Calculator Inputs
Use aligned variable order across labels, vectors, and covariance matrix rows and columns.
Example Data Table
This example matches the default values loaded into the calculator.
| Variable | Target Mean | Observed Sample Mean | Variance |
|---|---|---|---|
| Copper | 10.0 | 10.2 | 0.40 |
| Zinc | 15.0 | 14.8 | 0.60 |
| Lead | 8.0 | 8.5 | 0.50 |
Example covariance off-diagonal terms: Cu-Zn = 0.12, Cu-Pb = 0.05, Zn-Pb = 0.08.
Formula Used
T² = n(x̄ − μ₀)ᵀS⁻¹(x̄ − μ₀)
F = [(n − p) / (p(n − 1))] × T²
T²critical = [p(n − 1) / (n − p)] × F1−α, p, n−p
- x̄ is the sample mean vector.
- μ₀ is the hypothesized target mean vector.
- S is the sample covariance matrix.
- n is the sample size.
- p is the number of chemistry variables.
How to Use This Calculator
- Enter variable labels for each chemistry measurement.
- Provide the sample size used to estimate the mean vector.
- Enter the observed sample mean values in matching order.
- Enter the target or hypothesized mean vector.
- Paste the covariance matrix with one row per line.
- Choose alpha and display precision.
- Submit the form to view T2, the F statistic, p-value, and the decision.
- Review the contribution table and Plotly chart to see which variables drive the result.
- Use the CSV and PDF buttons to export the result summary.
Frequently Asked Questions
1. What does Hotelling T2 measure?
Hotelling T2 measures how far a multivariate sample mean is from a target mean after accounting for variance and covariance together. It is useful when several chemistry variables must be judged as a single profile rather than as separate univariate tests.
2. Why is covariance important here?
Covariance captures how chemistry variables move together. Ignoring covariance can overstate or understate the real multivariate distance. Hotelling T2 uses the inverse covariance matrix so correlated measurements are interpreted correctly.
3. Can I use this for laboratory quality control?
Yes. It is helpful for monitoring batches, method validation studies, instrument checks, and multielement composition targets. It works best when the mean vector and covariance matrix come from representative, stable laboratory data.
4. What if my covariance matrix is not invertible?
A singular covariance matrix cannot be used directly because the Hotelling T2 formula requires S⁻¹. This often happens when variables are redundant or sample size is too small. Remove highly collinear variables or improve data quality.
5. Why must sample size exceed the number of variables?
The F transformation used for inference needs n greater than p. When sample size is too small, the test becomes unstable and the reference distribution is not valid for the standard critical-value calculation.
6. How should I interpret the contribution chart?
The chart shows each variable's signed contribution to the total T2 value. Larger magnitudes indicate variables with stronger influence. Because covariance is included, a contribution can be negative while the total T2 remains positive.
7. Is this calculator for one-sample or two-sample testing?
This page performs the one-sample Hotelling T2 test. It compares one observed multivariate mean vector against a known or hypothesized target vector, which suits many chemistry control and compliance workflows.
8. What input format should I follow?
Keep the same variable order everywhere. Labels, sample means, target means, and covariance matrix rows and columns must match exactly. Any mismatch changes the matrix algebra and will produce incorrect multivariate results.