FreeStatisticalTools.com
Browse tools
Bayesian statistics

Bayesian Correlation Calculator

Test whether two variables are correlated with a Bayes factor (Ly, Verhagen & Wagenmakers, 2016), using a uniform (stretched beta) prior on ρ, and see the posterior distribution of the population correlation.

Free · No sign-upReplaces JASPWorks with your data
Updated · By Bibin Xavier

Data 90 × 12

Paste from Excel, or drop a CSV, TSV or Excel file here.
Example data 90 rows, 12 variables. Data stays in this browser tab.
IDtext · 90 levelsGrouptext · 3 levelsGendertext · 2 levelsAgenumericHours_StudynumericAnxietynumericMotivationnumericPre_ScorenumericPost_ScorenumericExam_ScorenumericPassednumeric · codedSatisfactionnumeric · coded
S001ControlMale2311.2294.660626712
S002WorkshopFemale233.9283.554705402
S003OnlineFemale1810.132455576603
S004ControlMale2613.6413.552555903
S005WorkshopFemale229.7273.656636504
S006OnlineFemale207.9323.865826502
S007ControlFemale206.328372836103
S008WorkshopMale228293.164726213

Variables and options

Results update automatically when you change an option.

Results

90 rows · example data
Table 1

Bayesian Pearson correlation

PairnPearson rBF10BF01EvidencePosterior median ρ95% credible interval
Hours_Study – Exam_Score90.36255.6050.018very strong evidence for H₁.353[.161, .520]

Note. H₁: ρ ≠ 0, stretched beta prior with κ = 1. Classical test for comparison: p < .001.

Prior and posterior of ρ

Exam_Score by Hours_Study

BF₁₀55.605
Posterior median ρ.35395% CrI [.161, .520]
P(ρ > 0 | data)1.000

Interpretation

The correlation between Hours_Study and Exam_Score is r = .362 (n = 90). BF10 = 55.605, very strong evidence for H₁. Given the data, the population correlation most likely lies between .161 and .520 (95% credible interval).

A Bayesian Pearson correlation (stretched beta prior, κ = 1) gave r = .362, BF10 = 55.605, posterior median ρ = .353, 95% CrI [.161, .520].

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 Correlation Calculator [Web application]. FreeStatisticalTools.com. https://freestatisticaltools.com/bayesian-correlation/

BibTeX 

@software{xavier_bayesiancorrelation_2026,
  author = {Xavier, Bibin},
  title = {Bayesian Correlation Calculator},
  publisher = {FreeStatisticalTools.com},
  year = {2026},
  url = {https://freestatisticaltools.com/bayesian-correlation/},
  note = {Web application}
}

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

What is a Bayesian correlation?

A Bayesian correlation test weighs the evidence for a non-zero population correlation ρ against ρ = 0. Following Ly, Verhagen and Wagenmakers (2016) and JASP, the prior on ρ is a stretched beta distribution, uniform from −1 to 1 by default. The result is a Bayes factor and a posterior distribution with a credible interval for ρ.

When to use a Bayesian correlation

Use it to quantify the evidence for or against a linear association between two continuous variables, including evidence that two variables are unrelated.

Assumptions

  • Bivariate normality and a linear relationship.
  • Independent pairs of observations.

How to set up your data

Two numeric columns, one row per case.

How to use this tool

  1. 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.
  2. Choose the variables. Select the variable x and variable y.
  3. Set the options. Fill in prior width κ (stretched beta) and alternative hypothesis.
  4. 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 correlation in APA style

“A Bayesian correlation showed extreme evidence for a positive association between study hours and exam score, r = .62, BF₁₀ = 2.4 × 10⁸, 95% CrI [.49, .72].”

Frequently asked questions

How do I report a Bayesian correlation?

Give r, the Bayes factor and the credible interval, for example “r = .30, BF₁₀ = 1.09, 95% CrI [.00, .53]”, and state the prior (stretched beta, κ = 1).

What does κ (kappa) mean?

κ sets the width of the prior on ρ. κ = 1 is uniform; smaller values concentrate prior mass on small correlations.

Can I test a directional hypothesis?

Yes. Choose ρ > 0 or ρ < 0 to get a one-sided Bayes factor.

Does it match JASP?

Yes, the Bayes factors match JASP and pingouin.

Related tools

Bayesian T-Test CalculatorJZS Bayes factor t-test for independent, paired or one-sample designs, with the posterior of the effect size and a robustness check.Bayesian One-Way ANOVABayes factor for differences among group means (JZS g-prior), with Bayesian post hoc comparisons and a robustness check.Bayesian Linear RegressionBayes factors for regression models and predictors, posterior inclusion probabilities, model averaging and coefficient credible intervals.Bayesian A/B Test CalculatorProbability that variant B beats A, expected uplift with a credible interval and expected loss, from conversions and visitors.Bayesian Binomial TestBayes factor for a proportion against a test value, with the Beta posterior, credible interval and HDI.Bayes Factor CalculatorConvert a published t statistic, correlation or BIC values into a Bayes factor, with a robustness plot.

Browse all categories

Data cleaning and preparation · Describe and explore · Compare means · Nonparametric tests · Correlation and regression · Bayesian statistics · Categorical data · Reliability and agreement · Multivariate analysis · Structural equation modeling · Econometrics and time series · Panel data and causal inference · Marketing and decision research · Survival, clinical and meta-analysis · Study planning and calculators · PLS-SEM