Probability Distribution Tables

These tables provide critical values for common probability distributions used in hypothesis testing and confidence interval construction. For interactive exploration, see the individual distribution pages.

Normal Distribution Critical Values

Upper-tail critical values z_alpha where P(Z > z_alpha) = alpha for the standard Normal Distribution.

alpha	0.10	0.05	0.025	0.01	0.005	0.001
z_alpha	1.282	1.645	1.960	2.326	2.576	3.090

Common two-sided intervals: 90% uses z = 1.645, 95% uses z = 1.960, 99% uses z = 2.576.

Student’s t-Distribution Critical Values

Upper-tail critical values t_(alpha,df) for the t-Distribution. Used in Confidence Limits and Two-Sample t-Test.

df	t_0.10	t_0.05	t_0.025	t_0.01	t_0.005
1	3.078	6.314	12.706	31.821	63.657
2	1.886	2.920	4.303	6.965	9.925
5	1.476	2.015	2.571	3.365	4.032
10	1.372	1.812	2.228	2.764	3.169
20	1.325	1.725	2.086	2.528	2.845
30	1.310	1.697	2.042	2.457	2.750
60	1.296	1.671	2.000	2.390	2.660
120	1.289	1.658	1.980	2.358	2.617
inf	1.282	1.645	1.960	2.326	2.576

As df increases, t-values converge to normal z-values (bottom row).

Chi-Square Distribution Critical Values

Upper-tail critical values chi2_(alpha,df) for the Chi-Square Distribution. Used in Chi-Square SD Test and Chi-Square GOF Test.

df	chi2_0.95	chi2_0.90	chi2_0.10	chi2_0.05	chi2_0.025	chi2_0.01
1	0.004	0.016	2.706	3.841	5.024	6.635
2	0.103	0.211	4.605	5.991	7.378	9.210
5	1.145	1.610	9.236	11.070	12.833	15.086
10	3.940	4.865	15.987	18.307	20.483	23.209
20	10.851	12.443	28.412	31.410	34.170	37.566
30	18.493	20.599	40.256	43.773	46.979	50.892
50	34.764	37.689	63.167	67.505	71.420	76.154
100	77.929	82.358	118.498	124.342	129.561	135.807

Left-tail values (chi2_0.95, chi2_0.90) are used for lower confidence bounds on variance.

F-Distribution Critical Values

Upper-tail F_(0.05, df1, df2) for the F-Distribution. Used in F-Test and ANOVA.

df1 \ df2	5	10	20	30	60	120
1	6.608	4.965	4.351	4.171	4.001	3.920
2	5.786	4.103	3.493	3.316	3.150	3.072
5	5.050	3.326	2.711	2.534	2.368	2.290
10	4.735	2.978	2.348	2.165	1.993	1.910
20	4.558	2.774	2.124	1.932	1.748	1.659
30	4.496	2.700	2.039	1.841	1.649	1.554

For alpha = 0.01 critical values, consult exact tables as the ratio F_0.01/F_0.05 varies substantially across degrees of freedom.

Using These Tables

• One-tailed test — compare test statistic directly to the table value at your chosen alpha
• Two-tailed test — use alpha/2 column (e.g., for 5% two-tailed, use the 0.025 column)
• Confidence intervals — use the appropriate percentile to construct bounds
• For distributions not tabulated here, see Related Distributions for transformation relationships