# Hotelling's T-squared distribution

**Hotelling's T-squared distribution** is the sampling distribution of a scalar statistic, Hotelling's \( T^2 \), that measures the squared statistical distance of a multivariate sample mean (or observation) from a hypothesized value, taking the correlations among variables into account. It is the multivariate counterpart of Student's t distribution: for a single variable, \( T^2 \) reduces to the square of the ordinary t statistic, and in general it is proportional to an F distribution.<sup>[1](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm)</sup><sup> • </sup><sup>[2](https://cir.nii.ac.jp/crid/1360011144703680000)</sup>

| Key fact | Detail |
|---|---|
| Origin | Introduced by Harold Hotelling in "The Generalization of Student's Ratio", *Annals of Mathematical Statistics* 2(3), 360–378, August 1931<sup>[2](https://cir.nii.ac.jp/crid/1360011144703680000)</sup> |
| One-sample statistic | \( T^2 = n(\bar{\mathbf{x}} - \boldsymbol{\mu}_0)' \mathbf{S}^{-1}(\bar{\mathbf{x}} - \boldsymbol{\mu}_0) \), where \( \mathbf{S}^{-1} \) is the inverse sample covariance matrix<sup>[1](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm)</sup> |
| Exact null distribution | \( \dfrac{n-p}{p(n-1)}T^2 \sim F_{p,\,n-p} \) for a p-variate sample of size n under \( H_0: \boldsymbol{\mu} = \boldsymbol{\mu}_0 \)<sup>[3](https://online.stat.psu.edu/stat505/Lesson07)</sup> |
| Large-sample form | For large n, \( T^2 \) is approximately chi-square distributed with p degrees of freedom<sup>[3](https://online.stat.psu.edu/stat505/Lesson07)</sup> |
| Relation to Mahalanobis distance | \( T^2 \) is a scaled squared Mahalanobis distance; the two statistics differ only by a constant<sup>[4](https://epubs.siam.org/doi/book/10.1137/1.9780898718461)</sup> |
| Main uses | One-sample and two-sample tests of mean vectors, and the basis of multivariate control charts in industrial process monitoring<sup>[1](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm)</sup><sup> • </sup><sup>[5](https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/hotell.htm)</sup> |
| Key limitation | When the dimension p equals or exceeds the sample size n, the sample covariance matrix is singular and the statistic is undefined<sup>[6](https://doi.org/10.1016/j.jmva.2021.104813)</sup> |

## Definition and the statistic

For a single p-variate sample of size n drawn from a multivariate normal population, the one-sample statistic is

\[ T^2 = n(\bar{\mathbf{x}} - \boldsymbol{\mu}_0)' \mathbf{S}^{-1}(\bar{\mathbf{x}} - \boldsymbol{\mu}_0), \]

where \( \bar{\mathbf{x}} \) is the sample mean vector, \( \boldsymbol{\mu}_0 \) the hypothesized mean vector, and \( \mathbf{S}^{-1} \) the inverse of the sample variance-covariance matrix, whose diagonal entries are the variances and whose off-diagonal entries are the covariances of the p variables.<sup>[1](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm)</sup> The quantity is a constant multiple of a quadratic form built from three factors: the vector of deviations between the observation and the mean, the inverse covariance matrix, and the deviation vector again.<sup>[7](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc341.htm)</sup>

Geometrically, \( T^2 \) is a scaled squared [Mahalanobis distance](https://www.edgepedia.org/mahalanobis-distance), the distance from the center of a distribution measured in units of the data's own spread and correlation structure. For independent variables the covariance matrix is diagonal and \( T^2 \) becomes proportional to the sum of squared standardized variables; with correlated variables, the inverse covariance matrix reweights the deviations so that directions of high variability contribute less.<sup>[7](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc341.htm)</sup> Richard G. Brereton, professor of chemistry at the [University of Bristol](https://www.edgepedia.org/university-of-bristol) who works on chemometrics, gives the explicit scaling: writing \( \Delta \) for the Mahalanobis distance, \( H = (n-k)\Delta^2 / (k(n-1)) \), where k is the number of variables; when k = 1 this reduces to the squared Mahalanobis distance itself.<sup>[8](https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/10.1002/cem.2763)</sup>

## Distribution and the F connection

Under the null hypothesis \( H_0: \boldsymbol{\mu} = \boldsymbol{\mu}_0 \), the scaled statistic

\[ F = \frac{n-p}{p(n-1)}\,T^2 \sim F_{p,\,n-p} \]

follows an F distribution with p and n−p degrees of freedom, so the test rejects at level \( \alpha \) when F exceeds the corresponding F-table critical value.<sup>[3](https://online.stat.psu.edu/stat505/Lesson07)</sup> The NIST handbook states the equivalent form \( T^2 \sim \frac{p(n-1)}{n-p}F_{(p,\,n-p)} \).<sup>[1](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm)</sup> If the population covariance matrix \( \Sigma \) were known and used in place of \( \mathbf{S} \), the resulting statistic \( n(\bar{\mathbf{x}}-\boldsymbol{\mu}_0)'\Sigma^{-1}(\bar{\mathbf{x}}-\boldsymbol{\mu}_0) \) would be exactly chi-square distributed with p degrees of freedom; the F form is preferred for small samples because the chi-square approximation ignores the variation from estimating \( \Sigma \) with \( \mathbf{S} \).<sup>[3](https://online.stat.psu.edu/stat505/Lesson07)</sup>

For the two-sample problem with independent samples of sizes \( n_1 \) and \( n_2 \) from populations sharing a common covariance matrix, the statistic compares the two mean vectors using the pooled covariance matrix \( S_p \), and under \( H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \) it satisfies \( T^2 \sim \frac{mp}{m-p+1}F_{p,\,m-p+1} \) with \( m = n_1 + n_2 - 2 \).<sup>[9](https://web.stanford.edu/class/stats305c/lectures/Hotelling_T2.html)</sup> This test is the multivariate analog of the two-sample t-test.<sup>[3](https://online.stat.psu.edu/stat505/Lesson07)</sup>

The scaling matters in practice. Brereton shows that for a very large univariate sample (n = 1001) the normal, chi-squared, t, and F approaches all agree, but with more than one variable the chi-squared and F values differ even for large samples: with two variables and a nominal distance of 4, the F distribution gives a cumulative probability of 0.981 while chi-squared gives 0.864. For modest sample sizes the difference between \( T^2 \) and chi-squared is largest in the tails, so the two can lead to different conclusions about whether an observation is an outlier; at the 99% confidence level with a sample size of 10 the difference is considerable, while the 50% limits are nearly identical.<sup>[8](https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/10.1002/cem.2763)</sup>

## History

[Harold Hotelling](https://www.edgepedia.org/harold-hotelling) (1895–1973) was an American mathematical statistician who contributed to both statistics and economics. His 1931 paper "The Generalization of Student's Ratio", published in *The Annals of Mathematical Statistics* shortly before he left Stanford for a professorship in the Economics Department of Columbia University, derived the distribution of the multivariate generalization of the t statistic; the paper also contains an early statement of the idea of confidence intervals.<sup>[2](https://cir.nii.ac.jp/crid/1360011144703680000)</sup><sup> • </sup><sup>[10](https://mathshistory.st-andrews.ac.uk/Biographies/Hotelling/)</sup>

Slightly before 1931, P. C. Mahalanobis proposed a similar statistic, later known as the Mahalanobis distance, for measuring the squared distance between two populations; the two statistics differ only by a constant factor.<sup>[4](https://epubs.siam.org/doi/book/10.1137/1.9780898718461)</sup> In 1947 Hotelling applied multivariate procedures to the analysis of bombsight data in a paper on multivariate quality control, becoming one of the first to treat correlated variables from a statistical control perspective; the charting statistic he used was later named Hotelling's \( T^2 \) in his honor.<sup>[4](https://epubs.siam.org/doi/book/10.1137/1.9780898718461)</sup>

## Use in hypothesis testing

The one-sample \( T^2 \) test is used when two or more response variables are measured and a set of means is tested simultaneously against hypothesized values; rejecting the null hypothesis leads to the conclusion that at least one of the p means differs from its hypothesized value. The same procedure covers paired designs, where the difference vectors between before and after measurements are analyzed as a one-sample problem. The test assumes the data are approximately multivariate normal with independent observations, and randomization (permutation) tests are available when that assumption is doubtful.<sup>[11](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Hotellings_One-Sample_T2.pdf)</sup>

For fixed or low-dimensional data, the Hotelling \( T^2 \) test for one-sample and two-sample mean problems is the uniformly most powerful test invariant under affine transformations.<sup>[12](https://pmc.ncbi.nlm.nih.gov/articles/PMC9996668/)</sup> The two-sample version is sensitive to violations of the assumption of equal covariance matrices, especially when the sample sizes are unequal.<sup>[3](https://online.stat.psu.edu/stat505/Lesson07)</sup>

## Multivariate control charts

In statistical process control, the Hotelling control chart is a multivariate extension of the univariate \( \bar{X} \) chart that takes the correlation between measured processes into account, which running k separate \( \bar{X} \) charts does not do.<sup>[5](https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/hotell.htm)</sup> Because \( T^2 \) distances lend themselves readily to graphical display, the \( T^2 \) chart is the most popular among multivariate control charts.<sup>[7](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc341.htm)</sup> When data are grouped, the \( T^2 \) chart for means is paired with a \( T_d^2 \) chart for dispersion, the multivariate counterpart of the univariate \( \bar{X} \) and S (or \( \bar{X} \) and R) chart pair.<sup>[13](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc34.htm)</sup>

Charts are built in two phases: Phase I uses historical data to establish control limits, and Phase II monitors future data against those limits, a distinction the NIST software documentation notes is typically ignored in univariate practice but should not be ignored for multivariate Hotelling charts.<sup>[5](https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/hotell.htm)</sup> A signal only says that the multivariate system is suspect; to find an assignable cause the analyst must fall back on the individual univariate control charts or another univariate procedure.<sup>[13](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc34.htm)</sup> In Phase I, multiple outliers can mask each other and go undetected by the \( T^2 \) chart, which has motivated robust alternatives based on minimum volume ellipsoid (MVE) estimators, [minimum covariance determinant](https://www.edgepedia.org/minimum-covariance-determinant) (MCD) estimators, reweighted MCD estimators, and trimmed estimators.<sup>[14](https://ideas.repec.org/a/taf/japsta/v36y2009i12p1385-1396.html)</sup>

## High-dimensional limitations and extensions

The statistic requires the inverse of the sample covariance matrix, so it is defined only when that matrix is full rank, which needs \( n - 1 > p \).<sup>[6](https://doi.org/10.1016/j.jmva.2021.104813)</sup> When the dimension p equals or exceeds the sample size n, the sample covariance matrix becomes singular and the \( T^2 \) test breaks down completely; even when p < n, its performance can degrade as p grows relative to n, because the sample covariance matrix is no longer a stable estimator of the population covariance matrix.<sup>[12](https://pmc.ncbi.nlm.nih.gov/articles/PMC9996668/)</sup>

A large literature has developed replacements. Bai and Saranadasa (1996) proposed a test statistic based on the \( L_2 \)-distance between sample and population means that avoids estimating \( \Sigma^{-1} \), inspiring many follow-up methods, though statistics that ignore correlations can lose power.<sup>[6](https://doi.org/10.1016/j.jmva.2021.104813)</sup> Other approaches include ridge-regularized versions of \( T^2 \), tests replacing the sample covariance matrix with nonlinear shrinkage estimators such as the Ledoit-Wolf estimator, and a pairwise Hotelling method that sums \( T^2 \) statistics over strongly correlated covariate pairs together with squared t-statistics for weakly correlated individual covariates.<sup>[15](https://arxiv.org/abs/2202.12725)</sup><sup> • </sup><sup>[16](https://www3.stat.sinica.edu.tw/statistica/J34N1/J34N111/J34N111.html)</sup>

## References

1. [Hotelling's T squared, NIST/SEMATECH e-Handbook of Statistical Methods, Section 6.5.4.3](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm)
2. [Harold Hotelling (1931). The Generalization of Student's Ratio. The Annals of Mathematical Statistics 2(3), 360–378.](https://cir.nii.ac.jp/crid/1360011144703680000)
3. [Inferences Regarding Multivariate Population Mean, STAT 505, Penn State Eberly College of Science](https://online.stat.psu.edu/stat505/Lesson07)
4. [Mason, R. L. and Young, J. C. Multivariate Statistical Process Control with Industrial Applications, SIAM](https://epubs.siam.org/doi/book/10.1137/1.9780898718461)
5. [Hotelling Multivariate Control Chart, NIST/SEMATECH e-Handbook Software Documentation](https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/hotell.htm)
6. [An overview of tests on high-dimensional means, Journal of Multivariate Analysis 188 (2022) 104813](https://doi.org/10.1016/j.jmva.2021.104813)
7. [Hotelling Control Charts, NIST/SEMATECH e-Handbook, Section 6.3.4.1](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc341.htm)
8. [Brereton, R. G. (2016). Hotelling's T squared distribution, its relationship to the F distribution and its use in multivariate space. Journal of Chemometrics 30(1), 18–21.](https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/10.1002/cem.2763)
9. [Hotelling T2, STATS305C lecture notes, Stanford University](https://web.stanford.edu/class/stats305c/lectures/Hotelling_T2.html)
10. [Harold Hotelling (1895–1973), MacTutor History of Mathematics, University of St Andrews](https://mathshistory.st-andrews.ac.uk/Biographies/Hotelling/)
11. [Hotelling's One-Sample T2, NCSS Statistical Software documentation, Chapter 405](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Hotellings_One-Sample_T2.pdf)
12. [Linear Hypothesis Testing in Linear Models With High-Dimensional Responses, PMC9996668](https://pmc.ncbi.nlm.nih.gov/articles/PMC9996668/)
13. [What are Multivariate Control Charts?, NIST/SEMATECH e-Handbook, Section 6.3.4](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc34.htm)
14. [Alfaro, J. L. and Ortega, J. Fco. (2009). A comparison of robust alternatives to Hotelling's T2 control chart. Journal of Applied Statistics 36(12), 1385–1396.](https://ideas.repec.org/a/taf/japsta/v36y2009i12p1385-1396.html)
15. [An Improvement on the Hotelling T² Test Using the Ledoit-Wolf Nonlinear Shrinkage Estimator, arXiv:2202.12725](https://arxiv.org/abs/2202.12725)
16. [Hu, Z., Tong, T. and Genton, M. G. (2024). A Pairwise Hotelling Method for Testing High-Dimensional Mean Vectors. Statistica Sinica 34, 229–256.](https://www3.stat.sinica.edu.tw/statistica/J34N1/J34N111/J34N111.html)

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