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Studentized Range Upper Quantiles

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Studentized Range Upper Quantiles A Table of Upper Quantile Points of the Studentized Range Distribution This is a table of upper quantile points of the studentized range distribution, values that are required by several statistical techniques including the Tukey wsd and the Newman-Keuls multiple comparison procedures. The table entries are given to four significant digits, in the style of the classic table for this distribution (Harter, 1960; Harter & Clemm, 1959). 2 Suppose that X(1) and X(r) are the smallest and largest order statistics in r independent N(µ, σ ) random variables. That is, X(1) and X(r) are the minimum and maximum values in a simple random sample of size r from the normal distribution with mean µ and variance σ2. The (externally) studentized range is a variable of the form X − X Q = (r) (1) S 2 2 2 where νS /σ is a χ (ν) random variable that is independent of X(r) − X(1). The p-th quantile qp(r; ν)ofthe distribution of Q satisfies Pr(Q ≤ qp(r; ν)) = p; where 0 <p<1: That is, the area to the right of qp(r; ν) under the density function of Q is 1 − p. This table includes 4944 values of qp(r; ν), at all combinations of the following parameters: • p =0:50; 0:75; 0:90; 0:95; 0:975; 0:99; 0:995; 0:999 • r = 2(1)20; 30; 40(20)100 • ν = 1(1)20; 24; 30; 40; 60; 120; ∞ except that no values are provided for ν =1atp =0:50 or p =0:75. The values of qp(r; ν) were computed using the Copenhaver & Holland (1988) algorithm. For p ≥ 0:90, values were checked against Harter’s (1960) table; discrepancies were rare and generally resolved by accepting the value from Harter’s table. References Copenhaver, M. D., & Holland, B. S. (1988). Computation of the distribution of the maximum studentized range statistic with application to multiple significance testing of simple effects. Journal of Statistical Computation and Simulation, 30, 1–15. Harter, H. L. (1960). Tables of range and studentized range. Annals of Mathematical Statistics, 31, 1122–1147. Harter, H. L., & Clemm, D. S. (1959). The probability integrals of the range and of the studentized range — probability integral, percentage points, and moments of the range. Dayton, OH: Wright Air Development Center Technical Report 58–484, Vol. I. Acknowledgments This table is provided compliments of the Stata Technical Bulletin (http://www.stata.com). The table was prepared by John R. Gleason, Syracuse University ([email protected]), who gratefully acknowledges the support of grant R01-MH54929 from the National Institute on Mental Health to Michael P. Carey. (July 20, 1998) Studentized Range 0.50 Quantiles, q0:50(r; ν) Number of variates (r) ν 2345678910111213ν 2 1.155 1.908 2.377 2.713 2.973 3.184 3.361 3.513 3.645 3.762 3.867 3.963 2 3 1.082 1.791 2.230 2.545 2.789 2.986 3.152 3.294 3.418 3.528 3.626 3.715 3 4 1.048 1.736 2.163 2.468 2.704 2.895 3.055 3.193 3.313 3.419 3.515 3.601 4 5 1.028 1.705 2.124 2.423 2.655 2.843 3.000 3.135 3.253 3.357 3.451 3.535 5 6 1.015 1.684 2.098 2.394 2.623 2.808 2.964 3.097 3.213 3.317 3.409 3.493 6 7 1.006 1.670 2.080 2.374 2.601 2.784 2.938 3.070 3.186 3.288 3.380 3.463 7 8 .9990 1.659 2.067 2.359 2.584 2.767 2.919 3.051 3.165 3.267 3.358 3.440 8 9 .9938 1.651 2.057 2.347 2.571 2.753 2.905 3.035 3.149 3.250 3.341 3.423 9 10 .9897 1.645 2.049 2.338 2.561 2.742 2.893 3.023 3.137 3.237 3.328 3.409 10 11 .9863 1.639 2.042 2.330 2.553 2.733 2.884 3.013 3.127 3.227 3.317 3.398 11 12 .9836 1.635 2.037 2.324 2.546 2.726 2.876 3.005 3.118 3.218 3.308 3.389 12 13 .9812 1.631 2.032 2.319 2.540 2.719 2.869 2.998 3.111 3.210 3.300 3.381 13 14 .9792 1.628 2.028 2.314 2.535 2.714 2.864 2.992 3.105 3.204 3.293 3.374 14 15 .9775 1.625 2.025 2.310 2.531 2.709 2.859 2.987 3.099 3.199 3.288 3.368 15 16 .9760 1.623 2.022 2.307 2.527 2.705 2.855 2.983 3.095 3.194 3.283 3.363 16 17 .9747 1.621 2.019 2.304 2.524 2.702 2.851 2.979 3.090 3.189 3.278 3.359 17 18 .9735 1.619 2.017 2.301 2.521 2.699 2.848 2.975 3.087 3.186 3.274 3.354 18 19 .9724 1.617 2.015 2.299 2.518 2.696 2.845 2.972 3.084 3.182 3.271 3.351 19 20 .9715 1.616 2.013 2.297 2.516 2.693 2.842 2.969 3.081 3.179 3.268 3.348 20 24 .9685 1.611 2.007 2.290 2.508 2.685 2.833 2.960 3.071 3.170 3.258 3.337 24 30 .9656 1.606 2.001 2.283 2.501 2.677 2.825 2.951 3.062 3.160 3.248 3.327 30 40 .9626 1.602 1.996 2.277 2.494 2.669 2.816 2.942 3.053 3.150 3.238 3.317 40 60 .9597 1.597 1.990 2.270 2.486 2.661 2.808 2.933 3.043 3.140 3.227 3.306 60 120 .9568 1.592 1.984 2.263 2.479 2.653 2.799 2.924 3.034 3.130 3.217 3.296 120 1 .9539 1.588 1.978 2.257 2.472 2.645 2.791 2.915 3.024 3.121 3.207 3.285 1 Number of variates (r) ν 14 15 16 17 18 19 20 30 40 60 80 100 ν 2 4.049 4.129 4.203 4.271 4.335 4.394 4.451 4.878 5.165 5.549 5.810 6.006 2 3 3.796 3.871 3.940 4.004 4.064 4.120 4.172 4.573 4.842 5.202 5.447 5.630 3 4 3.680 3.752 3.819 3.881 3.939 3.993 4.044 4.432 4.693 5.043 5.279 5.457 4 5 3.613 3.684 3.749 3.810 3.867 3.920 3.970 4.351 4.608 4.951 5.184 5.358 5 6 3.569 3.639 3.704 3.764 3.820 3.873 3.922 4.299 4.552 4.891 5.121 5.294 6 7 3.538 3.608 3.672 3.732 3.787 3.840 3.889 4.262 4.513 4.849 5.077 5.249 7 8 3.515 3.584 3.648 3.708 3.763 3.815 3.863 4.234 4.484 4.818 5.045 5.215 8 9 3.498 3.566 3.630 3.689 3.744 3.796 3.844 4.213 4.461 4.794 5.020 5.189 9 10 3.484 3.552 3.615 3.674 3.729 3.780 3.829 4.196 4.443 4.775 5.000 5.168 10 11 3.472 3.540 3.603 3.662 3.717 3.768 3.816 4.182 4.429 4.759 4.983 5.152 11 12 3.463 3.531 3.594 3.652 3.706 3.757 3.805 4.171 4.417 4.746 4.970 5.138 12 13 3.455 3.522 3.585 3.643 3.698 3.749 3.796 4.161 4.406 4.735 4.958 5.126 13 14 3.448 3.515 3.578 3.636 3.690 3.741 3.789 4.153 4.397 4.726 4.948 5.115 14 15 3.442 3.509 3.572 3.630 3.684 3.735 3.782 4.145 4.390 4.718 4.940 5.107 15 16 3.436 3.504 3.566 3.624 3.678 3.729 3.776 4.139 4.383 4.710 4.932 5.099 16 17 3.432 3.499 3.561 3.619 3.673 3.724 3.771 4.133 4.377 4.704 4.926 5.092 17 18 3.428 3.495 3.557 3.615 3.669 3.719 3.767 4.128 4.372 4.698 4.920 5.086 18 19 3.424 3.491 3.553 3.611 3.665 3.715 3.763 4.124 4.367 4.693 4.914 5.080 19 20 3.421 3.488 3.550 3.607 3.661 3.712 3.759 4.120 4.363 4.688 4.909 5.075 20 24 3.410 3.477 3.539 3.596 3.650 3.700 3.747 4.107 4.349 4.674 4.894 5.060 24 30 3.400 3.466 3.528 3.585 3.638 3.688 3.735 4.094 4.335 4.659 4.878 5.043 30 40 3.389 3.455 3.517 3.574 3.627 3.677 3.723 4.080 4.321 4.643 4.862 5.027 40 60 3.378 3.444 3.505 3.562 3.615 3.665 3.711 4.067 4.306 4.627 4.845 5.009 60 120 3.367 3.433 3.494 3.550 3.603 3.652 3.699 4.052 4.290 4.610 4.827 4.990 120 1 3.356 3.422 3.482 3.538 3.591 3.640 3.686 4.037 4.274 4.591 4.806 4.968 1 Studentized Range 0.75 Quantiles, q0:75(r; ν) Number of variates (r) ν 2345678910111213ν 2 2.268 3.308 3.969 4.451 4.828 5.136 5.395 5.618 5.814 5.988 6.144 6.285 2 3 2.012 2.884 3.431 3.829 4.140 4.395 4.609 4.794 4.956 5.101 5.230 5.348 3 4 1.901 2.701 3.199 3.559 3.841 4.071 4.266 4.433 4.580 4.711 4.828 4.935 4 5 1.840 2.600 3.069 3.409 3.674 3.890 4.072 4.230 4.368 4.491 4.601 4.702 5 6 1.801 2.535 2.987 3.312 3.566 3.774 3.948 4.099 4.231 4.349 4.455 4.551 6 7 1.774 2.491 2.930 3.246 3.492 3.693 3.862 4.008 4.136 4.250 4.352 4.445 7 8 1.754 2.458 2.888 3.197 3.437 3.633 3.798 3.940 4.065 4.176 4.276 4.367 8 9 1.739 2.433 2.856 3.159 3.395 3.587 3.749 3.888 4.011 4.120 4.218 4.306 9 10 1.727 2.414 2.831 3.130 3.362 3.551 3.710 3.847 3.968 4.075 4.171 4.258 10 11 1.718 2.398 2.810 3.106 3.335 3.521 3.678 3.814 3.932 4.038 4.133 4.219 11 12 1.710 2.385 2.793 3.086 3.312 3.497 3.652 3.786 3.903 4.007 4.101 4.186 12 13 1.703 2.374 2.779 3.069 3.293 3.476 3.630 3.762 3.878 3.982 4.074 4.159 13 14 1.697 2.365 2.767 3.055 3.277 3.459 3.611 3.742 3.857 3.960 4.052 4.135 14 15 1.692 2.357 2.757 3.042 3.264 3.443 3.595 3.725 3.839 3.941 4.032 4.115 15 16 1.688 2.350 2.748 3.032 3.251 3.430 3.580 3.710 3.823 3.924 4.014 4.097 16 17 1.684 2.343 2.740 3.022 3.241 3.419 3.568 3.696 3.809 3.909 3.999 4.081 17 18 1.681 2.338 2.733 3.014 3.231 3.408 3.557 3.685 3.797 3.896 3.986 4.067 18 19 1.678 2.333 2.726 3.006 3.223 3.399 3.547 3.674 3.785 3.884 3.973 4.054 19 20 1.676 2.329 2.721 3.000 3.215 3.391 3.538 3.664 3.775 3.874 3.962 4.043 20 24 1.667 2.315 2.703 2.979 3.192 3.364 3.510 3.634 3.743 3.841 3.928 4.007 24 30 1.659 2.301 2.685 2.958 3.168 3.338 3.481 3.604 3.712 3.807 3.893 3.971 30 40 1.651 2.288 2.668 2.937 3.144 3.312 3.453 3.574 3.680 3.774 3.858 3.935 40 60 1.643 2.275 2.650 2.916 3.121 3.286 3.425 3.544 3.648 3.740 3.823 3.898 60 120 1.635 2.261 2.633 2.896 3.098 3.261 3.397 3.514 3.616 3.707 3.788 3.862 120 1 1.627 2.248 2.616 2.875 3.074 3.235 3.369 3.484 3.584 3.673 3.752 3.825 1 Number of variates (r) ν 14 15 16 17 18 19 20 30 40 60 80 100 ν 2 6.415 6.534 6.644 6.747 6.842 6.932 7.016 7.661 8.096 8.681 9.079 9.378 2 3 5.456 5.555 5.646 5.731 5.811 5.886 5.956 6.494 6.857 7.347 7.680 7.931 3 4 5.033 5.123 5.206 5.283 5.356 5.423 5.487 5.976 6.307 6.754 7.058 7.287 4 5 4.793 4.878 4.956 5.029 5.097 5.161 5.221 5.682 5.994 6.415 6.702 6.918 5 6 4.639 4.720 4.795 4.865 4.930 4.991 5.049 5.490 5.790 6.194 6.470 6.677 6 7 4.530 4.609 4.681 4.749 4.812 4.871 4.927 5.355 5.645 6.038 6.305 6.507 7 8 4.450 4.526 4.597 4.663 4.725 4.783 4.837 5.255 5.538 5.921 6.182 6.379 8 9 4.388 4.463 4.532 4.597 4.657 4.714 4.767 5.176 5.454 5.830 6.086 6.279 9 10 4.338 4.412 4.480 4.544 4.603 4.659 4.711 5.114 5.387 5.756 6.008 6.199 10 11 4.298 4.370 4.438 4.500 4.559 4.614 4.666 5.062 5.332 5.696 5.945 6.133 11 12 4.264 4.336 4.402 4.464 4.522 4.576 4.628 5.020 5.286 5.646 5.892 6.077 12 13 4.236 4.307 4.373 4.434 4.491 4.545 4.595 4.983 5.247 5.603 5.846 6.030 13 14 4.211 4.282 4.347 4.408 4.464 4.517 4.568 4.952
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