Appendix D: Table D

Table D: Critical Values for the Distribution of Chi-Square (x²)

How to use this table: The top row of the chi-square critical values table lists the most common levels of significance. The column of figures on the left indicates the degrees of freedom.

For the chi-square statistic: df = (r − 1) (c − 1)
where r and c represent the number of rows and columns, respectively, in the contingency table.

Example: Suppose a researcher is using a 3 3 contingency table and is testing the null hypothesis of independence at the .01 level of significance. Using the formula above, degrees of freedom for a 3 3 table equal 4. Turn to page 620 and locate 4 in the df column. Move 5 columns to the right of 4 df and the critical value is 13.277. Therefore, if the researcher's chi-square statistic were equal to or greater than the critical value of 13.277, the null hypothesis of independence would be rejected at the .01 level of significance, and the alternative hypothesis would be supported.

For the goodness-of-fit statistic: df = K − 1
where K represents the number of attributes (cells) that comprise the variable of interest.

Example: Suppose a researcher is interested in studying the frequency of occurrence of accidents that occur on an hourly basis during an 8-hour shift. Using the formula above, degrees of freedom for a variable comprised of 8 attributes would be equal to 7, that is, (K − 1 = 8 − 1 = 7). If the researcher were testing the null hypothesis of no significant difference between observed and expected frequencies at the .10 level of significance, the critical value would be 12.017. Turn to page 620 and locate 7 in the df column. Move 2 columns to the right of 7 df and the critical value is 12.017. Therefore, if the researcher's chi-square statistic for this goodness-of-fit test were equal to or greater than the critical value of 12.017, the null hypothesis of no difference would be rejected at the .10 level of significance, and the alternative hypothesis would be supported.

Note

For chi-square and goodness-of-fit tests, all tests of significance are essentially two-tailed tests. This is because all chi-square and goodness-of-fit tests are nondirectional.

Critical Values for the Distribution of Chi Square (x²)

df	Levels of significance
	.20	.10	.05	.025	.01	.005	.001
1	1.644	2.706	3.842	5.024	6.635	7.879	10.828
2	3.219	4.605	5.991	7.378	9.210	10.597	13.816
3	4.642	6.251	7.815	9.348	11.345	12.838	16.265
4	5.989	7.779	9.488	11.143	13.277	14.860	18.459
5	7.289	9.236	11.070	12.832	15.086	16.750	20.515
6	8.558	10.645	12.592	14.449	16.812	18.547	22.458
7	9.803	12.017	14.067	16.013	18.475	20.278	24.321
8	11.030	13.362	15.507	17.534	20.090	21.955	26.124
9	12.242	14.684	16.919	19.023	21.666	23.589	27.876
10	13.442	15.987	18.307	20.483	23.209	25.187	29.586
11	14.631	17.275	19.675	21.920	24.724	26.756	31.261
12	15.812	18.549	21.026	23.336	26.216	28.298	32.906
13	16.985	19.812	22.362	24.735	27.687	29.817	34.524
14	18.151	21.064	23.685	26.118	29.140	31.317	36.117
15	19.311	22.307	24.995	27.488	30.578	32.798	37.690
16	20.465	23.542	26.296	28,845	32.000	34.267	39.244
17	21.615	24.769	27.587	30.191	33.409	35.718	40.780
18	22.760	25.989	28.869	31,526	34.805	37.156	42.301
19	23.900	27.203	30.144	32.852	36.191	38.582	43.807
20	25.037	28.412	31.410	34.170	37.566	39.997	45.300
21	26.171	29.615	32.671	35.479	38.932	41.401	46.797
22	27.301	30.813	33.924	36.781	40.289	42.796	48.268
23	28.429	32.007	35.172	38.076	41.638	44.181	49.728
24	29.553	33.196	36.415	39.364	42.980	45.558	51.178
25	30.675	34.382	37.653	40.646	44.314	46.928	52.620
26	31.795	35.563	38.885	41.923	45.642	48.290	54.052
27	32.912	36.741	40.113	43.195	46.963	49.645	55.476
28	34.027	37.916	41.337	44.461	48.278	50.993	56.892
29	35.139	39.087	42.557	45.722	49.588	52.336	58.301
30	36.250	40.256	43.773	46.979	50.892	53.672	59.703
Critical values computed by John Timko. Used by permission.