Statistics Reference

Chi-Square Table for α = 0.05 (χ² Critical Values)

The workhorse chi-square column: upper-tail critical values at the conventional 5% significance level for every degree of freedom from 1 to 100, plus the matching lower critical values. Standard goodness-of-fit and independence tests are judged against the upper column.

How to Read This Table

Rows are degrees of freedom from 1 to 100 — every integer, finer than the stepped rows of the main table. The upper critical value leaves probability 0.05 above it (the standard rejection cutoff); the lower critical value leaves the same probability below it (for lower-tailed tests and two-sided procedures).

Test of independence on a 2×2 table (df = 1) at α = 0.05:

Row df = 1, upper critical column: 3.841.
Reject independence when χ² exceeds 3.841.
Sanity check: 3.841 = 1.96² — a chi-square with one degree of freedom is a squared standard normal.

Common uses of this level:

Goodness-of-fit and independence tests at the conventional 5% level
90% confidence intervals for a variance (0.05 in each tail)
The df = 1 value 3.841 is the square of the z critical value 1.96

χ² Critical Values, α = 0.05, df 1–100

Chi-square critical values at tail area 0.05 for degrees of freedom 1 to 100
df	Lower critical (area 0.05 below)	Upper critical (area 0.05 above)
1	0.004	3.841
2	0.103	5.991
3	0.352	7.815
4	0.711	9.488
5	1.145	11.070
6	1.635	12.592
7	2.167	14.067
8	2.733	15.507
9	3.325	16.919
10	3.940	18.307
11	4.575	19.675
12	5.226	21.026
13	5.892	22.362
14	6.571	23.685
15	7.261	24.996
16	7.962	26.296
17	8.672	27.587
18	9.390	28.869
19	10.117	30.144
20	10.851	31.410
21	11.591	32.671
22	12.338	33.924
23	13.091	35.172
24	13.848	36.415
25	14.611	37.652
26	15.379	38.885
27	16.151	40.113
28	16.928	41.337
29	17.708	42.557
30	18.493	43.773
31	19.281	44.985
32	20.072	46.194
33	20.867	47.400
34	21.664	48.602
35	22.465	49.802
36	23.269	50.998
37	24.075	52.192
38	24.884	53.384
39	25.695	54.572
40	26.509	55.758
41	27.326	56.942
42	28.144	58.124
43	28.965	59.304
44	29.787	60.481
45	30.612	61.656
46	31.439	62.830
47	32.268	64.001
48	33.098	65.171
49	33.930	66.339
50	34.764	67.505
51	35.600	68.669
52	36.437	69.832
53	37.276	70.993
54	38.116	72.153
55	38.958	73.311
56	39.801	74.468
57	40.646	75.624
58	41.492	76.778
59	42.339	77.931
60	43.188	79.082
61	44.038	80.232
62	44.889	81.381
63	45.741	82.529
64	46.595	83.675
65	47.450	84.821
66	48.305	85.965
67	49.162	87.108
68	50.020	88.250
69	50.879	89.391
70	51.739	90.531
71	52.600	91.670
72	53.462	92.808
73	54.325	93.945
74	55.189	95.081
75	56.054	96.217
76	56.920	97.351
77	57.786	98.484
78	58.654	99.617
79	59.522	100.749
80	60.391	101.879
81	61.261	103.010
82	62.132	104.139
83	63.004	105.267
84	63.876	106.395
85	64.749	107.522
86	65.623	108.648
87	66.498	109.773
88	67.373	110.898
89	68.249	112.022
90	69.126	113.145
91	70.003	114.268
92	70.882	115.390
93	71.760	116.511
94	72.640	117.632
95	73.520	118.752
96	74.401	119.871
97	75.282	120.990
98	76.164	122.108
99	77.046	123.225
100	77.929	124.342

Other Significance Levels

The chi-square table overview carries the classic multi-column grid, degrees-of-freedom rules, and the variance-interval walkthrough. The other levels each have a dedicated page:

Frequently Asked Questions

Is this the chi-square value used for standard tests?

Yes - unless another level is stated, goodness-of-fit and independence tests are judged at α = 0.05 against this page's upper critical column. Find your degrees of freedom (k - 1 categories, or (r-1)(c-1) for a table) and reject when the statistic exceeds the value in that row.

Why does df = 1 read 3.841, and what is special about it?

A chi-square variable with one degree of freedom is a squared standard normal, so its 5% critical value is exactly 1.96² = 3.8415. Every 2×2 independence test is judged against this single number, which makes it the most-used cell of the entire table.

Which column do I use for a 90% variance confidence interval?

Both. A 90% CI leaves 5% in each tail: divide (n-1)s² by this page's upper critical value for the lower bound and by the lower critical value for the upper bound, at df = n - 1. For the common 95% CI, use the α = 0.025 page instead.