TableOfCriticalValuesOfChiSquaredDistributions 2005-02-22 18:25:13 2005-02-23 17:29:07 Data Structure chi-squared random variable \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{pstricks} \usepackage{pst-plot} \usepackage{supertabular} \usepackage{psfrag} \usepackage{graphicx} \usepackage{amsthm} \usepackage{xypic} Below is a table of the critical values $c$ of the \PMlinkname{chi-squared distribution}{ChiSquaredRandomVariable} corresponding to various degrees $d$ of freedom (first column) and p-values $p$ (first row in red). \begin{supertabular}{|c||r|r|r|r|r|r|r|r|r|r|} \hline $d$ & \textbf{\red0.995} & \red0.990 & \red0.975 & \textbf{\red0.950} & \red0.900 & \red0.100 & \textbf{\red0.050} & \red0.025 & \red0.010 & \textbf{\red0.005} \\ \hline \hline 1 & \textbf{0.000} & 0.000 & 0.001 & \textbf{0.004} & 0.016 & 2.706 & \textbf{3.841} & 5.024 & 6.635 & \textbf{7.879} \\ \hline 2 & \textbf{0.010} & 0.020 & 0.051 & \textbf{0.103} & 0.211 & 4.605 & \textbf{5.991} & 7.378 & 9.210 & 10.597 \\ \hline \blue3 & \textbf{\blue0.072} & \blue0.115 & \blue0.216 & \textbf{\blue0.352} & \blue0.584 & \blue6.251 & \textbf{\blue7.815} & \blue9.348 & \blue11.345 & \textbf{\blue12.838} \\ \hline 4 & \textbf{0.207} & 0.297 & 0.484 & \textbf{0.711} & 1.064 & 7.779 & \textbf{9.488} & 11.143 & 13.277 & \textbf{14.860} \\ \hline 5 & \textbf{0.412} & 0.554 & 0.831 & \textbf{1.145} & 1.610 & 9.236 & \textbf{11.070} & 12.833 & 15.086 & \textbf{16.750} \\ \hline \blue6 & \textbf{\blue0.676} & \blue0.872 & \blue1.237 & \textbf{\blue1.635} & \blue2.204 & \blue10.645 & \textbf{\blue12.592} & \blue14.449 & \blue16.812 & \textbf{\blue18.548} \\ \hline 7 & \textbf{0.989} & 1.239 & 1.690 & \textbf{2.167} & 2.833 & 12.017 & \textbf{14.067} & 16.013 & 18.475 & \textbf{20.278} \\ \hline 8 & \textbf{1.344} & 1.646 & 2.180 & \textbf{2.733} & 3.490 & 13.362 & \textbf{15.507} & 17.535 & 20.090 & \textbf{21.955} \\ \hline \blue9 & \textbf{\blue1.735} & \blue2.088 & \blue2.700 & \textbf{\blue3.325} & \blue4.168 & \blue14.684 & \textbf{\blue16.919} & \blue19.023 & \blue21.666 & \textbf{\blue23.589} \\ \hline 10 & \textbf{2.156} & 2.558 & 3.247 & \textbf{3.940} & 4.865 & 15.987 & \textbf{18.307} & 20.483 & 23.209 & \textbf{25.188} \\ \hline 11 & \textbf{2.603} & 3.053 & 3.816 & \textbf{4.575} & 5.578 & 17.275 & \textbf{19.675} & 21.920 & 24.725 & \textbf{26.757} \\ \hline \blue12 & \textbf{\blue3.074} & \blue3.571 & \blue4.404 & \textbf{\blue5.226} & \blue6.304 & \blue18.549 & \textbf{\blue21.026} & \blue23.337 & \blue26.217 & \textbf{\blue28.300} \\ \hline 13 & \textbf{3.565} & 4.107 & 5.009 & \textbf{5.892} & 7.042 & 19.812 & \textbf{22.362} & 24.736 & 27.688 & \textbf{29.819} \\ \hline 14 & \textbf{4.075} & 4.660 & 5.629 & \textbf{6.571} & 7.790 & 21.064 & \textbf{23.685} & 26.119 & 29.141 & \textbf{31.319} \\ \hline \blue15 & \textbf{\blue4.601} & \blue5.229 & \blue6.262 & \textbf{\blue7.261} & \blue8.547 & \blue22.307 & \textbf{\blue24.996} & \blue27.488 & \blue30.578 & \textbf{\blue32.801} \\ \hline 16 & \textbf{5.142} & 5.812 & 6.908 & \textbf{7.962} & 9.312 & 23.542 & \textbf{26.296} & 28.845 & 32.000 & \textbf{34.267} \\ \hline 17 & \textbf{5.697} & 6.408 & 7.564 & \textbf{8.672} & 10.085 & 24.769 & \textbf{27.587} & 30.191 & 33.409 & \textbf{35.718} \\ \hline \blue18 & \textbf{\blue6.265} & \blue7.015 & \blue8.231 & \textbf{\blue9.390} & \blue10.865 & \blue25.989 & \textbf{\blue28.869} & \blue31.526 & \blue34.805 & \textbf{\blue37.156} \\ \hline 19 & \textbf{6.844} & 7.633 & 8.907 & \textbf{10.117} & 11.651 & 27.204 & \textbf{30.144} & 32.852 & 36.191 & \textbf{38.582} \\ \hline 20 & \textbf{7.434} & 8.260 & 9.591 & \textbf{10.851} & 12.443 & 28.412 & \textbf{31.410} & 34.170 & 37.566 & \textbf{39.997} \\ \hline \blue21 & \textbf{\blue8.034} & \blue8.897 & \blue10.283 & \textbf{\blue11.591} & \blue13.240 & \blue29.615 & \textbf{\blue32.671} & \blue35.479 & \blue38.932 & \textbf{\blue41.401} \\ \hline 22 & \textbf{8.643} & 9.542 & 10.982 & \textbf{12.338} & 14.041 & 30.813 & \textbf{33.924} & 36.781 & 40.289 & \textbf{42.796} \\ \hline 23 & \textbf{9.260} & 10.196 & 11.689 & \textbf{13.091} & 14.848 & 32.007 & \textbf{35.172} & 38.076 & 41.638 & \textbf{44.181} \\ \hline \blue24 & \textbf{\blue9.886} & \blue10.856 & \blue12.401 & \textbf{\blue13.848} & \blue15.659 & \blue33.196 & \textbf{\blue36.415} & \blue39.364 & \blue42.980 & \textbf{\blue45.559} \\ \hline 25 & \textbf{10.520} & 11.524 & 13.120 & \textbf{14.611} & 16.473 & 34.382 & \textbf{37.652} & 40.646 & 44.314 & \textbf{46.928} \\ \hline 26 & \textbf{11.160} & 12.198 & 13.844 & \textbf{15.379} & 17.292 & 35.563 & \textbf{38.885} & 41.923 & 45.642 & \textbf{48.290} \\ \hline \blue27 & \textbf{\blue11.808} & \blue12.879 & \blue14.573 & \textbf{\blue16.151} & \blue18.114 & \blue36.741 & \textbf{\blue40.113} & \blue43.195 & \blue46.963 & \textbf{\blue49.645} \\ \hline 28 & \textbf{12.461} & 13.565 & 15.308 & \textbf{16.928} & 18.939 & 37.916 & \textbf{41.337} & 44.461 & 48.278 & \textbf{50.993} \\ \hline 29 & \textbf{13.121} & 14.256 & 16.047 & \textbf{17.708} & 19.768 & 39.087 & \textbf{42.557} & 45.722 & 49.588 & \textbf{52.336} \\ \hline \blue30 & \textbf{\blue13.787} & \blue14.953 & \blue16.791 & \textbf{\blue18.493} & \blue20.599 & \blue40.256 & \textbf{\blue43.773} & \blue46.979 & \blue50.892 & \textbf{\blue53.672} \\ \hline 40 & \textbf{20.707} & 22.164 & 24.433 & \textbf{26.509} & 29.051 & 51.805 & \textbf{55.758} & 59.342 & 63.691 & \textbf{66.766} \\ \hline 50 & \textbf{27.991} & 29.707 & 32.357 & \textbf{34.764} & 37.689 & 63.167 & \textbf{67.505} & 71.420 & 76.154 & \textbf{79.490} \\ \hline \blue60 & \textbf{\blue35.534} & \blue37.485 & \blue40.482 & \textbf{\blue43.188} & \blue46.459 & \blue74.397 & \textbf{\blue79.082} & \blue83.298 & \blue88.379 & \textbf{\blue91.952} \\ \hline 70 & \textbf{43.275} & 45.442 & 48.758 & \textbf{51.739} & 55.329 & 85.527 & \textbf{90.531} & 95.023 & 100.425 & \textbf{104.215} \\ \hline 80 & \textbf{51.172} & 53.540 & 57.153 & \textbf{60.391} & 64.278 & 96.578 & \textbf{101.879} & 106.629 & 112.329 & \textbf{116.321} \\ \hline \blue90 & \textbf{\blue59.196} & \blue61.754 & \blue65.647 & \textbf{\blue69.126} & \blue73.291 & \blue107.565 & \textbf{\blue113.145} & \blue118.136 & \blue124.116 & \textbf{\blue128.299} \\ \hline 100 & \textbf{67.328} & 70.065 & 74.222 & \textbf{77.929} & 82.358 & 118.498 & \textbf{124.342} & 129.561 & 135.807 & \textbf{140.169} \\ \hline \end{supertabular} These values can be related by the equation: $$p=P(X\ge c)\qquad\mbox{where }X\sim\chi^2_{(d)}.$$ Graphically, this looks like \\ \begin{center} \includegraphics[scale=0.8]{chisquared0} \end{center} where the curve is the probability density function of the chi-squared distribution and $p$ is the darkened area.