Statistics Reference

Chi-Square Table for α = 0.10 (χ² Critical Values)

The α = 0.10 column of the chi-square distribution, for every degree of freedom from 1 to 100. The upper critical values serve lenient one-sided tests at the 10% level; the lower/upper pair together brackets the middle 80% of the distribution, the pair used by 80% confidence intervals for a variance.

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 0.10 above it (the standard rejection cutoff); the lower critical value leaves the same probability below it (for lower-tailed tests and two-sided procedures).

Goodness-of-fit screen over 4 categories (df = 3) at α = 0.10:

Row df = 3, upper critical column: 6.251.
Flag the fit for follow-up when χ² exceeds 6.251.
At the stricter α = 0.05 the same row reads 7.815 — the lenient level trades false alarms for sensitivity.

Common uses of this level:

Screening-level goodness-of-fit and independence tests at α = 0.10
80% confidence intervals for a variance (0.10 in each tail)
Lower-tailed tests that variability is below a target, at the 10% level

χ² Critical Values, α = 0.10, df 1–100

Chi-square critical values at tail area 0.1 for degrees of freedom 1 to 100
df	Lower critical (area 0.1 below)	Upper critical (area 0.1 above)
1	0.016	2.706
2	0.211	4.605
3	0.584	6.251
4	1.064	7.779
5	1.610	9.236
6	2.204	10.645
7	2.833	12.017
8	3.490	13.362
9	4.168	14.684
10	4.865	15.987
11	5.578	17.275
12	6.304	18.549
13	7.042	19.812
14	7.790	21.064
15	8.547	22.307
16	9.312	23.542
17	10.085	24.769
18	10.865	25.989
19	11.651	27.204
20	12.443	28.412
21	13.240	29.615
22	14.041	30.813
23	14.848	32.007
24	15.659	33.196
25	16.473	34.382
26	17.292	35.563
27	18.114	36.741
28	18.939	37.916
29	19.768	39.087
30	20.599	40.256
31	21.434	41.422
32	22.271	42.585
33	23.110	43.745
34	23.952	44.903
35	24.797	46.059
36	25.643	47.212
37	26.492	48.363
38	27.343	49.513
39	28.196	50.660
40	29.051	51.805
41	29.907	52.949
42	30.765	54.090
43	31.625	55.230
44	32.487	56.369
45	33.350	57.505
46	34.215	58.641
47	35.081	59.774
48	35.949	60.907
49	36.818	62.038
50	37.689	63.167
51	38.560	64.295
52	39.433	65.422
53	40.308	66.548
54	41.183	67.673
55	42.060	68.796
56	42.937	69.919
57	43.816	71.040
58	44.696	72.160
59	45.577	73.279
60	46.459	74.397
61	47.342	75.514
62	48.226	76.630
63	49.111	77.745
64	49.996	78.860
65	50.883	79.973
66	51.770	81.085
67	52.659	82.197
68	53.548	83.308
69	54.438	84.418
70	55.329	85.527
71	56.221	86.635
72	57.113	87.743
73	58.006	88.850
74	58.900	89.956
75	59.795	91.061
76	60.690	92.166
77	61.586	93.270
78	62.483	94.374
79	63.380	95.476
80	64.278	96.578
81	65.176	97.680
82	66.076	98.780
83	66.976	99.880
84	67.876	100.980
85	68.777	102.079
86	69.679	103.177
87	70.581	104.275
88	71.484	105.372
89	72.387	106.469
90	73.291	107.565
91	74.196	108.661
92	75.100	109.756
93	76.006	110.850
94	76.912	111.944
95	77.818	113.038
96	78.725	114.131
97	79.633	115.223
98	80.541	116.315
99	81.449	117.407
100	82.358	118.498

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

When is the α = 0.10 chi-square column appropriate?

For screening analyses where missing a real deviation costs more than a false alarm - early data-quality checks, pilot studies, and diagnostic looks that a stricter confirmatory test will follow. Results at this level are leads, not conclusions.

What do the two columns on this page mean?

The upper critical value leaves 10% of the distribution above it - the rejection cutoff for a standard upper-tailed test at α = 0.10. The lower critical value leaves 10% below it, used for lower-tailed tests and as the other half of two-sided procedures. Between the two columns sits the middle 80% of the distribution.

How do these values relate to an 80% confidence interval for a variance?

An 80% CI puts 10% in each tail, so both bounds come straight from this page: (n-1)s² divided by the upper critical value gives the interval's lower bound, and divided by the lower critical value gives the upper bound, both at df = n - 1.