Dr. Mark Gardener

 
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Statistics for Ecologists Edition 2 Cover
Available soon from
Pelagic Publishing


Statistics for Ecologists Using R and Excel (Edition 2)

Data Collection, Exploration, Analysis and Presentation

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.

Support Index | Exercises Index | Outline & TOC | Data files


 

Exercise 10.2a

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Table of Contents


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

Top

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

Top

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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Here is a list of keywords: it is by no means complete!

T-test, U-test, Kruskal-Wallis, Analysis of Variance, Spearman Rank, Correlation, Regression, Logistic Regression, Curved linear regression, histogram, scatter plot, bar chart, box-whisker plot, pie chart, Mean, Median, Mode, Standard Deviation, Standard Error, Range, Max, Min, Inter-quartile Range, IQR

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