Bayesian statistics

Bayesian T-Test Calculator

Quantify the evidence for and against a difference in means. The JZS Bayes factor (Rouder et al., 2009) compares H₁, an effect with a Cauchy prior on Cohen’s δ, against H₀, no effect, and the posterior shows the plausible effect sizes.

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

Design

Dependent variable

Grouping variable

Group 1

Group 2

Prior width on δ (Cauchy scale)

Alternative hypothesis

Results update automatically when you change an option.

Results

90 rows · example data

Table 1

Group statistics

Group	n	M	SD	SE	95% CI
Female	58	66.10	10.38	1.36	[63.37, 68.83]
Male	32	63.25	12.13	2.14	[58.88, 67.62]

Table 2

Bayesian t-test

Hypothesis test	BF₁₀	BF₀₁	ln BF₁₀	Evidence	t	df
H₁: δ ≠ 0 vs H₀: δ = 0	0.418	2.392	-0.872	anecdotal evidence for H₀	1.175	88

Note. Cauchy prior on δ with scale r = 0.707. BF₁₀ > 1 favours H₁; BF₀₁ = 1 / BF₁₀ favours H₀.

Table 3

Posterior of the effect size δ

Median	Mean	95% credible interval	95% HDI	P(δ > 0 \| data)	Sample Cohen’s d
0.224	0.226	[-0.178, 0.639]	[-0.180, 0.636]	0.862	0.259

Note. Posterior under the two-sided H₁. The posterior median is shrunk slightly towards zero by the prior.

Prior and posterior of δ

Bayes factor robustness check

Table 4

Robustness of BF₁₀

Prior	r	BF₁₀	Evidence
Medium	0.707	0.418	anecdotal evidence for H₀
Wide	1	0.319	moderate evidence for H₀
Ultrawide	1.414	0.236	moderate evidence for H₀

BF₁₀0.418

Evidenceanecdotal for H₀

Posterior median δ0.2295% CrI [-0.18, 0.64]

Interpretation

For Female vs Male on Post_Score, the Bayes factor is BF₁₀ = 0.418: the data are 2.39 times more likely under H₀ (no effect) than under H₁, which is anecdotal evidence for H₀. The posterior median effect size is δ = 0.22 (95% credible interval [-0.18, 0.64]).

A Bayesian independent-samples t-test with a Cauchy prior (r = 0.707) gave BF₁₀ = 0.418, anecdotal evidence for H₀; posterior median δ = 0.22, 95% CrI [-0.18, 0.64].

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 T-Test Calculator [Web application]. FreeStatisticalTools.com. https://freestatisticaltools.com/bayesian-t-test/

BibTeX

@software{xavier_bayesianttest_2026,
  author = {Xavier, Bibin},
  title = {Bayesian T-Test Calculator},
  publisher = {FreeStatisticalTools.com},
  year = {2026},
  url = {https://freestatisticaltools.com/bayesian-t-test/},
  note = {Web application}
}

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

What is a Bayesian t-test?

A Bayesian t-test compares two hypotheses: H₀, that the mean difference is zero, and H₁, that there is an effect whose standardized size δ follows a Cauchy prior. The Bayes factor BF₁₀ is how much more likely the observed data are under H₁ than under H₀. Unlike a p-value, it can provide evidence for the null hypothesis, and the posterior distribution of δ shows which effect sizes are plausible.

When to use a Bayesian t-test

Use it instead of, or alongside, a classical t-test when you want to state how strongly the data support an effect or the absence of an effect. Unlike a p-value, a Bayes factor can show evidence for the null hypothesis.

Assumptions

The same assumptions as the t-test: independent observations and roughly normal data (or differences, for paired designs).
The Cauchy prior width r encodes how large effects are expected to be; check robustness to r.

How to set up your data

Independent design: one outcome column and one group column. Paired design: two columns measured on the same cases. One-sample design: one column and a test value.

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, grouping variable, group 1 and group 2.
Set the options. Fill in design, prior width on δ (cauchy scale) and alternative hypothesis.
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 t-test in APA style

“A Bayesian independent-samples t-test (Cauchy prior r = 0.707) indicated moderate evidence for a difference, BF₁₀ = 4.82; posterior median δ = 0.46, 95% CrI [0.05, 0.88].”

Frequently asked questions

How do I interpret a Bayes factor?

BF₁₀ = 5 means the data are five times more likely under H₁ than H₀. Values of 1–3 are anecdotal, 3–10 moderate, 10–30 strong, 30–100 very strong and above 100 extreme evidence (Lee & Wagenmakers, 2013). BF₀₁ = 1 / BF₁₀ expresses evidence for H₀.

What prior does the Bayesian t-test use?

A Cauchy distribution on the effect size δ with scale r = 0.707 by default (the JZS prior of Rouder et al., 2009), the same default as JASP and the BayesFactor R package. A robustness plot shows how BF₁₀ changes with r.

Can a Bayes factor support the null hypothesis?

Yes. BF₁₀ below 1/3 (BF₀₁ above 3) is moderate evidence that there is no effect, which a non-significant p-value cannot show.

What is the difference between a credible interval and a confidence interval?

A 95% credible interval contains the effect size with 95% probability given the data and prior; a confidence interval is defined by long-run coverage of repeated samples.

Does this match JASP?

Yes. The two-sided JZS Bayes factor matches JASP, the BayesFactor package and pingouin to at least five significant digits.

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