TableOfCriticalValuesOfTDistributions 2005-02-23 17:14:46 2005-02-23 17:14:46 Data Structure t distribution % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{pstricks} \usepackage{pst-plot} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Below is a table of the \emph{two-tailed} critical values $c$ of the t distribution corresponding to various degrees $d$ of freedom (first column) and p-values $p$ (first row in red). \begin{tabular}{|c||r|r|r|r|r|r|r|r|} \hline $d$ & \textbf{\red0.2} & \red0.1 & \red0.05 & \textbf{\red0.02} & \red0.01 & \red0.005 & \textbf{\red0.002} & \red0.001 \\ \hline\hline 1 & \textbf{3.078} & 6.314 & 12.706 & \textbf{31.821} & 63.657 & 127.321 & \textbf{318.309} & 636.619 \\ \hline 2 & \textbf{1.886} & 2.920 & 4.303 & \textbf{6.965} & 9.925 & 14.089 & \textbf{22.327} & 31.599 \\ \hline \blue3 & \textbf{\blue1.638} & \blue2.353 & \blue3.182 & \textbf{\blue4.541} & \blue5.841 & \blue7.453 & \textbf{\blue10.215} & \blue12.924 \\ \hline 4 & \textbf{1.533} & 2.132 & 2.776 & \textbf{3.747} & 4.604 & 5.598 & \textbf{7.173} & 8.610 \\ \hline 5 & \textbf{1.476} & 2.015 & 2.571 & \textbf{3.365} & 4.032 & 4.773 & \textbf{5.893} & 6.869 \\ \hline \blue6 & \textbf{\blue1.440} & \blue1.943 & \blue2.447 & \textbf{\blue3.143} & \blue3.707 & \blue4.317 & \textbf{\blue5.208} & \blue5.959 \\ \hline 7 & \textbf{1.415} & 1.895 & 2.365 & \textbf{2.998} & 3.499 & 4.029 & \textbf{4.785} & 5.408 \\ \hline 8 & \textbf{1.397} & 1.860 & 2.306 & \textbf{2.896} & 3.355 & 3.833 & \textbf{4.501} & 5.041 \\ \hline \blue9 & \textbf{\blue1.383} & \blue1.833 & \blue2.262 & \textbf{\blue2.821} & \blue3.250 & \blue3.690 & \textbf{\blue4.297} & \blue4.781 \\ \hline 10 & \textbf{1.372} & 1.812 & 2.228 & \textbf{2.764} & 3.169 & 3.581 & \textbf{4.144} & 4.587 \\ \hline 11 & \textbf{1.363} & 1.796 & 2.201 & \textbf{2.718} & 3.106 & 3.497 & \textbf{4.025} & 4.437 \\ \hline \blue12 & \textbf{\blue1.356} & \blue1.782 & \blue2.179 & \textbf{\blue2.681} & \blue3.055 & \blue3.428 & \textbf{\blue3.930} & \blue4.318 \\ \hline 13 & \textbf{1.350} & 1.771 & 2.160 & \textbf{2.650} & 3.012 & 3.372 & \textbf{3.852} & 4.221 \\ \hline 14 & \textbf{1.345} & 1.761 & 2.145 & \textbf{2.624} & 2.977 & 3.326 & \textbf{3.787} & 4.140 \\ \hline \blue15 & \textbf{\blue1.341} & \blue1.753 & \blue2.131 & \textbf{\blue2.602} & \blue2.947 & \blue3.286 & \textbf{\blue3.733} & \blue4.073 \\ \hline 16 & \textbf{1.337} & 1.746 & 2.120 & \textbf{2.583} & 2.921 & 3.252 & \textbf{3.686} & 4.015 \\ \hline 17 & \textbf{1.333} & 1.740 & 2.110 & \textbf{2.567} & 2.898 & 3.222 & \textbf{3.646} & 3.965 \\ \hline \blue18 & \textbf{\blue1.330} & \blue1.734 & \blue2.101 & \textbf{\blue2.552} & \blue2.878 & \blue3.197 & \textbf{\blue3.610} & \blue3.922 \\ \hline 19 & \textbf{1.328} & 1.729 & 2.093 & \textbf{2.539} & 2.861 & 3.174 & \textbf{3.579} & 3.883 \\ \hline 20 & \textbf{1.325} & 1.725 & 2.086 & \textbf{2.528} & 2.845 & 3.153 & \textbf{3.552} & 3.850 \\ \hline \blue21 & \textbf{\blue1.323} & \blue1.721 & \blue2.080 & \textbf{\blue2.518} & \blue2.831 & \blue3.135 & \textbf{\blue3.527} & \blue3.819 \\ \hline 22 & \textbf{1.321} & 1.717 & 2.074 & \textbf{2.508} & 2.819 & 3.119 & \textbf{3.505} & 3.792 \\ \hline 23 & \textbf{1.319} & 1.714 & 2.069 & \textbf{2.500} & 2.807 & 3.104 & \textbf{3.485} & 3.768 \\ \hline \blue24 & \textbf{\blue1.318} & \blue1.711 & \blue2.064 & \textbf{\blue2.492} & \blue2.797 & \blue3.091 & \textbf{\blue3.467} & \blue3.745 \\ \hline 25 & \textbf{1.316} & 1.708 & 2.060 & \textbf{2.485} & 2.787 & 3.078 & \textbf{3.450} & 3.725 \\ \hline 26 & \textbf{1.315} & 1.706 & 2.056 & \textbf{2.479} & 2.779 & 3.067 & \textbf{3.435} & 3.707 \\ \hline \blue27 & \textbf{\blue1.314} & \blue1.703 & \blue2.052 & \textbf{\blue2.473} & \blue2.771 & \blue3.057 & \textbf{\blue3.421} & \blue3.690 \\ \hline 28 & \textbf{1.313} & 1.701 & 2.048 & \textbf{2.467} & 2.763 & 3.047 & \textbf{3.408} & 3.674 \\ \hline 29 & \textbf{1.311} & 1.699 & 2.045 & \textbf{2.462} & 2.756 & 3.038 & \textbf{3.396} & 3.659 \\ \hline \blue30 & \textbf{\blue1.310} & \blue1.697 & \blue2.042 & \textbf{\blue2.457} & \blue2.750 & \blue3.030 & \textbf{\blue3.385} & \blue3.646 \\ \hline 40 & \textbf{1.303} & 1.684 & 2.021 & \textbf{2.423} & 2.704 & 2.971 & \textbf{3.307} & 3.551 \\ \hline 50 & \textbf{1.299} & 1.676 & 2.009 & \textbf{2.403} & 2.678 & 2.937 & \textbf{3.261} & 3.496 \\ \hline \blue60 & \textbf{\blue1.296} & \blue1.671 & \blue2.000 & \textbf{\blue2.390} & \blue2.660 & \blue2.915 & \textbf{\blue3.232} & \blue3.460 \\ \hline 70 & \textbf{1.294} & 1.667 & 1.994 & \textbf{2.381} & 2.648 & 2.899 & \textbf{3.211} & 3.435 \\ \hline 80 & \textbf{1.292} & 1.664 & 1.990 & \textbf{2.374} & 2.639 & 2.887 & \textbf{3.195} & 3.416 \\ \hline \blue90 & \textbf{\blue1.291} & \blue1.662 & \blue1.987 & \textbf{\blue2.368} & \blue2.632 & \blue2.878 & \textbf{\blue3.183} & \blue3.402 \\ \hline 100 & \textbf{1.290} & 1.660 & 1.984 & \textbf{2.364} & 2.626 & 2.871 & \textbf{3.174} & 3.390 \\ \hline \end{tabular} These values can be related by the equation: $$p=P(\mid\! X\!\mid\ \ge c)\qquad\mbox{where }X\sim t(d).$$ Graphically, this looks like \\ \begin{center} \includegraphics{t_dist} \end{center} where the curve is the probability density function of the t distribution and $p$ is the darkened area. It is evident from the graph where the name \emph{two-tailed} comes from. The \emph{one-tailed} critical value for the t distribution (with degrees of freedom $=d$) is defined to be the value $c$ such that $$p=P(X\ge c)\qquad\mbox{where }X\sim t(d),$$ for any given p-value $p$. One can use the same table to find the one-tailed critical value $c$ using the following two rules: \begin{enumerate} \item if the p-value $p\le0.5$ then $c$ is the same as the critical value for the two-tailed case with p-value $2p$. \item if $p>0.5$ then $c$ corresponds to the negative of the one-tailed critical value with p-value $1-p$, which then, is the same as the negative of the critical value for the two-tailed case with p-value $2(1-p)$. \end{enumerate}