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Chi-squared distribution
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=== Table of {{math|''Ο''<sup>2</sup>}} values vs {{math|''p''}}-values === The [[p-value|<math display="inline">p</math>-value]] is the probability of observing a test statistic ''at least'' as extreme in a chi-squared distribution. Accordingly, since the [[cumulative distribution function]] (CDF) for the appropriate degrees of freedom ''(df)'' gives the probability of having obtained a value ''less extreme'' than this point, subtracting the CDF value from 1 gives the ''p''-value. A low ''p''-value, below the chosen significance level, indicates [[statistical significance]], i.e., sufficient evidence to reject the null hypothesis. A significance level of 0.05 is often used as the cutoff between significant and non-significant results. The table below gives a number of ''p''-values matching to <math> \chi^2 </math> for the first 10 degrees of freedom. {| class="wikitable" ! Degrees of freedom (df) !colspan=11| <math> \chi^2 </math> value<ref>[http://www2.lv.psu.edu/jxm57/irp/chisquar.html Chi-Squared Test] {{Webarchive|url=https://web.archive.org/web/20131118011437/http://www2.lv.psu.edu/jxm57/irp/chisquar.html |date=2013-11-18 }} Table B.2. Dr. Jacqueline S. McLaughlin at The Pennsylvania State University. In turn citing: R. A. Fisher and F. Yates, Statistical Tables for Biological Agricultural and Medical Research, 6th ed., Table IV. Two values have been corrected, 7.82 with 7.81 and 4.60 with 4.61</ref> |- | style="text-align:center;" | 1 | 0.004 | 0.02 | 0.06 | 0.15 | 0.46 | 1.07 | 1.64 | 2.71 | 3.84 | 6.63 | 10.83 |- | style="text-align:center;" | 2 | 0.10 | 0.21 | 0.45 | 0.71 | 1.39 | 2.41 | 3.22 | 4.61 | 5.99 | 9.21 | 13.82 |- | style="text-align:center;" | 3 | 0.35 | 0.58 | 1.01 | 1.42 | 2.37 | 3.66 | 4.64 | 6.25 | 7.81 | 11.34 | 16.27 |- | style="text-align:center;" | 4 | 0.71 | 1.06 | 1.65 | 2.20 | 3.36 | 4.88 | 5.99 | 7.78 | 9.49 | 13.28 | 18.47 |- | style="text-align:center;" | 5 | 1.14 | 1.61 | 2.34 | 3.00 | 4.35 | 6.06 | 7.29 | 9.24 | 11.07 | 15.09 | 20.52 |- | style="text-align:center;" | 6 | 1.63 | 2.20 | 3.07 | 3.83 | 5.35 | 7.23 | 8.56 | 10.64 | 12.59 | 16.81 | 22.46 |- | style="text-align:center;" | 7 | 2.17 | 2.83 | 3.82 | 4.67 | 6.35 | 8.38 | 9.80 | 12.02 | 14.07 | 18.48 | 24.32 |- | style="text-align:center;" | 8 | 2.73 | 3.49 | 4.59 | 5.53 | 7.34 | 9.52 | 11.03 | 13.36 | 15.51 | 20.09 | 26.12 |- | style="text-align:center;" | 9 | 3.32 | 4.17 | 5.38 | 6.39 | 8.34 | 10.66 | 12.24 | 14.68 | 16.92 | 21.67 | 27.88 |- | style="text-align:center;" | 10 | 3.94 | 4.87 | 6.18 | 7.27 | 9.34 | 11.78 | 13.44 | 15.99 | 18.31 | 23.21 | 29.59 13.25 |- ! scope="row" style="text-align:right;" | ''p''-value (probability) | style="background: #ffa2aa" | 0.95 | style="background: #efaaaa" | 0.90 | style="background: #e8b2aa" | 0.80 | style="background: #dfbaaa" | 0.70 | style="background: #d8c2aa" | 0.50 | style="background: #cfcaaa" | 0.30 | style="background: #c8d2aa" | 0.20 | style="background: #bfdaaa" | 0.10 | style="background: #b8e2aa" | 0.05 | style="background: #afeaaa" | 0.01 | style="background: #a8faaa" | 0.001 |} These values can be calculated evaluating the [[quantile function]] (also known as "inverse CDF" or "ICDF") of the chi-squared distribution;<ref>{{Cite web|url=https://www.r-tutor.com/elementary-statistics/probability-distributions/chi-squared-distribution|title=Chi-squared Distribution | R Tutorial|website=www.r-tutor.com}}</ref> e. g., the {{math|Ο<sup>2</sup>}} ICDF for {{math|1=''p'' = 0.05}} and {{math|1=df = 7}} yields {{math|2.1673 β 2.17}} as in the table above, noticing that {{math|1 β ''p''}} is the [[p-value|''p''-value]] from the table.
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