Statistics Reference

Chi-Square Table for α = 0.025 (χ² Critical Values)

The per-tail column behind the most common two-sided procedure: a 95% confidence interval for a variance puts 2.5% in each tail, and this page lists both required critical values side by side for every degree of freedom from 1 to 100.

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

95% CI for a variance from n = 11 observations (df = 10), s² = 8:

Row df = 10: lower critical 3.247, upper critical 20.483.
Lower bound: (n−1)s²/upper = 10×8/20.483 ≈ 3.91.
Upper bound: (n−1)s²/lower = 10×8/3.247 ≈ 24.64 — the interval is (3.91, 24.64).

Common uses of this level:

95% confidence intervals for a variance or standard deviation (0.025 per tail)
One-sided tests at the 2.5% significance level
Two-sided variance tests at the overall 5% level

χ² Critical Values, α = 0.025, df 1–100

Chi-square critical values at tail area 0.025 for degrees of freedom 1 to 100
df	Lower critical (area 0.025 below)	Upper critical (area 0.025 above)
1	0.001	5.024
2	0.051	7.378
3	0.216	9.348
4	0.484	11.143
5	0.831	12.833
6	1.237	14.449
7	1.690	16.013
8	2.180	17.535
9	2.700	19.023
10	3.247	20.483
11	3.816	21.920
12	4.404	23.337
13	5.009	24.736
14	5.629	26.119
15	6.262	27.488
16	6.908	28.845
17	7.564	30.191
18	8.231	31.526
19	8.907	32.852
20	9.591	34.170
21	10.283	35.479
22	10.982	36.781
23	11.689	38.076
24	12.401	39.364
25	13.120	40.646
26	13.844	41.923
27	14.573	43.195
28	15.308	44.461
29	16.047	45.722
30	16.791	46.979
31	17.539	48.232
32	18.291	49.480
33	19.047	50.725
34	19.806	51.966
35	20.569	53.203
36	21.336	54.437
37	22.106	55.668
38	22.878	56.896
39	23.654	58.120
40	24.433	59.342
41	25.215	60.561
42	25.999	61.777
43	26.785	62.990
44	27.575	64.201
45	28.366	65.410
46	29.160	66.617
47	29.956	67.821
48	30.755	69.023
49	31.555	70.222
50	32.357	71.420
51	33.162	72.616
52	33.968	73.810
53	34.776	75.002
54	35.586	76.192
55	36.398	77.380
56	37.212	78.567
57	38.027	79.752
58	38.844	80.936
59	39.662	82.117
60	40.482	83.298
61	41.303	84.476
62	42.126	85.654
63	42.950	86.830
64	43.776	88.004
65	44.603	89.177
66	45.431	90.349
67	46.261	91.519
68	47.092	92.689
69	47.924	93.856
70	48.758	95.023
71	49.592	96.189
72	50.428	97.353
73	51.265	98.516
74	52.103	99.678
75	52.942	100.839
76	53.782	101.999
77	54.623	103.158
78	55.466	104.316
79	56.309	105.473
80	57.153	106.629
81	57.998	107.783
82	58.845	108.937
83	59.692	110.090
84	60.540	111.242
85	61.389	112.393
86	62.239	113.544
87	63.089	114.693
88	63.941	115.841
89	64.793	116.989
90	65.647	118.136
91	66.501	119.282
92	67.356	120.427
93	68.211	121.571
94	69.068	122.715
95	69.925	123.858
96	70.783	125.000
97	71.642	126.141
98	72.501	127.282
99	73.361	128.422
100	74.222	129.561

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

Why does a 95% confidence interval use the 0.025 columns?

A two-sided 95% procedure leaves 5% of probability split across both tails - 2.5% below the lower critical value and 2.5% above the upper one. The two columns on this page are exactly that pair, which is why variance CIs quote χ²(0.025) and χ²(0.975) values.

Why is the larger critical value used for the interval's lower bound?

The variance estimate sits in the numerator of (n-1)s²/χ², so dividing by a bigger chi-square value produces a smaller bound. The inversion surprises many students: upper critical → lower bound, lower critical → upper bound. The worked example on this page shows the full arithmetic.

Are these the same values as the 0.975 column in other tables?

The lower critical column here (area 0.025 below) is what many textbooks label as the 'upper-tail area 0.975' column - identical numbers, different labeling convention. Checking one cell against a known value (df = 10 lower ≈ 3.247) instantly confirms which convention a table uses.