Results summary
| Test | Statistic | Degrees of freedom | P-value | Decision |
|---|
Dataset summary
Recommended interpretation
Numerical stability
Descriptive statistics
| Variable | Count | Mean | Median | Std. dev. | Variance | Minimum | Maximum | Range | Skewness | Kurtosis |
|---|
Matrix results
Mean vector
Covariance matrix
Correlation matrix
Inverse covariance matrix
Eigenvalue summary
Matrix diagnostics
Multivariate outlier analysis
| Observation | Mahalanobis D² | Chi-square cutoff | Percentile | Status |
|---|
Graphs and diagnostics
Dataset and test settings
Formula used
The calculator estimates a mean vector and covariance matrix. Mahalanobis distance measures each observation’s joint separation. Larger values may indicate multivariate outliers.
The null hypothesis states joint normality. Small p-values challenge that assumption. Visual diagnostics should support every final conclusion.
How to use this calculator
- Paste numeric data with observations arranged in rows.
- Keep variables in separate columns with clear names.
- Select tests, preprocessing, precision, and significance level.
- Run the analysis and inspect all warnings carefully.
- Review tests, matrices, plots, and detected outliers.
- Export the report or cleaned dataset when finished.
Example data table
| Height | Weight | Age |
|---|---|---|
| 170 | 65 | 28 |
| 165 | 59 | 31 |
| 180 | 80 | 35 |
| 175 | 72 | 29 |
Assumptions and limitations
Multivariate tests react differently to sample size. Outliers can dominate several statistics. Always combine formal and visual evidence.
Browser calculations use stable numerical approximations. Results may differ slightly from specialist software. Confirm high-stakes analyses independently before reporting.
Uploaded files remain inside your browser session. This page sends no dataset server-side. Local processing helps protect confidential information.
Frequently asked questions
What is multivariate normality?
It describes joint normal behavior across variables. Every linear combination should appear normally distributed. This assumption supports many multivariate procedures.
Which test should I prefer?
Henze-Zirkler provides broad general-purpose detection. Mardia separates skewness and kurtosis departures. Review several methods before deciding.
Does a large p-value prove normality?
No test can prove exact normality. A large p-value shows insufficient contrary evidence. Diagnostics still remain important for interpretation.
Why does sample size matter?
Small samples may hide real departures. Large samples detect minor harmless deviations. Practical context should guide final decisions.
What causes a singular covariance matrix?
Perfectly related variables create singularity. Constant columns cause the same issue. Too many variables can also destabilize inversion.
Should outliers be removed automatically?
Automatic deletion can distort valid findings. Investigate measurement and data-entry causes first. Report every exclusion transparently and carefully.
When should variables be standardized?
Standardization helps when scales differ greatly. Mahalanobis distance already uses covariance scaling. Transformations may still improve distribution shape.
Can missing values be analyzed?
Complete-case analysis removes incomplete observations. Mean imputation preserves rows but reduces variability. Advanced studies should use principled imputation methods.
Why do tests disagree?
Each method targets different distribution departures. Power also changes with dataset dimensions. Use disagreement as a diagnostic signal.