Tables of P-Values for t-and Chi-Square Reference Distributions - Statistical Report No. 194

Oct 12, 2026

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Statistical Technical Report No. 194 provides tables of P-values for significance testing when the null reference distribution is standard normal, and more specifically when it follows central t and chi-square (χ²) distributions. It defines P-values in terms of probabilities of observing results at least as extreme as the test statistic under H0, then explains how to use the tables by matching the observed positive test statistic (u or c) and degrees of freedom (df) parameters.

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Introduction

P-Values

References

Table 1: One-sided P-values from t(df) distribution

TABLES OF P-VALUES FOR t-AND CHI-SQUARE REFERENCE DISTRIBUTIONS by W. W. Piegorsch University of South Carolina Statistics Technical Report No. 194 62Q05-3 August 2002 Department of Statistics University of South Carolina Columbia, SC 29208 TABLES OF P-VALUES FOR t-AND CHI-SQUARE REFERENCE DISTRIBUTIONS Walter W. Piegorsch Department of Statistics University of South Carolina Columbia, SC INTRODUCTION An important area of statistical practice involves determination of P-values when performing significance testing. If the null reference distribution is standard normal, then many standard statistical texts provide a table of probabilities that may be used to determine the P-value; examples include Casella and Berger (1990), Hogg and Tanis (2001), Iman (1994), Moore and McCabe (1993), Neter et al. (1996), Snedecor and Cochran (1980), Sokaland Rohlf (1995), and Steel and Torrie (1980), among many others. If the null reference distribution is slightly more complex, however, such as a t-distribution or a 􀀂 2-distribution, most standard textbooks give only upper-􀀃 critical points rather than actual P-values. With the advent of modern statistical computing power, this is not a major concern; most statistical computing packages can output P-values associated with the test statistics they calculate, and can even give upper tail areas (which are often equivalent to or components of a required P-value) . Nonetheless, it is useful to have available a table of P-values for settings where computer access may not be available. Towards that end, this work provides a short set of tables for t-and 􀀂 2-based P-values. P-VALUES Defined simply, a P-value is a data-based measure that helps indicate departure from a specified null hypothesis, Ho, in the direction of a specified alternative Ha. Formally, it is the probability of recovering a response as extreme as or more extreme than that actually observed, when Ho is true. (Note that ‘more extreme’ is defined in the context of Ha. For example, when testing Ho: 􀀄 = 􀀄o vs. Ha:􀀄 > 􀀄o ,‘more extreme’ corresponds to values of the test statistic supporting 􀀄 > 􀀄o .) In Tables 1 and 2, below, P-values are given for upper tail areas for central t- and 􀀂 2-distributions, respectively. These have the form P[t(􀀅) > u] for the t-tail areas and P[􀀂2(􀀅) > c] for the 􀀂 2-tail areas, where 􀀅 is the degree of freedom parameter for the corresponding reference distribution. Enter the tables with the argument u or c as the observed (positive) value of the test statistic and with degrees of freedom 􀀅. REFERENCES Casella, G., and Berger, R. L. (1990) . Statistical Inference, 1st Edn. Belmont, CA: Duxbury Press. Hogg, R. V., and Tanis, E. A. (2001) . Probability and Statistical Inference, 6th Edn. Upper Saddle River, New Jersey: Prentice Hall. Iman, R. L. (1994) . A Data-Based Approach to Statistics. Belmont, CA: Duxbury Press. Moore, D. S., and McCabe, G. P. (1993) . Introduction to the Practice of Statistics, 2nd Edn. New York: W.H. Freeman & Co. Neter, J., Kutner, M. H., Nachtsheim, C. J., and Wasserman, W. (1996) . Applied Linear Statistical Models, 4th Edn. Chicago: R.D. Irwin. Snedecor, G. W., and Cochran, W. G. (1980) . Statistical Methods, 7th Edn. Ames: Iowa State University Press. Sokal, R. R., and Rohlf, F. J. (1995) . Biometry, 3rd Edn. New York: W.H. Freeman & Co. Steel, R. G. D., and Torrie, J. H. (1980) . Principles and Procedures of Statistics: A Biometric Approach, 2nd Edn. New York: McGraw-Hill. 2 Table 1. One-sided P-values from t(􀀅) distribution: P[t(􀀅) > u] . df = 􀀅 u 1 2 3 4 5 6 7 8 9 10 11 12 | 1.30 | 0.209 | 0.162 | 0.142 | 0.132 | 0.125 | 0.121 | 0.117 | 0.115 | 0.113 | 0.111 | 0.110 | 0.109 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 1.32 | 0.206 | 0.159 | 0.139 | 0.129 | 0.122 | 0.117 | 0.114 | 0.112 | 0.110 | 0.108 | 0.107 | 0.106 | | 1.34 | 0.204 | 0.156 | 0.136 | 0.126 | 0.119 | 0.114 | 0.

FAQs

What is a P-value according to the report?

A P-value is the probability, under the null hypothesis H0, of recovering a response as extreme as or more extreme than what was actually observed, defined relative to the alternative Ha.

How do you use the t-distribution P-value table (Table 1)?

Use the tables with u as the observed positive value of the test statistic for the upper tail area, and match the degrees of freedom parameter df for the corresponding column.

What types of distributions are covered by the tables?

The report provides upper-tail P-value tables for central t-distributions and chi-square (χ²) distributions, with entries expressed as P[t(df) > u] and P[χ²(df) > c].

Oct 12, 2026

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