Welch's ANOVA

Welch's ANOVA builds an F value that weights each group by its own variance and its own sample size, and pulls the within-groups degrees of freedom down accordingly. The classical analysis of variance pools one common variance estimate across all groups, and when the variances genuinely differ that pooled estimate understates the influence of small groups with wide spread, pushing the p value in the wrong direction. The correction compensates by reducing the degrees of freedom to a fractional number, which is why the output shows a value such as 58.42 rather than a whole number. The panel offers the correction in two forms: the Welch procedure by default and the Brown-Forsythe procedure as an alternative. YouReply Analyze also displays the p value from Levene's test for information, without letting that result change the test it runs.

• Does average order value differ across three sales regions whose sample sizes are very unequal?
• Do four occupational groups differ in average monthly spending when their spending spreads are clearly unequal?
• Do average session lengths differ between users arriving through three channels?
• Do exam score means differ across five campuses of markedly different size?

• When the means of three or more independent groups are compared and equal variances cannot be defended.
• When group sizes are clearly unbalanced, the situation in which unequal variances damage the classical F test most.
• When Levene's test came out significant during a classical one-way ANOVA run and the same question needs a sturdier test.
• When the dependent variable is continuous and the group distributions are roughly symmetric, with spread being the real problem.

• Dependent variable: a single continuous numeric column, with its measurement level set on the Variable tab.
• Independent variable: a single grouping column carrying three or more categories.
• One row per participant; if several conditions were measured on the same person, this method does not apply.

Independence of observations

Each participant belongs to one group only and observations do not influence each other. This is secured by the sampling design and cannot be repaired afterwards.

Continuous dependent variable

Because the test compares group means, the dependent variable is expected to be measured at interval or ratio level.

Normality within groups

The correction handles unequal variances, not a distorted distributional shape. Each group is still expected to be roughly symmetric.

Equal variances are not required

The whole point of the method is the case where group variances may differ, so equality is not a precondition.

• Independence of observations: The panel does not check this assumption automatically; the researcher evaluates it.
• Continuous dependent variable: The method picker asks for a numeric column in this field; selections that do not fit dim the method and the reason is stated in plain language.
• Normality within groups: The panel does not test normality while running this method; inspecting the distributions means running the normality tests separately.
• Equal variances are not required: Levene's test is computed and reported with its p value and a flag for whether the variances count as equal. That is informational only: the corrected test runs whatever the result.

1. 1Upload your data file and check on the Data tab that the numeric columns were read as numbers.
2. 2On the Variable tab mark the grouping column as categorical and the measurement to be compared as continuous.
3. 3Pick Welch's ANOVA in the method picker on the Analysis tab; the data requirements box shows the columns with three or more categories.
4. 4In the parameter form name the grouping column and the value column, switching the correction to Brown-Forsythe if you need it.
5. 5Run the analysis. The F value, both degrees of freedom, the p value, the group statistics table and the Levene information open as separate sections.
6. 6Download the group means bar chart as a PNG and the result tables as Excel; the computation credits card names the library call and its version.

Corrected F statistic

The F value computed with group variances as weights, coming from either the Welch or the Brown-Forsythe procedure depending on the correction chosen.

Between-groups degrees of freedom

The number of groups minus one, obtained exactly as in the classical analysis of variance.

Within-groups degrees of freedom

The fractional value produced by the correction, written out with its decimal places when reporting.

p value and significance flag

The probability under the hypothesis of equal group means, plus a marker for whether the result passes the threshold.

Group statistics

Sample size, mean and standard deviation for each group; the size of the variance gap becomes visible in this table.

Levene information

Levene's p value and whether the variances count as equal; it does not alter the procedure that runs.

Group means chart

A bar chart placing the means side by side is drawn above the result sections and can be saved as a PNG.

No effect size is computed

This method returns no effect size at all: neither eta squared nor omega squared appears in the output. If you need to report a proportion of variance explained, run the classical one-way ANOVA method on the same variables and take its effect size, or derive your own measure from the means and standard deviations in the group statistics table. Either way you are expected to state which test produced the p value and which calculation produced the effect size.

No confidence interval comes back from this method, neither for the group means nor for differences between them. The means and standard deviations in the group statistics table are point estimates, and an interval has to be computed separately if one is needed.

The numbers below are a representative example, not data from a real study or a real user.

Research question

In a customer study, do users on three subscription tiers differ in average monthly usage hours when the tiers differ markedly in both size and spread?

Variables

Dependent variable: monthly usage in hours (continuous) · Independent variable: subscription tier (three categories)

Example result

Group statistics: basic tier n = 112, mean 3.91, standard deviation 0.62; mid tier n = 47, mean 4.38, standard deviation 1.14; top tier n = 29, mean 4.61, standard deviation 1.31. Total observations: 188. Levene's p = 0.004, so the variances are not treated as equal. F(2, 58.42) = 6.83, p = 0.002.

Interpretation

With standard deviations differing by more than a factor of two, leaning on the classical F test would have been hard to justify, and the corrected test finds a significant difference in means across tiers. The within-groups degrees of freedom falling to 58.42 against 188 observations shows how cautious the correction is. Since no effect size comes with this output, the practical size of the gap has to be judged from the 0.7 hour spread between means and from the standard deviations.

Research question: Does mean income differ by education level even when the variances are unequal?

Grouping variable

education

Measured variable

income

Correction applied

welch

Computation credits: scipy 1.18.0 · statsmodels 0.14.6 · scikit-learn 1.9.0 · numpy 2.5.1 · pandas 3.0.5 · semopy 2.3.11 · 89fc29a · Data seed: 20260914

Monthly usage differed significantly by subscription tier, F(2, 58.42) = 6.83, p = .002, with the correction for unequal variances; the means were 3.91, 4.38 and 4.61 hours (Levene's p = .004).

An example sentence close to APA style; the numbers are representative.

• Unnecessary with two groups, where the independent samples t-test with its unequal-variance option answers the question directly.
• Not for designs with repeated measurements on the same participants, since independence between groups is a basic requirement here.
• Because the output carries no effect size, running this method alone falls short for a reviewer who asks for a proportion of variance explained.
• A significant result does not say which tiers separate, and this method produces no pairwise comparisons.
• When distributions are strongly skewed or heavy with outliers the correction does not help, because the problem is shape rather than spread.
• Once some groups drop to single-digit sample sizes the fractional degrees of freedom shrink sharply and power erodes with them.

• One-Way ANOVAWhy: When the variances can be treated as equal; the classical F test has more power and also returns eta squared and omega squared.
• Kruskal-Wallis H TestWhy: When the real issue is distributional shape or an ordinal measurement level rather than spread; the comparison rests on ranks instead of means.
• Independent Samples t-TestWhy: When the number of categories being compared comes down to two; unequal variances are covered there as well.

• One-Way ANOVA
• Independent Samples t-Test
• Kruskal-Wallis H Test

Levene's test was not significant, can I still use this method?

You can. The correction still yields a valid result when the variances are equal, at the cost of a little power. Levene's test, meanwhile, frequently misses unequal variances in small samples, so going straight to the corrected test is a defensible choice whenever group sizes are unbalanced.

Why are the degrees of freedom not a whole number?

The correction weights each group's variance and sample size separately in order to approximate the distribution of the test statistic. A non-integer within-groups value is the consequence of that approximation. When reporting, keep the decimal places shown in the panel rather than rounding to an integer.

How do I choose between Welch and Brown-Forsythe?

Both deal with unequal variances. The Welch procedure is the dominant choice in the literature and usually keeps the error rate better protected. The Brown-Forsythe procedure is an option when the largest variance belongs to the largest group; whichever you pick, name the procedure in your write-up.

How can I see which tiers differ?

This method runs no pairwise comparisons. With unequal variances the appropriate pairwise procedure is Games-Howell, which the panel computes inside the post-hoc block when the classical one-way ANOVA comes out significant; if you read that block, say which test your overall verdict came from.

• Welch, B. L. (1951). On the Comparison of Several Mean Values: An Alternative Approach
• Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics
• Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2005). Applied Linear Statistical Models
• statsmodels.stats.oneway.anova_oneway documentation