Dr. Mark Gardener

Available soon from
Pelagic Publishing

# Statistics for Ecologists Using R and Excel (Edition 2)

### by: Mark Gardener

Available soon from Pelagic Publishing

Welcome to the support pages for Statistics for Ecologists. These pages provide information and support material for the book. You should be able to find an outline and table of contents as well as support datafiles and additional material.

## Exercise 10.2a

Critical values for Kruskal-Wallis are produced for general and explicit scenarios

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### 10.2a. Critical values for the Kruskal-Wallis test

These notes are concerned with the Kruskal-Wallis test (Section 10.2). The notes give the critical values for the Kruskal-Wallis test under various scenarios.

#### Introduction

The Kruskal-Wallis test is appropriate when you have non-parametric data and one predictor variable (Section 10.2). It it analogous to a one-way ANOVA but uses ranks of items in various groups to determine the likely significance.

If there are at least 5 replicates in each group the critical values are close to the Chi-Squared distribution. There are exact critical values computed when you have equal group sizes. There are also exact critical values for situation where you have un-equal group sizes.

When you have at least 5 replicates per group you can use the Chi Squared distribution as a critical value for Kruskal-Wallis.

Degrees of freedom are number of groups -1

Exceed the tabulated value for a significant result.

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### General critical values: sample sizes >= 5

When you have at least 5 observations in each group the Kruskal-Wallis critical value is approximately the same as Chi Squared. You need to determine the degrees of freedom, which are the number of groups minus 1. You can reject the null hypothesis if your calculaed value of H is bigger than the tabulated value.

 degrees of Significance level freedom 5% 2% 1% 0.01% 1 3.841 5.412 6.635 10.830 2 5.991 7.824 9.210 13.820 3 7.815 9.837 11.341 16.270 4 9.488 11.668 13.277 18.470 5 11.070 13.388 15.086 20.510 6 12.592 15.033 16.812 22.460 7 14.067 16.622 18.475 24.320 8 15.507 18.168 20.090 26.130 9 16.909 19.679 21.666 27.880 10 18.307 21.161 23.209 29.590 11 19.675 22.618 24.725 31.260 12 21.026 24.054 26.217 32.910 13 22.362 25.472 27.688 34.530 14 23.685 26.873 29.141 36.120 15 24.996 28.259 30.578 37.700 16 26.296 29.633 32.000 39.250 17 27.587 30.995 33.409 40.790 18 28.869 32.346 34.805 42.310 19 30.144 33.687 36.191 43.820 20 31.410 35.020 37.566 45.310 21 32.671 36.343 38.932 46.800 22 33.924 37.659 40.289 48.270 23 35.172 38.968 41.638 49.730 24 36.415 40.270 42.980 51.180 25 37.652 41.566 44.314 52.620 26 38.885 42.856 45.642 54.050 27 40.113 44.140 46.963 55.480 28 41.337 45.419 48.278 56.890 29 42.557 46.693 49.588 58.300 30 43.773 47.962 50.892 59.700

Use this table if you have three groups and equal sample replication.

Reject the null hypothesis if your value of H is >= the tabulated value.

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### Exact values: Equal sample sizes

If you have the same number of replicates for each group you can use a table of exact values, instead of the Chi Squared approximation. There are separate tables for different numbers of groups. Reject the null hypothesis if your calculated value of H is equal or greater than the tabulated value for the appropriate sample size.

#### Groups = 3

 Significance level Sample 5% 2% 1% size Groups = 3 2 - - - 3 5.600 6.489 7.200 4 5.692 6.962 7.654 5 5.780 7.220 8.000 6 5.801 7.240 8.222 7 5.819 7.332 8.378 8 5.805 7.355 8.465 9 5.831 7.418 8.529 10 5.853 7.453 8.607 11 5.885 7.489 8.648 12 5.872 7.523 8.712 13 5.901 7.551 8.735 14 5.896 7.566 8.754 15 5.902 7.582 8.821 16 5.909 7.596 8.822 17 5.915 7.609 8.856 18 5.932 7.622 8.865 19 5.923 7.634 8.887 20 5.926 7.641 8.905 21 5.930 7.652 8.918 22 5.932 7.657 8.928 23 5.937 7.664 8.947 24 5.936 7.670 8.964 25 5.942 7.682 8.975 infinity 5.991 7.824 9.210

Use this table if you have four groups and equal sample replication.

Reject the null hypothesis if your value of H is >= the tabulated value.

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#### Groups = 4

 Significance level Sample 5% 2% 1% size Groups = 4 2 6.167 6.667 6.667 3 7.000 7.872 8.538 4 7.235 8.515 9.287 5 7.377 8.863 9.789 6 7.453 9.027 10.090 7 7.501 9.152 10.250 8 7.534 9.250 10.420 9 7.557 9.316 10.530 10 7.586 9.376 10.620 11 7.623 9.422 10.690 12 7.629 9.458 10.750 13 7.645 9.481 10.800 14 7.658 9.508 10.840 15 7.676 9.531 10.870 16 7.678 9.550 10.900 17 7.682 9.568 10.920 18 7.698 9.583 10.950 19 7.701 9.595 10.980 20 7.703 9.606 10.980 21 7.709 9.623 11.010 22 7.714 9.629 11.030 23 7.719 9.640 11.030 24 7.724 9.652 11.060 25 7.727 9.659 11.070 infinity 7.815 9.837 11.340

Use this table if you have five groups and equal sample replication.

Reject the null hypothesis if your value of H is >= the tabulated value.

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#### Groups = 5

 Significance level Sample 5% 2% 1% size Groups = 5 2 7.418 8.073 8.291 3 8.333 9.467 10.200 4 8.685 10.130 11.070 5 8.876 10.470 11.570 6 9.002 10.720 11.910 7 9.080 10.870 12.140 8 9.126 10.990 12.290 9 9.166 11.060 12.410 10 9.200 11.130 12.500 11 9.242 11.190 12.580 12 9.274 11.220 12.630 13 9.303 11.270 12.690 14 9.307 11.290 12.740 15 9.302 11.320 12.770 16 9.313 11.340 12.790 17 9.325 11.360 12.830 18 9.334 11.380 12.850 19 9.342 11.400 12.870 20 9.353 11.410 12.910 21 9.356 11.430 12.920 22 9.362 11.430 12.920 23 9.368 11.440 12.940 24 9.375 11.450 12.960 25 9.377 11.460 12.960 infinity 9.488 11.670 13.280

Use this table if you have six groups and equal sample replication.

Reject the null hypothesis if your value of H is >= the tabulated value.

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#### Groups = 6

 Significance level Sample 5% 2% 1% size Groups = 6 2 8.846 9.538 9.846 3 9.789 11.030 11.820 4 10.140 11.710 12.720 5 10.360 12.070 13.260 6 10.500 12.330 13.600 7 10.590 12.500 13.840 8 10.660 12.620 13.990 9 10.710 12.710 14.130 10 10.750 12.780 14.240 11 10.760 12.840 14.320 12 10.790 12.900 14.380 13 10.830 12.930 14.440 14 10.840 12.980 14.490 15 10.860 13.010 14.530 16 10.880 13.030 14.560 17 10.880 13.040 14.600 18 10.890 13.060 14.630 19 10.900 13.070 14.640 20 10.920 13.090 14.670 21 10.930 13.110 14.700 22 10.940 13.120 14.720 23 10.930 13.130 14.740 24 10.930 13.140 14.740 25 10.940 13.150 14.770 infinity 11.070 13.390 15.090

Exact critical values for various combinations of group size.

Reject the null hypothesis if your calculated value of H is equal to or greater than the tabulated value.

In the first column read the group sizes as single digits...

e.g. 422211

Equivalent to 6 groups with sizes: 4, 2, 2, 2, 1, 1

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### Exact values: Unequal sample sizes

When you have different sample sizes you can look up exact critical values using the following table(s).

#### Max group size = 1 to 5

The sample sizes are given in the first column, so 222110 is equivalent to 5 groups with replicates of 2, 2, 2, 1 and 1. Reject the null hypothesis if your calculated value of H is equal or grater than the tabulated value.

 Sample Significance level sizes 5% 2% 1% 222110 6.750 - - 222111 7.600 7.800 - 222200 6.167 6.667 6.667 222210 5.679 - - 222210 7.133 7.533 7.533 222211 8.018 8.455 8.618 222220 7.418 8.073 8.291 222221 8.455 9.000 9.227 222222 8.846 9.538 9.846 311100 - - - 311111 - - - 321100 - - - 321110 6.583 - - 321111 7.467 - - 322000 4.714 - - 322100 5.833 6.500 - 322110 6.800 7.400 7.600 322111 7.954 8.345 8.509 322200 6.333 6.978 7.133 322210 7.309 7.836 8.127 322211 8.348 8.939 9.136 322220 7.682 8.303 8.682 322221 8.731 9.346 9.692 322222 9.033 9.813 10.220 331000 5.143 - - 331100 6.333 - - 331110 7.111 7.467 - 331111 7.909 8.564 8.564 332000 5.361 6.250 - 332100 6.244 6.689 7.200 332110 7.200 7.782 8.073 332111 8.303 8.803 9.045 332200 6.527 7.182 7.636 332210 7.591 8.258 8.576 332211 8.615 9.269 9.628 332220 7.910 8.667 9.115 332221 8.923 9.714 10.150 333000 5.600 6.489 7.200 333100 6.600 7.109 7.400 333110 7.576 8.242 8.424 333111 8.641 9.205 9.564 333200 6.727 7.636 8.015 333210 7.769 8.590 9.051 333211 8.835 9.670 10.080 333220 8.044 9.011 9.505 333300 7.000 7.872 8.538 333310 8.000 8.879 9.451 411000 - - - 411111 7.333 - - 421000 - - - 421100 5.833 - - 421110 6.733 7.267 - 421111 7.827 8.236 8.400 422000 5.333 6.000 - 422100 6.133 6.667 7.000 422110 7.145 7.636 7.936 422111 8.205 8.727 9.000 422200 6.545 7.091 7.391 422210 7.500 8.205 8.545 422211 8.558 9.192 9.538 422220 7.846 8.673 9.077 422221 8.868 9.643 10.070 431000 5.208 - - 431100 6.178 6.711 7.067 431110 7.073 7.691 8.236 431111 8.053 8.758 9.023 432000 5.444 6.144 6.444 432100 6.309 7.018 7.455 432110 7.439 8.091 8.394 432111 8.429 9.115 9.506 432200 6.621 7.530 7.871 432210 7.679 8.545 8.962 432211 8.742 9.577 10.010 432220 7.984 8.918 9.429 433000 5.791 6.564 6.745 433100 6.545 7.485 7.758 433110 7.660 8.513 8.891 433111 8.654 9.495 9.934 433200 6.795 7.763 8.333 433210 7.874 8.830 9.374 433300 6.984 7.995 8.659 441000 4.967 6.667 6.667 441100 5.945 7.091 7.909 441110 7.114 8.182 8.636 441111 8.231 9.096 9.538 442000 5.455 6.600 7.036 442100 6.386 7.364 7.909 442110 7.500 8.385 8.885 442111 8.571 9.445 9.940 442200 6.731 7.750 8.346 442210 7.797 8.802 9.330 443000 5.598 6.712 7.144 443100 6.635 7.660 8.231 443110 7.714 8.742 9.247 443200 6.874 7.951 8.621 443300 7.038 8.181 8.876 444000 5.692 6.962 7.654 444100 6.725 7.879 8.588 444200 6.957 8.157 8.871 511000 - - - 511110 6.667 - - 511111 7.909 - - 521000 5.000 - - 521100 5.960 6.600 - 521110 6.905 7.418 7.855 521111 7.891 8.473 8.682 522000 5.160 6.000 6.533 522100 6.109 6.927 7.276 522110 7.273 7.909 8.318 522111 8.308 8.938 9.362 522200 6.564 7.364 7.773 522210 7.600 8.408 8.831 522211 8.624 9.442 9.890 531000 4.960 6.044 - 531100 6.004 6.964 7.400 531110 7.925 8.782 9.316 531111 8.169 9.062 9.503 532000 5.251 6.124 6.909 532100 6.364 7.285 7.758 532110 7.164 7.939 8.303 532111 8.495 9.371 9.837 532200 6.664 7.626 8.203 532210 7.462 8.272 8.756 533000 5.648 6.533 7.079 533100 6.641 7.656 8.128 533110 7.756 8.705 9.251 533200 6.822 7.912 8.607 533300 7.019 8.124 8.848 541000 4.985 6.431 6.955 541100 6.041 7.182 7.909 541110 7.684 8.651 9.187 541110 7.154 8.235 8.831 541111 8.242 9.234 9.841 542000 5.273 6.505 7.205 542100 6.419 7.477 8.173 542110 7.520 8.492 9.152 542200 6.725 7.849 8.473 543000 5.656 6.676 7.445 543100 6.685 7.793 8.409 543200 6.926 8.069 8.802 544000 5.657 6.953 7.760 544100 6.760 7.986 8.726 551000 5.127 6.145 7.309 551100 6.077 7.308 8.108 551110 7.226 8.334 9.152 552000 5.338 6.446 7.338 552100 6.541 7.536 8.327 552200 6.777 7.943 8.634 553000 5.705 6.886 7.578 553100 6.745 7.857 8.611 554000 5.666 7.000 7.823 555000 5.780 7.220 8.000

Exact critical values for various combinations of group size.

Reject the null hypothesis if your calculated value of H is equal to or greater than the tabulated value.

In the first column read the group sizes as single digits...

e.g. 742100

Equivalent to 4 groups with sizes: 7, 4, 2, 1

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#### Max group size 6 to 9

The sample sizes are given in the first column, so 611000 is equivalent to 3 groups with replicates of 6, 1 and 1. Reject the null hypothesis if your calculated value of H is equal or grater than the tabulated value.

 Sample Significance level sizes 5% 2% 1% 611000 - - - 611100 5.667 - - 611110 7.091 - - 611111 7.879 8.409 - 621000 4.822 - - 621100 5.964 6.600 7.036 621110 6.909 7.682 8.000 621111 8.013 8.692 9.051 622000 5.345 6.182 6.665 622100 6.242 7.000 7.500 622110 7.308 8.051 8.628 622111 8.374 9.165 9.604 622200 6.538 7.513 7.923 622210 7.593 8.549 9.077 631000 4.855 6.236 6.873 631100 6.045 7.091 7.621 631110 7.051 8.064 8.590 631111 8.209 9.176 9.659 632000 5.348 6.227 6.970 632100 6.397 7.321 7.885 632110 7.505 8.407 9.000 632200 6.703 7.758 8.363 633000 5.615 6.590 7.410 633100 6.637 7.725 8.220 633200 6.876 7.962 8.695 641000 4.947 6.174 7.106 641100 6.071 7.250 8.000 641110 7.187 8.286 9.033 642000 5.340 6.571 7.340 642100 6.489 7.516 8.302 642200 6.743 7.929 8.610 643000 5.610 6.725 7.500 643100 6.710 7.819 8.538 644000 5.681 6.900 7.795 651000 4.990 6.138 7.182 651100 6.110 7.218 8.141 652000 5.338 6.585 7.376 652100 6.541 7.598 8.389 653000 5.602 6.829 7.590 654000 5.661 7.018 7.936 655000 5.729 7.110 8.028 661000 4.945 6.286 7.121 661100 6.133 7.276 8.181 662000 5.410 6.667 7.467 663000 5.625 6.900 7.725 664000 5.724 7.107 8.000 665000 5.765 7.152 8.124 666000 5.801 7.240 8.222 711000 - - - 711100 5.945 - - 711110 6.831 7.455 - 711111 7.791 8.846 8.846 721000 4.706 5.891 - 721100 6.006 6.786 7.273 721110 6.984 7.753 8.231 721111 8.119 8.821 9.268 722000 5.143 6.058 7.000 722100 6.319 7.011 7.626 722110 7.356 8.143 8.689 722200 6.565 7.568 8.053 731000 4.952 6.043 7.030 731100 6.070 7.037 7.652 731110 7.152 8.119 8.779 732000 5.357 6.339 6.839 732100 6.466 7.383 8.005 732200 6.718 7.759 8.407 733000 5.620 6.656 7.228 733100 6.671 7.721 8.352 741000 4.986 6.319 6.986 741100 6.104 7.222 8.032 742000 5.376 6.447 7.321 742100 6.543 7.531 8.337 743000 5.623 6.780 7.550 744000 5.650 6.962 7.814 751000 5.064 6.194 7.061 751100 6.113 7.318 8.148 752000 5.393 6.477 7.450 753000 5.607 6.874 7.697 754000 5.733 7.084 7.931 755000 5.708 7.101 8.108 761000 5.067 6.214 7.254 762000 5.357 6.587 7.490 763000 5.689 6.930 7.756 764000 5.706 7.086 8.039 765000 5.770 7.191 8.157 766000 5.730 7.197 8.257 771000 4.986 6.300 7.157 772000 5.398 6.693 7.491 773000 5.688 7.003 7.810 774000 5.766 7.145 8.142 775000 5.746 7.247 8.257 811000 - - - 811100 6.182 - - 811110 6.538 7.769 - 811111 7.788 8.712 9.231 821000 4.909 6.000 - 821100 5.933 6.692 7.423 821110 6.997 7.821 8.308 822000 5.356 5.962 6.663 822100 6.305 7.096 7.648 822200 6.571 7.600 8.207 831000 4.881 6.179 6.804 831100 6.099 7.154 7.788 832000 5.316 6.371 7.022 832100 6.464 7.455 8.114 833000 5.617 6.683 7.350 841000 5.044 6.140 6.973 841100 6.143 7.271 8.029 842000 5.393 6.536 7.350 843000 5.623 6.854 7.585 844000 5.779 7.075 7.853 851000 4.869 6.257 7.110 852000 5.415 6.571 7.440 853000 5.614 6.932 7.706 854000 5.718 7.051 7.992 855000 5.769 7.159 8.116 861000 5.015 6.336 7.256 862000 5.404 6.618 7.522 863000 5.678 6.980 7.796 864000 5.743 7.120 8.045 865000 5.750 7.221 8.226 871000 5.041 6.366 7.308 872000 5.403 6.619 7.571 873000 5.698 7.021 7.827 874000 5.759 7.153 8.118 881000 5.039 6.294 7.314 882000 5.408 6.711 7.654 883000 5.734 7.021 7.889 911000 - - - 911100 5.701 6.385 - 911110 6.755 7.458 8.044 921000 4.842 5.662 6.346 921100 5.919 6.725 7.326 922000 5.260 6.095 6.897 922100 6.292 7.130 7.692 931000 4.952 6.095 6.886 931100 6.105 7.171 7.768 932000 5.340 6.359 7.006 933000 5.589 6.800 7.422 941000 5.071 6.130 7.171 942000 5.400 6.518 7.364 943000 5.652 6.882 7.614 944000 5.704 6.990 7.910 951000 5.040 6.349 7.149 952000 5.396 6.596 7.447 953000 5.670 6.972 7.733 954000 5.713 7.121 8.025 955000 5.770 7.213 8.173 961000 5.049 6.255 7.248 962000 5.392 6.614 7.566 963000 5.671 6.975 7.823 964000 5.745 7.130 8.109 971000 5.042 6.397 7.282 972000 5.429 6.679 7.637 973000 5.656 6.998 7.861 981000 4.985 6.351 7.394 982000 5.420 6.679 7.642 991000 4.961 6.407 7.333

Exact critical values for various combinations of group size.

Reject the null hypothesis if your calculated value of H is equal to or greater than the tabulated value.

In the first column read the group sizes as single digits except...

First two digits give size of grp 1 so...

e.g. 1153000

Equivalent to 3 groups with sizes: 11, 5, 3

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#### Max group size 10 to 17

The sample sizes are given in the first column (the first two digits are values 10-17), so 1011000 is equivalent to 3 groups with replicates of 10, 1 and 1. Reject the null hypothesis if your calculated value of H is equal or grater than the tabulated value.

 Sample Significance level sizes 5% 2% 1% 1011000 4.654 - - 1011100 5.908 6.560 - 1021000 4.840 5.776 6.429 1021100 5.937 6.794 7.251 1022000 5.120 6.034 6.537 1031000 5.076 6.053 6.851 1032000 5.362 6.375 7.042 1033000 5.588 6.784 7.372 1041000 5.018 6.158 7.105 1042000 5.345 6.492 7.357 1043000 5.661 6.905 7.617 1044000 5.716 7.065 7.907 1051000 4.954 6.318 7.178 1052000 5.420 6.612 7.514 1053000 5.636 6.938 7.752 1054000 5.744 7.135 8.048 1061000 5.042 6.383 7.316 1062000 5.406 6.669 7.588 1063000 5.656 7.002 7.882 1071000 4.986 6.370 7.252 1072000 5.377 6.652 7.641 1081000 5.038 6.414 7.359 1111000 4.747 - - 1111100 5.457 6.714 - 1121000 4.816 5.834 6.600 1122000 5.164 6.050 6.766 1131000 5.030 6.030 6.818 1132000 5.374 6.379 7.094 1133000 5.583 6.776 7.418 1141000 4.988 6.111 7.090 1142000 5.365 6.553 7.396 1143000 5.660 6.881 7.679 1144000 5.740 7.036 7.945 1151000 5.020 6.284 7.130 1152000 5.374 6.648 7.507 1153000 5.646 6.962 7.807 1161000 5.062 6.304 7.261 1162000 5.408 6.693 7.564 1171000 4.985 6.409 7.330 1211000 4.829 - - 1221000 4.875 5.550 6.229 1222000 5.173 5.967 6.761 1231000 4.930 6.018 6.812 1232000 5.350 6.412 7.134 1233000 5.576 6.746 7.471 1241000 4.931 6.225 7.108 1242000 5.442 6.547 7.389 1243000 5.661 6.903 7.703 1251000 4.977 6.326 7.215 1252000 5.395 6.649 7.512 1261000 5.005 6.371 7.297 1311000 4.900 - - 1321000 4.819 5.727 6.312 1322000 5.199 6.134 6.792 1331000 5.024 6.081 6.846 1332000 5.371 6.407 7.138 1333000 5.613 6.755 7.449 1341000 4.963 6.325 7.052 1342000 5.368 6.587 7.434 1351000 4.993 6.288 7.238 1411000 4.963 - - 1421000 4.863 5.737 6.356 1422000 5.193 6.045 6.812 1431000 4.977 6.029 6.811 1432000 5.383 6.413 7.218 1441000 4.991 6.265 7.176 1511000 5.020 - - 1521000 4.827 5.599 6.053 1522000 5.184 6.044 6.760 1531000 5.019 6.139 6.813 1611000 4.511 5.070 - 1621000 4.849 5.600 6.189 1711000 4.581 5.116 -

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