Distribution tables takeaway

Standard Normal distribution

Let us note the random variable $Z$ that follows the standard normal distribution, i.e. which is such that:

$$Z\sim\mathcal{N}(0,1)$$

We note $z_\alpha$ as follows:

$$z_\alpha=\Phi_Z^{-1}(1-\alpha)\quad\textrm{i.e}\quad \alpha=\int_{z_{\alpha}}^{+\infty}f_Z(z)dz=1-\Phi_Z(z_\alpha)$$

with $\Phi_Z$ the cumulative distribution of $Z$

For one-tailed tests, the value of interest is $z_{\alpha}$, whereas for two-tailed tests, we have to look at $z_{\frac{\alpha}{2}}$

Confidence level	80%	85%	90%	95%	98%	99%	99.5%	99.9%
$\alpha$	0.20	0.15	0.10	0.05	0.02	0.01	0.005	0.001
$z_{\alpha}$	0.842	1.036	1.282	1.645	2.054	2.326	2.576	3.090
$z_{\frac{\alpha}{2}}$	1.282	1.440	1.645	1.960	2.326	2.576	2.807	3.291

Let us note the random variable $T$ that follows a $t$ distribution of $n$ degrees of freedom, i.e. which is such that:

$$T\sim t_n$$

We note $t_\alpha$ as follows:

$$t_\alpha=\Phi_T^{-1}(1-\alpha)\quad\textrm{i.e}\quad \alpha=\int_{t_{\alpha}}^{+\infty}f_T(t)dt=1-\Phi_T(t_\alpha)$$

with $\Phi_T$ the cumulative distribution of $T$

$\alpha$ \ $n$	1	2	3	4	5	6	7	8	9	10	11	12	13
0.20	1.38	1.06	0.98	0.94	0.92	0.91	0.90	0.89	0.88	0.88	0.88	0.87	0.87
0.10	3.08	1.89	1.64	1.53	1.48	1.44	1.41	1.40	1.38	1.37	1.36	1.36	1.35
0.05	6.31	2.92	2.35	2.13	2.02	1.94	1.89	1.86	1.83	1.81	1.80	1.78	1.77
0.025	12.7	4.30	3.18	2.78	2.57	2.45	2.36	2.31	2.26	2.23	2.20	2.18	2.16
0.01	31.8	6.96	4.54	3.75	3.36	3.14	3.00	2.90	2.82	2.76	2.72	2.68	2.65
0.005	63.7	9.92	5.84	4.60	4.03	3.71	3.50	3.36	3.25	3.17	3.11	3.05	3.01
0.001	318.3	22.3	10.2	7.17	5.89	5.21	4.79	4.50	4.30	4.14	4.03	3.93	3.85

$\alpha$ \ $n$	15	18	20	22	24	26	28	30	40	50	100	200	$+\infty$
0.20	0.87	0.86	0.86	0.86	0.86	0.86	0.85	0.85	0.85	0.85	0.85	0.84	0.84
0.10	1.34	1.33	1.33	1.32	1.32	1.31	1.31	1.31	1.30	1.30	1.29	1.29	1.28
0.05	1.75	1.73	1.72	1.72	1.71	1.71	1.70	1.70	1.68	1.68	1.66	1.65	1.65
0.025	2.13	2.10	2.09	2.07	2.06	2.06	2.05	2.04	2.02	2.01	1.98	1.97	1.96
0.01	2.60	2.55	2.53	2.51	2.49	2.48	2.47	2.46	2.42	2.40	2.36	2.35	2.33
0.005	2.95	2.88	2.85	2.82	2.80	2.78	2.76	2.75	2.70	2.68	2.63	2.60	2.58
0.001	3.73	3.61	3.55	3.50	3.47	3.43	3.41	3.39	3.31	3.26	3.17	3.13	3.09

Let us note the random variable $K$ that follows a $\chi^2$ distribution of $n$ degrees of freedom, i.e. which is such that:

$$K\sim \chi_n^2$$

We note $q$ the quantile of the distribution.

$q$ \ $n$	1	2	3	4	5	6	7	8	9	10	11	13
0.005	0.00	0.01	0.07	0.21	0.41	0.68	0.99	1.34	1.73	2.16	2.60	3.57
0.01	0.00	0.02	0.11	0.30	0.55	0.87	1.24	1.65	2.09	2.56	3.05	4.11
0.025	0.00	0.05	0.22	0.48	0.83	1.24	1.69	2.18	2.70	3.25	3.82	5.01
0.05	0.00	0.10	0.35	0.71	1.15	1.64	2.17	2.73	3.33	3.94	4.57	5.89
0.95	3.84	5.99	7.81	9.49	11.07	12.59	14.07	15.51	16.92	18.31	19.68	22.36
0.975	5.02	7.38	9.35	11.14	12.83	14.45	16.01	17.53	19.02	20.48	21.92	24.74
0.99	6.63	9.21	11.34	13.28	15.09	16.81	18.48	20.09	21.67	23.21	24.72	27.69
0.995	7.88	10.60	12.84	14.86	16.75	18.55	20.28	21.95	23.59	25.19	26.76	29.82

$q$ \ $n$	15	18	20	22	24	26	28	30	40	50	100
0.005	4.60	6.26	7.43	8.64	9.89	11.16	12.46	13.79	20.71	27.99	67.33
0.01	5.23	7.01	8.26	9.54	10.86	12.20	13.56	14.95	22.16	29.71	70.06
0.025	6.26	8.23	9.59	10.98	12.40	13.84	15.31	16.79	24.43	32.36	74.22
0.05	7.26	9.39	10.85	12.34	13.85	15.38	16.93	18.49	26.51	34.76	77.93
0.95	25.00	28.87	31.41	33.92	36.42	38.89	41.34	43.77	55.76	67.50	124.34
0.975	27.49	31.53	34.17	36.78	39.36	41.92	44.46	46.98	59.34	71.42	129.56
0.99	30.58	34.81	37.57	40.29	42.98	45.64	48.28	50.89	63.69	76.15	135.81
0.995	32.80	37.16	40.00	42.80	45.56	48.29	50.99	53.67	66.77	79.49	140.17