Hotelling's T² Test Calculator

Hotelling's T² test is a multivariate statistical test used to compare the mean vectors of two groups simultaneously across multiple correlated outcome variables. It is the multivariate extension of the two-sample Student's t-test, developed by Harold Hotelling in 1931. It is widely used in psychology, biology, finance, and quality control.

A standard t-test compares means on a single variable between two groups. Hotelling's T² extends this to p variables simultaneously, taking into account the correlations among them. Running multiple separate t-tests inflates the Type I error rate, whereas Hotelling's T² controls it by testing all variables jointly in a single test.

The null hypothesis (H₀) states that the mean vectors of the two groups are equal: μ₁ = μ₂. The alternative hypothesis (H₁) states that they are not equal: μ₁ ≠ μ₂. Rejecting H₀ means there is significant evidence that the groups differ on at least one of the variables.

The T² statistic is converted to an F statistic using the formula: F = [(n₁ + n₂ − p − 1) / (p(n₁ + n₂ − 2))] × T². This F statistic follows an F-distribution with degrees of freedom df₁ = p and df₂ = n₁ + n₂ − p − 1, allowing a p-value to be calculated from standard F-distribution tables.

The test assumes that the data come from multivariate normal distributions, that the two groups have equal population covariance matrices (homogeneity of covariance), and that observations are independent. When sample sizes are large, the test is reasonably robust to mild violations of normality.

The pooled covariance matrix Sₚ combines the within-group covariance matrices of both samples, weighted by their degrees of freedom: Sₚ = [(n₁−1)S₁ + (n₂−1)S₂] / (n₁ + n₂ − 2). It estimates the common population covariance matrix under the assumption of equal covariances across groups.

If the p-value is less than your chosen significance level α (commonly 0.05), you reject the null hypothesis and conclude that the mean vectors of the two groups are significantly different. If p ≥ α, you fail to reject H₀ and do not have sufficient evidence of a difference between group means.

When assumptions are violated or you have more than two groups, consider MANOVA (Multivariate Analysis of Variance). When covariance matrices are unequal across groups, the James test or Yao's test may be more appropriate. For non-normal data, permutation-based multivariate tests can provide robust alternatives.