropriate answer because variance tests have their own limitations and the overall procedure can distort error rates 20.

What Welch's t-Test Does Differently

Welch's t-test, also called the unequal variances t-test, adjusts the test statistic and the degrees of freedom to account for the fact that the two groups may have different variances. Instead of pooling the variances from both groups into a single estimate, Welch's test uses a separate variance estimate for each group.

The Welch test statistic is calculated as the difference between the two sample means divided by the standard error of that difference. The standard error incorporates the sample variance and sample size from each group separately instead of a pooled estimate.

The degrees of freedom for Welch's t-test are adjusted downward from the degrees of freedom used in Student's t-test. This adjustment, sometimes called the Welch-Satterthwaite correction, produces a fractional degrees of freedom value that reflects how unequal the variances and sample sizes are. When variances and sample sizes are equal across groups, the Welch degrees of freedom approach those of Student's t-test. When they differ, the degrees of freedom decrease, which makes the test more conservative and reduces the risk of false positives.

The Welch test is sometimes referred to as the Satterthwaite approximate F test in its ANOVA extension. When the underlying population distributions are normal with unequal and unknown variances, either Welch's t-statistic or Satterthwaite's approximate F test is suggested as an appropriate procedure 8.

Assumptions of Welch's t-Test

Welch's t-test requires fewer assumptions than Student's t-test, but it is not assumption-free. Understanding these assumptions is essential for correct application.

Independence of Observations

The two samples must be independent of each other, and the observations within each sample must be independent of one another. This means that the value of one observation does not influence the value of another observation. In animal research, this requires that animals are randomly assigned to treatment groups and that measurements from one animal do not affect measurements from another animal. Littermate effects, shared housing, and repeated measurements on the same animal all violate the independence assumption.

Normality of the Data

Welch's t-test assumes that the data within each group are drawn from a normally distributed population. The test is considered robust to moderate violations of normality, especially when sample sizes are reasonably large. However, severe departures from normality can cause problems.

Simulation work has shown that Welch's t-test can have an inflated Type I error rate when the normality assumption is violated and the data follow a Poisson distribution, particularly when sample sizes are unequal 19. In this scenario, Welch's test can be systematically biased and prone to producing extremely low p-values. Student's t-test performs better in this specific situation, but a likelihood ratio test based on logistic regression performs better still 19.

More recent simulation research confirms that Welch's test inflates Type I error when distributions are skewed. The false positive rate reached approximately 6 percent at population skewness of 1, 7.5 percent at skewness of 2, and 9 percent at skewness of 3, with a nominal error rate of 5 percent 23. When sample sizes were equal, both classical tests held nominal Type I error, but under unequal sample sizes with skewness, both lost power to robust alternatives 23.

Unequal Variances Are Allowed

Unlike Student's t-test, Welch's t-test does not require equal variances between the two groups. This is the defining feature of the test. The test accommodates situations where one group is more variable than the other.

Sample Size Considerations

Welch's t-test performs well with moderate to large sample sizes. However, with very small sample sizes, the test can show Type I error inflation even when assumptions are met. A bibliographic record on this topic documents Type I error inflation of the separate-variances Welch t-test with very small sample sizes when assumptions are met 26.

Simulation research using empirical data from murine genetic models of obesity found Type I error inflation for all common tests, including Welch's t-test, with small samples of five or fewer animals per group. Type I error inflation decreased as sample size increased to eight or more per group but remained present. The bootstrap test was recommended for small samples to avoid Type I error inflation, but this benefit came at the cost of lower power. When sample size is large enough, Welch's t-test is recommended because of high power with minimal Type I error inflation 11.

At a Glance: Choosing Between Welch's t-Test and Alternatives

The following table summarizes the key considerations for choosing between Welch's t-test and common alternatives.

When to Use Welch's t-Test

Welch's t-test should be used as the default strategy for comparing two independent group means in most research situations. The recommendation to use Welch's test by default is supported by simulation evidence showing that it provides better control of Type I error rates when the homogeneity of variance assumption is not met, and it loses little robustness compared to Student's t-test when the assumptions are met 20.

The same logic extends to analysis of variance. The classical F-test for one-way ANOVA relies on assumptions of normality and equal variances between groups. Under realistic deviations from the assumption of equal variances, the classical F-test can yield severely biased results and lead to invalid statistical inferences. Simulation studies show that Welch's ANOVA is a better alternative across a range of realistic scenarios, and it should be used by default when comparing means across more than two groups 22.

Specific Applications in Research

Welch's t-test has been applied across many research domains. In environmental toxicology, the U.S. Environmental Protection Agency and state agencies evaluate the toxicity of effluent and surface water samples using the test of significant toxicity analysis, which uses Welch's t-test to compare organism responses in a sample with responses in a control or site sample. Evaluation of data from over 2,000 whole effluent toxicity tests showed that moderately unequal variances have little effect on coverage of the t-test or Welch's t-test for normally distributed data. For nonnormally distributed data similar in distribution to whole effluent toxicity data, the test of significant toxicity using Welch's t-test has close to nominal coverage based on simulations with up to a ninefold difference in variance between the effluent and control groups 6.

In stress research, a study evaluating a multicomponent program to manage academic stress used Welch's t-test to compare Resilience to Stress Index scores between pre-test and post-test phases. The test showed a difference in mean scores with a t-value of -2.30 and a p-value of 0.025 9.

In dental research, a study of orofacial esthetics perception among laypersons in Switzerland applied Welch's t-test along with descriptive analyses and the Fisher exact test to compare responses across age and gender groups 10.

In sports science, a study of Polish National Paralympic Team wheelchair fencers used Welch's t-test to compare muscle activation time and co-activation between two disability-level groups. The normal distribution of analyzed features was checked with the Shapiro-Wilk test before applying Welch's t-test 12.

In medical imaging research, a retrospective longitudinal MRI analysis of pediatric patients with tuberous sclerosis complex used paired t-test analysis for within-group comparisons and Welch's two-sample t-test for direct comparison between age subgroups 13.

In medical education research, a pilot study comparing online learning modules with in-person lectures on nutritional medicine used permutation Welch t-tests for group differences in test scores 17.

How to Perform Welch's t-Test

Step-by-Step Calculation

The Welch t-test statistic is calculated using the following formula:

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Where:

• x̄₁ and x̄₂ are the sample means of group 1 and group 2
• s₁² and s₂² are the sample variances of group 1 and group 2
• n₁ and n₂ are the sample sizes of group 1 and group 2

The degrees of freedom are calculated using the Welch-Satterthwaite equation:

df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁ - 1) + (s₂²/n₂)²/(n₂ - 1)]

The resulting degrees of freedom is typically not a whole number. The p-value is obtained by comparing the calculated t-statistic to the t-distribution with the calculated degrees of freedom.

Worked Example

Consider a study comparing weight gain in two groups of growing pigs. Group A receives a standard diet and Group B receives a diet supplemented with a test ingredient. The researcher records weight gain in kilograms over a 60-day period.

Group A (standard diet): n₁ = 12, mean = 32.5 kg, variance = 18.4 Group B (supplemented diet): n₂ = 10, mean = 38.2 kg, variance = 42.7

The variances differ substantially between the groups, so Welch's t-test is appropriate.

Step 1: Calculate the standard error of the difference SE = √(18.4/12 + 42.7/10) = √(1.533 + 4.270) = √5.803 = 2.409

Step 2: Calculate the t-statistic t = (32.5 - 38.2) / 2.409 = -5.7 / 2.409 = -2.366

Step 3: Calculate the degrees of freedom df = (1.533 + 4.270)² / [(1.533)²/11 + (4.270)²/9] df = (5.803)² / [2.350/11 + 18.233/9] df = 33.675 / [0.214 + 2.026] df = 33.675 / 2.240 = 15.03

Step 4: Determine the p-value With t = -2.366 and df = 15.03, the two-tailed p-value is approximately 0.032. This would be considered statistically significant at the conventional alpha level of 0.05.

Using Statistical Software

Most statistical software packages include Welch's t-test as an option. In R, the t.test function uses Welch's t-test by default. In SPSS, the independent samples t-test output includes both Student's t-test results and Welch's t-test results. In Python, the scipy.stats.ttest_ind function has an equal_var parameter that should be set to False for Welch's t-test.

The key point is that researchers should not need to manually calculate the test statistic in most cases. The important decision is selecting the correct test and interpreting the output correctly.

Reporting Effect Sizes with Welch's t-Test

When reporting the results of Welch's t-test, researchers should also report an appropriate effect size measure. The most commonly used effect size for two-group comparisons is Cohen's d, which divides the sample mean difference by the pooled standard deviation. However, computing the pooled error term is not valid when both groups do not share common population variances 21.

For Welch's t-test, the recommended effect size is Hedges' g*, which is based on the non-pooled standard deviation. Simulation comparisons show that Hedges' g* shows better properties than other estimators including Glass's g, Shieh's g, and the standard Hedges' g with pooled standard deviation. This recommendation applies under normality as well as under realistic deviations from the assumptions of normality and equal variances 21.

Reporting effect sizes with confidence intervals is important for interpreting the practical significance of findings. A statistically significant result with a small effect size may not be practically meaningful, while a nonsignificant result with a moderate effect size may warrant further investigation with a larger sample.

Welch's t-Test in the Context of Statistical Fragility

The choice of statistical test can affect the fragility of research findings. The reverse continuous fragility index is a measure of how many data points would need to be moved from the lower mean group to the higher mean group to alter the significance of a result. A systematic review and meta-analysis of arthroscopic rotator cuff repair studies calculated the reverse continuous fragility index for the Welch t-test, Student t-test, and Wilcoxon rank-sum test under various data assumptions. Using the Welch t-test, the median reverse continuous fragility index across all study outcomes was 20. For the Student t-test, the median was 14, and for the Wilcoxon rank-sum test, the median was 14. While using the Welch t-test, 64 percent of study outcomes had a reverse continuous fragility index greater than the loss to follow-up, compared with 32 percent when using the other tests. The fragility of these studies was largely dependent on the statistical test used 7.

This finding illustrates that the choice of statistical test can materially affect the robustness of research conclusions. Studies that appear robust under one test may appear fragile under another.

Common Failure Patterns in Applying Welch's t-Test

Failure to Check Normality

A common error is applying Welch's t-test without checking the normality assumption. The test is robust to moderate violations of normality, but severe skewness can inflate Type I error rates, especially with unequal sample sizes 23. Researchers should examine the distribution of data within each group using histograms, Q-Q plots, or formal normality tests such as the Shapiro-Wilk test.

In the wheelchair fencing study, the normal distribution of analyzed statistical features was checked with the Shapiro-Wilk test before applying Welch's t-test 12. This is an appropriate workflow.

Using a Pretest for Variance Equality

A common but flawed strategy is to test for equality of variances first and then choose between Student's t-test and Welch's t-test based on the outcome. This approach often fails to provide an appropriate answer because the pretest itself has limited power and the overall procedure can distort error rates 20. The recommended approach is to use Welch's t-test by default.

Applying Welch's t-Test to Very Small Samples

Welch's t-test can show Type I error inflation with very small sample sizes even when assumptions are met 26. Simulation research using empirical data from murine genetic models found Type I error inflation for all common tests with samples of five or fewer animals per group 11. Researchers working with very small samples should consider alternative approaches such as the bootstrap test.

Ignoring Severe Nonnormality

Welch's t-test is not robust to all forms of nonnormality. When data follow a Poisson distribution and sample sizes are unequal, Welch's test can have an inflated Type I error rate, be systematically biased, and produce extremely low p-values 19. Researchers working with count data or other clearly nonnormal distributions should consider alternatives such as logistic regression or robust tests.

Misinterpreting Nonsignificant Results

A nonsignificant result from Welch's t-test does not prove that the two group means are equal. It only indicates that the observed difference is not large enough to be distinguished from random variation given the sample sizes and variability. Researchers should report confidence intervals for the mean difference to communicate the range of plausible effect sizes.

Limitations of Welch's t-Test

Sensitivity to Nonnormal Distributions

Welch's procedure is non-robust under most non-normal distributions 8. While the test handles unequal variances well, it does not handle all violations of normality equally well. Skewed distributions can cause inflated Type I error rates, particularly when sample sizes are unequal 23.

Performance with Small Samples

The test can show Type I error inflation with very small sample sizes even when assumptions are met 26. Researchers should be cautious when applying Welch's t-test to samples of five or fewer observations per group.

Not a Test of Distribution Equality

Welch's t-test specifically tests for differences in means. It does not test whether two distributions are identical. Two groups can have identical means but different variances, different shapes, or different distributions. If the research question concerns whether two distributions differ instead of whether their means differ specifically, other tests such as the Anderson-Darling test may be more appropriate 23.

Assumption of Independence

Like all standard t-tests, Welch's t-test assumes independence of observations. This assumption is often violated in practice when animals are housed together, when repeated measurements are taken from the same subject, or when data are clustered in some way. Violations of independence can seriously distort test results.

Alternatives to Welch's t-Test

Permutation-Based Welch's Test

A permutation-based Welch's test holds the nominal Type I error across a factorial design varying sample-size imbalance, variances, and skewness while preserving the original measurement scale. This makes it a defensible default when the research question concerns equality of means and the data may violate assumptions 23.

Bootstrap Test

The bootstrap test is recommended for small samples to avoid Type I error inflation, but this benefit comes at the cost of lower power 11. The bootstrap approach resamples the data many times to estimate the sampling distribution of the test statistic.

Yuen-Welch Test

The Yuen-Welch test is a robust alternative that trims a proportion of observations from each tail of the distribution before applying a Welch-type test. This approach reduces the influence of outliers and heavy-tailed distributions.

Logistic Regression

A likelihood ratio test based on logistic regression models performs better than either Student's t-test or Welch's t-test for data that follow a Poisson distribution 19. Logistic regression may be a preferable method to test for a difference in distributions between two samples in some situations.

Nonparametric Tests

Nonparametric tests such as the Mann-Whitney U test and the Wilcoxon rank-sum test do not require normality assumptions. However, the Wilcoxon test should be avoided when distributions are heterogeneous because it can produce misleading results 11. The Mann-Whitney U test may have lower power than parametric tests when sample sizes are small 24.

Anderson-Darling Test

The Anderson-Darling test attained relatively high power when two skewed populations differed simultaneously in mean and variability. It is a strong candidate when the research question concerns whether two distributions differ instead of whether their means differ specifically 23.

Practical Workflow for Comparing Two Group Means

The following workflow provides a practical approach to comparing two independent group means.

Step 1: Define the Research Question

Clearly state whether the research question concerns a difference in means, a difference in distributions, or a difference in some other parameter. This decision determines which statistical test is appropriate.

Step 2: Examine the Data

Plot the data for each group using histograms, box plots, or Q-Q plots. Calculate descriptive statistics including means, standard deviations, and sample sizes. Check for outliers and missing values.

Step 3: Assess Normality

Use the Shapiro-Wilk test or other normality tests to assess whether the data within each group are approximately normally distributed. Visual inspection of Q-Q plots is also valuable. The Shapiro-Wilk test was used in the wheelchair fencing study before applying Welch's t-test 12.

Step 4: Assess Sample Sizes

Determine whether sample sizes are adequate for the chosen test. Very small samples of five or fewer per group require special consideration 11.

Step 5: Select the Test

For approximately normal data with moderate to large samples, use Welch's t-test by default. For skewed data with unequal sample sizes, consider a permutation-based Welch's test or robust alternatives. For very small samples, consider the bootstrap test. For count data, consider logistic regression.

Step 6: Perform the Test and Report Results

Report the test statistic, degrees of freedom, p-value, and an appropriate effect size with confidence interval. For Welch's t-test, use Hedges' g* based on the non-pooled standard deviation 21.

Step 7: Interpret Results in Context

Interpret the results in the context of the research question, the study design, and the limitations of the data. A statistically significant result should be evaluated for practical significance, and a nonsignificant result should be evaluated for the adequacy of the sample size to detect meaningful effects.

Records and Measurements for Research Integrity

Maintaining accurate records of statistical decisions is essential for research integrity. The following records should be documented for any study using Welch's t-test.

Data Collection Records

Document how data were collected, including the sampling method, the measurement instruments, and any data cleaning procedures. Record the raw data in a format that allows verification of the analysis.

Assumption Checking Records

Document the results of normality checks and any other assumption assessments. Record the Shapiro-Wilk test statistics and p-values if used. Document any transformations applied to the data.

Analysis Records

Record the statistical software and version used, the specific test performed, and the output including the test statistic, degrees of freedom, and p-value. Document the effect size measure and its confidence interval.

Decision Records

Document the rationale for choosing Welch's t-test over alternatives. This is particularly important if the choice was based on the outcome of a variance equality test, which is generally not recommended 20.

Quality Controls for Statistical Analysis

Quality control in statistical analysis involves verifying that the analysis is appropriate for the data and that the results are reproducible.

Verification of Data Entry

Verify that data were entered correctly by checking a random sample of records against the original data source. Errors in data entry can produce misleading results.

Reproducibility Checks

Run the analysis twice or have a colleague independently verify the results. Document the exact code or commands used so that the analysis can be reproduced.

Sensitivity Analysis

Consider how sensitive the results are to the choice of statistical test. If Welch's t-test and Student's t-test produce different conclusions, this is a signal that the data may violate assumptions and the results should be interpreted with caution.

Peer Review

Have the statistical analysis reviewed by a colleague with statistical expertise. The EQUATOR Network provides reporting guidelines that can help ensure that statistical methods are reported completely and accurately.

Professional Escalation Criteria

Researchers should seek additional statistical expertise in the following situations.

Complex Data Structures

If the data have a complex structure such as clustering, repeated measures, or hierarchical organization, standard t-tests may not be appropriate. Consult a statistician for guidance on mixed models or other appropriate methods.

Severe Assumption Violations

If the data show severe violations of normality that are not addressed by transformations or robust alternatives, consult a statistician for guidance on appropriate methods.

Small Sample Sizes

If sample sizes are very small, particularly five or fewer per group, consult a statistician about appropriate methods such as the bootstrap test 11.

Regulatory or High-Stakes Decisions

If the statistical analysis will be used for regulatory decisions, such as environmental toxicity testing, or for decisions with significant consequences, consult a statistician to ensure that the analysis meets all applicable standards. The National Institute of Standards and Technology provides resources on research data frameworks that can support rigorous analysis.

Study Design Questions

If you are uncertain about the appropriate sample size, randomization scheme, or experimental design, consult a statistician before collecting data. The NC3Rs Experimental Design Assistant provides guidance on experimental design for animal research.

Welfare and Safety Context

The choice of statistical test has implications for animal welfare and research ethics. Using an inappropriate statistical test can lead to incorrect conclusions about treatment effects, which can have consequences for animal welfare.

Avoiding False Positives

An inflated Type I error rate means that researchers may conclude that a treatment has an effect when it does not. This can lead to unnecessary follow-up studies, wasted resources, and potentially harmful treatments being recommended. Using Welch's t-test by default helps control Type I error rates when variances are unequal 20.

Avoiding False Negatives

A test with low power may fail to detect a real treatment effect. This can lead to effective treatments being discarded. Welch's t-test maintains high power when sample sizes are adequate 11.

Reducing Animal Numbers

Using an appropriate statistical test with adequate power can help reduce the number of animals needed in research studies. The NC3Rs Experimental Design Assistant provides tools for optimizing experimental design and sample size calculations.

Ethical Reporting

Researchers have an ethical obligation to report their statistical methods accurately and completely. The EQUATOR Network provides reporting guidelines that help ensure transparency in research reporting.

Frequently Asked Questions

What is the main difference between Welch's t-test and Student's t-test?

Welch's t-test does not assume that the two populations have equal variances, while Student's t-test does. Welch's test uses separate variance estimates for each group and adjusts the degrees of freedom to account for unequal variances. This makes Welch's test more reliable when variances differ between groups.

When should I use Welch's t-test instead of Student's t-test?

Use Welch's t-test by default when comparing two independent group means. Simulation evidence shows that Welch's test provides better control of Type I error rates when the homogeneity of variance assumption is not met, and it loses little robustness compared to Student's t-test when the assumptions are met 20.

Does Welch's t-test require equal sample sizes?

No, Welch's t-test does not require equal sample sizes. The test accommodates unequal sample sizes and unequal variances simultaneously. However, very unequal sample sizes combined with skewed distributions can cause inflated Type I error rates 23.

What are the assumptions of Welch's t-test?

Welch's t-test assumes independence of observations, approximate normality of the data within each group, and that the two samples are independent of each other. It does not require equal variances between groups.

How do I report the results of Welch's t-test?

Report the test statistic, degrees of freedom, p-value, and an appropriate effect size with confidence interval. For Welch's t-test, use Hedges' g* based on the non-pooled standard deviation instead of Cohen's d based on the pooled standard deviation 21.

Can Welch's t-test be used with very small sample sizes?

Welch's t-test can show Type I error inflation with very small sample sizes even when assumptions are met 26. For samples of five or fewer per group, the bootstrap test is recommended to avoid Type I error inflation, though this comes at the cost of lower power 11.

What should I do if my data are not normally distributed?

If the data are moderately nonnormal, Welch's t-test may still be acceptable, especially with larger sample sizes. If the data are severely skewed, consider a permutation-based Welch's test, the Yuen-Welch test, or other robust alternatives 23.

Is Welch's t-test the same as the unequal variances t-test?

Yes, Welch's t-test is also called the unequal variances t-test or the separate-variances t-test. It is sometimes referred to as the Welch-Satterthwaite test because of the degrees of freedom correction.

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References and Further Reading

• Research Data Framework. National Institute of Standards and Technology.
• EQUATOR Network. EQUATOR Network.
• Experimental Design Assistant. NC3Rs.
• NCBI Literature Resources. National Center for Biotechnology Information.
• PubMed. National Library of Medicine.
• Evaluation of whole effluent toxicity data characteristics and use of Welch's T-test in the test of significant toxicity analysis.. Environmental toxicology and chemistry, 2013.
• The Statistical Fragility of Functional Outcomes for Arthroscopic Rotator Cuff Repair With and Without Acromioplasty: A Systematic Review and Meta-analysis.. The American journal of sports medicine, 2025.
• Solutions to the Behrens-Fisher problem.. Computer methods and programs in biomedicine, 2003.
• Measuring the Effectiveness of a Multicomponent Program to Manage Academic Stress through a Resilience to Stress Index.. Sensors (Basel, Switzerland), 2023.
• The influence of age and gender on perception of orofacial esthetics among laypersons in Switzerland.. Journal of esthetic and restorative dentistry : official publication of the American Academy of Esthetic Dentistry ... [et al.], 2022.
• Murine genetic models of obesity: type I error rates and the power of commonly used analyses as assessed by plasmode-based simulation.. International journal of obesity (2005), 2020.
• Movement Patterns of Polish National Paralympic Team Wheelchair Fencers with Regard To Muscle Activity and Co-Activation Time.. Journal of human kinetics, 2022.
• Cortical Tubers' Transformation in Pediatric Patients Diagnosed with Tuberous Sclerosis Complex: A Retrospective Longitudinal MRI Analysis.. Journal of clinical medicine, 2025.
• Effect of locally transplanted bone marrow derived mesenchymal stem cells on the lingual filiform and fungiform papillae of ovariectomized albino rat model.. 2026.
• Impact of educational level on medical students' ability to learn abdominal point-of-care ultrasonography: a pilot study.. 2026.
• An empirical study on exercise addiction and grit among college students: based on 35 exercise sessions.. 2026.
• In-Person vs. Virtual: A Comparative Study of Teaching Methods in Nutritional Medicine.. 2026.
• The Predictors of COMLEX-USA (Comprehensive Osteopathic Medical Licensing Examination of the United States) Level 1 Success Among Osteopathic Medical Students: The Role of Study Habits and Post-baccalaureate Background.. 2025.
• Welch’s t test is more sensitive to real world violations of distributional assumptions than student’s t test but logistic regression is more robust than either. Statistical Papers, 2024.
• Why Psychologists Should by Default Use Welch's t-test Instead of Student's t-test with Unequal Group Sizes. 2017.
• Why Hedges’ g*s based on the non-pooled standard deviation should be reported with Welch’s t-test. 2021.
• Taking Parametric Assumptions Seriously: Arguments for the Use of Welch’s F-test instead of the Classical F-test in One-Way ANOVA. International Review of Social Psychology, 2019.
• Neither the t-test nor Welch’s test: a case for robust alternatives in two-sample comparisons with non-ideal data. Quality &amp, Quantity, 2026.
• Comparison of Student - t, Welch s t, and Mann - Whitney U Tests in Terms of Type I Error Rate and Test Power. Selcuk journal of agriculture and food sciences, 2023.
• Challenging Assumptions of Normality in AES s-Box Configurations under Side-Channel Analysis. Journal of Cybersecurity and Privacy, 2023.
• Type I error inflation of the separate-variances Welch t test with very small sample sizes when assumptions are met. Journal of Modern Applied Statistical Methods, 2011.

This article is educational and does not replace institutional policy, professional advice, or applicable safety and regulatory requirements.