Bayesian statistics

Bayesian One-Way ANOVA

Compare the means of two or more groups with a default Bayes factor (Rouder et al., 2012). BF₁₀ tells you how much more likely the data are if the groups differ than if all means are equal.

Free · No sign-upReplaces JASP · SPSSWorks with your data

Updated 13 October 2026 · By Bibin Xavier

Data 90 × 12

Paste from Excel, or drop a CSV, TSV or Excel file here.Upload file

Example data 90 rows, 12 variables. Data stays in this browser tab.

IDtext · 90 levels	Grouptext · 3 levels	Gendertext · 2 levels	Agenumeric	Hours_Studynumeric	Anxietynumeric	Motivationnumeric	Pre_Scorenumeric	Post_Scorenumeric	Exam_Scorenumeric	Passednumeric · coded	Satisfactionnumeric · coded
S001	Control	Male	23	11.2	29	4.6	60	62	67	1	2
S002	Workshop	Female	23	3.9	28	3.5	54	70	54	0	2
S003	Online	Female	18	10.1	32	4	55	57	66	0	3
S004	Control	Male	26	13.6	41	3.5	52	55	59	0	3
S005	Workshop	Female	22	9.7	27	3.6	56	63	65	0	4
S006	Online	Female	20	7.9	32	3.8	65	82	65	0	2
S007	Control	Female	20	6.3	28	3	72	83	61	0	3
S008	Workshop	Male	22	8	29	3.1	64	72	62	1	3

Variables and options

Dependent variable

Factor (groups)

Prior scale on effects

Results update automatically when you change an option.

Results

90 rows · example data

Table 1

Descriptive statistics

Group	n	M	SD	SE	95% CI
Control	30	61.03	10.34	1.89	[57.17, 64.89]
Online	30	65.53	9.75	1.78	[61.89, 69.17]
Workshop	30	68.70	11.93	2.18	[64.25, 73.15]

Table 2

Bayesian ANOVA

Model	BF₁₀	BF₀₁	ln BF₁₀	Evidence	Classical F	p
Group vs null (equal means)	2.086	0.479	0.735	anecdotal evidence for H₁	3.88	.024

Note. JZS fixed-effects prior on standardized effects with scale r = 0.5 (Rouder et al., 2012), the default in JASP and the BayesFactor package. The null model has one common mean.

Table 3

Bayesian post hoc comparisons

Comparison	Mean difference	BF₁₀	Evidence	Median δ	95% CrI for δ
Control – Online	-4.50	0.916	anecdotal evidence for H₀	-0.379	[-0.876, 0.092]
Control – Workshop	-7.67	4.686	moderate evidence for H₁	-0.598	[-1.118, -0.103]
Online – Workshop	-3.17	0.446	anecdotal evidence for H₀	-0.242	[-0.725, 0.220]

Note. Bayesian independent-samples t-tests with r = 0.707 for each pair. These Bayes factors are not adjusted for multiplicity; interpret them as a set.

Group means with 95% CI

Bayes factor robustness check

BF₁₀2.086

Evidenceanecdotal for H₁

Groups3N = 90

Interpretation

The Bayes factor for a difference among the 3 Group groups is BF₁₀ = 2.086, anecdotal evidence for H₁. The data are 2.09 times more likely if the group means differ.

A Bayesian one-way ANOVA (r = 0.5) showed anecdotal evidence for H₁ for an effect of Group on Post_Score, BF₁₀ = 2.086.

How to cite this tool

Using these results in a thesis, journal article, report or class assignment? Please cite this free tool so others can find it. It keeps the site free.

APA 7

Xavier, B. (2026). Bayesian One-Way ANOVA [Web application]. FreeStatisticalTools.com. https://freestatisticaltools.com/bayesian-anova/

BibTeX

@software{xavier_bayesiananova_2026,
  author = {Xavier, Bibin},
  title = {Bayesian One-Way ANOVA},
  publisher = {FreeStatisticalTools.com},
  year = {2026},
  url = {https://freestatisticaltools.com/bayesian-anova/},
  note = {Web application}
}

Suggested sentence for your methods section: “The analysis was carried out with the free Bayesian One-Way ANOVA tool on FreeStatisticalTools.com (Xavier, 2026).”

What is a Bayesian ANOVA?

A Bayesian ANOVA compares a model in which group means differ with a model in which they are all equal. It uses the default JZS prior on standardized effects of Rouder, Morey, Speckman and Province (2012), as in JASP and the BayesFactor package, and returns a Bayes factor that can favour either model.

When to use a Bayesian ANOVA

Use it to compare three or more group means when you want evidence for or against group differences, for example to support a claim that conditions did not differ.

Assumptions

Independent observations, roughly normal residuals and similar variances across groups.

How to set up your data

One numeric outcome column and one column with the group labels.

How to use this tool

Enter your data. Paste columns from Excel, Google Sheets or SPSS with variable names in the first row, or upload a CSV or Excel file. The example data shows the expected layout.
Choose the variables. Select the dependent variable and factor (groups).
Set the options. Fill in prior scale on effects.
Read the results. Results update automatically. Read the plain-English interpretation, then use Copy table to paste APA-style tables into Word and Copy for the results sentence.

How to report a Bayesian ANOVA in APA style

“A Bayesian ANOVA indicated strong evidence for differences between teaching methods, BF₁₀ = 18.6.”

Frequently asked questions

How is a Bayesian ANOVA different from a classical ANOVA?

Instead of a p-value for the null hypothesis, it gives a Bayes factor comparing the evidence for group differences with the evidence for equal means, so it can also support the null.

What prior scale should I use?

The default r = 0.5 (“medium”) suits most research. Wider priors expect larger effects and are more cautious about small differences.

Are post hoc Bayes factors corrected for multiple comparisons?

No. They are Bayesian t-tests for each pair; interpret them as a set, or compare models with prior odds that reflect the number of comparisons.

Does it match JASP?

With two groups, the result equals the Bayesian t-test with r = 0.707, as in JASP and BayesFactor.

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