The chi-squared distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. Some of the statistical methods involve summarizing a probability distribution by means of covering a sample space region of a specified probability. To select the sample space region it is required to contain points of relatively high density. These Highest density regions also referred to as HDR are particularly useful for displaying multimodal distributions.
| ν | 50% | 60% | 67% | 70% | 75% | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 0.259 | 2.543 | 0.170 | 3.061 | 0.120 | 3.486 | 0.099 | 3.731 | 0.070 | 4.155 |
| 4 | 0.871 | 3.836 | 0.684 | 4.411 | 0.565 | 4.876 | 0.506 | 5.143 | 0.420 | 5.603 |
| 5 | 1.576 | 5.097 | 1.315 | 5.730 | 1.141 | 6.238 | 1.052 | 6.527 | 0.918 | 7.023 |
| 6 | 2.327 | 6.330 | 2.004 | 7.016 | 1.783 | 7.563 | 1.669 | 7.874 | 1.493 | 8.404 |
| 7 | 3.107 | 7.540 | 2.729 | 8.276 | 2.467 | 8.860 | 2.331 | 9.190 | 2.118 | 9.753 |
| 8 | 3.907 | 8.732 | 3.480 | 9.515 | 3.181 | 10.133 | 3.025 | 10.482 | 2.779 | 11.075 |
| 9 | 4.724 | 9.911 | 4.251 | 10.737 | 3.917 | 11.387 | 3.742 | 11.754 | 3.465 | 12.376 |
| 10 | 5.552 | 11.079 | 5.037 | 11.946 | 4.671 | 12.626 | 4.479 | 13.009 | 4.174 | 13.658 |
| 11 | 6.391 | 12.238 | 5.836 | 13.143 | 5.440 | 13.851 | 5.231 | 14.250 | 4.899 | 14.925 |
| 12 | 7.238 | 13.388 | 6.645 | 14.330 | 6.221 | 15.066 | 5.997 | 15.480 | 5.639 | 16.179 |
| 13 | 8.093 | 14.532 | 7.464 | 15.508 | 7.013 | 16.271 | 6.774 | 16.699 | 6.392 | 17.422 |
| 14 | 8.954 | 15.669 | 8.290 | 16.679 | 7.814 | 17.467 | 7.560 | 17.909 | 7.155 | 18.654 |
| 15 | 9.821 | 16.801 | 9.124 | 17.844 | 8.622 | 18.656 | 8.355 | 19.111 | 7.928 | 19.878 |
| 16 | 10.692 | 17.929 | 9.963 | 19.003 | 9.438 | 19.838 | 9.158 | 20.306 | 8.709 | 21.094 |
| 17 | 11.568 | 19.052 | 10.809 | 20.156 | 10.260 | 21.014 | 9.968 | 21.494 | 9.497 | 22.303 |
| 18 | 12.448 | 20.171 | 11.659 | 21.304 | 11.088 | 22.184 | 10.783 | 22.677 | 10.293 | 23.505 |
| 19 | 13.331 | 21.287 | 12.514 | 22.448 | 11.921 | 23.349 | 11.605 | 23.854 | 11.095 | 24.701 |
| 20 | 14.218 | 22.399 | 13.373 | 23.589 | 12.759 | 24.510 | 12.431 | 25.026 | 11.902 | 25.892 |
| 21 | 15.108 | 23.509 | 14.235 | 24.725 | 13.602 | 25.667 | 13.262 | 26.194 | 12.715 | 27.078 |
| 22 | 16.001 | 24.615 | 15.102 | 25.858 | 14.448 | 26.820 | 14.098 | 27.357 | 13.532 | 28.259 |
| 23 | 16.897 | 25.719 | 15.971 | 26.987 | 15.298 | 27.969 | 14.937 | 28.517 | 14.354 | 29.436 |
| 24 | 17.794 | 26.821 | 16.844 | 28.114 | 16.152 | 29.114 | 15.780 | 29.672 | 15.180 | 30.609 |
| 25 | 18.695 | 27.921 | 17.719 | 29.238 | 17.008 | 30.257 | 16.627 | 30.825 | 16.010 | 31.778 |
| 26 | 19.597 | 29.018 | 18.597 | 30.360 | 17.868 | 31.396 | 17.477 | 31.974 | 16.843 | 32.944 |
| 27 | 20.501 | 30.113 | 19.478 | 31.479 | 18.731 | 32.533 | 18.329 | 33.121 | 17.680 | 34.106 |
| 28 | 21.407 | 31.207 | 20.361 | 32.595 | 19.596 | 33.667 | 19.185 | 34.264 | 18.520 | 35.265 |
| 29 | 22.315 | 32.299 | 21.246 | 33.710 | 20.464 | 34.798 | 20.044 | 35.405 | 19.363 | 36.421 |
| 30 | 23.225 | 33.389 | 22.133 | 34.822 | 21.335 | 35.927 | 20.905 | 36.543 | 20.209 | 37.575 |
| 35 | 27.795 | 38.818 | 26.597 | 40.357 | 25.718 | 41.541 | 25.245 | 42.201 | 24.477 | 43.304 |
| 40 | 32.396 | 44.216 | 31.099 | 45.853 | 30.146 | 47.112 | 29.632 | 47.812 | 28.797 | 48.981 |
| 45 | 37.023 | 49.588 | 35.633 | 51.318 | 34.611 | 52.646 | 34.059 | 53.383 | 33.161 | 54.616 |
| 50 | 41.670 | 54.940 | 40.194 | 56.757 | 39.106 | 58.150 | 38.518 | 58.924 | 37.560 | 60.216 |
| 55 | 46.336 | 60.275 | 44.776 | 62.174 | 43.626 | 63.629 | 43.004 | 64.437 | 41.990 | 65.784 |
| 60 | 51.017 | 65.593 | 49.378 | 67.572 | 48.169 | 69.086 | 47.514 | 69.926 | 46.446 | 71.328 |
