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Hotelling’s T 2Statistic

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International Encyclopedia of Statistical Science

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The univariate t statistic is well-known to most data analysts. If a random sample of n observations is taken from a normal distribution with mean μ and variance σ 2, the form of the statistic is given by

$$t = \frac{\overline{x} - \mu } {s/\sqrt{n}},$$
(1)

where \(\overline{x}\) is the sample mean and s is the sample standard deviation. This statistic has a Student t-distribution with (n − 1) degrees of freedom. Squaring the statistic, we obtain

$${t}^{2} = n(\overline{x} - \mu )'{({s}^{2})}^{-1}(\overline{x} - \mu ).$$

This value can be defined as the squared Euclidean distance between \(\overline{x}\) and μ.

A multivariate analogue to the t statistic can easily be constructed. Suppose a random sample of n observation vectors, given by x 1, x 2, …, x n , where x i ′ = (x i1, x i2, …, x ip )′, is taken from a p-variate multivariate normal distribution (see Multivariate Normal Distributions) with mean vector u and covariance matrix, Σ. Then the multivariate version of the tstatistic...

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References and Further Reading

  • Fisher RA (1936) The use of multiple measurements in taxonomic problems. Ann Eugen 8:376–378

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  • Hotelling H (1931) The generalization of Student’s ratio. Ann Math Stat 2:360–378

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  • Hotelling H (1947) Multivariate quality control-illustrated by the air testing of sample bombsights. Tech Stat Anal (Eisenhart C, Hastay MW, Wallis WA (eds)) McGraw-Hill, New York, pp 111–184

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  • Johnson RA, DW Wichern J (2008) Applied multivariate statistical analysis, 6th edn. Prentice Hall, New Jersey

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  • Mahalanobis PC (1930) On tests and measures of group divergence. I.F. Proc. Asiat. Soc. Bengal 26:541

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  • Mason RL, Young JC (2002) Multivariate statistical process control with industrial applications. ASA-SIAM, Philadelphia, PA

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Author information

Authors and Affiliations

  1. Southwest Research Institute, San Antonio, TX, USA

    Robert L. Mason

  2. 1750 Bilbo Street, 70609-2340, Lake Charles, LA, USA

    John C. Young

Authors
  1. Robert L. Mason
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  2. John C. Young
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Editor information

Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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© 2011 Springer-Verlag Berlin Heidelberg

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Mason, R.L., Young, J.C. (2011). Hotelling’s T 2Statistic. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_292

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