Quantile table

A quantile table is a table in stochastics that contains numerically calculated quantiles of certain probability distributions .

Quantile tables are used in many places in mathematical statistics . For example, they are used to determine confidence intervals . Furthermore, in the case of normally distributed random variables with a given variance and a given expected value, probabilities can be determined directly via the Z-transformation in combination with the corresponding quantile table.

Framework

A p-quantile of a probability distribution with probability density function .

A real number is called a p-quantile of a probability distribution on the real numbers , so that ${\displaystyle P}$ ${\displaystyle x_{p}}$

${\displaystyle P((-\infty ,x_{p}])\geq p}$ and ${\displaystyle P([x_{p},+\infty ))\geq 1-p}$

is. Here is . If the probability distribution has a continuous distribution function , it is equivalent to ${\displaystyle p\in (0,1)}$ ${\displaystyle F}$

${\displaystyle F(x_{p})=p}$.

Is the distribution function strictly increasing monotonically . so is clearly determined. The p-quantile then separates the real numbers into two parts: the part less than that receives the probability and the part greater than that receives the probability . ${\displaystyle x_{p}}$ ${\displaystyle x_{p}}$ ${\displaystyle p}$ ${\displaystyle x_{p}}$ ${\displaystyle 1-p}$

In many statistical applications one often needs the quantiles of certain probability distributions. These distributions include:

• Normal distribution
• Chi-square distribution
• Student distribution
• Fisher distribution

All quantiles of these distributions are unique. However, for some distributions there is no closed representation of the distribution function (normal distribution) or this closed representation is very complex or the solution of the equation

${\displaystyle F(x_{p})=p}$

is not practical. Therefore, the important quantiles of these distributions are numerically determined with the necessary precision and summarized in tables. So they can be looked up without having to be numerically determined each time.

Exactly which values ​​the table contains and how many decimal places it contains depends on the respective distribution and the context in which it is required. For example, only the standard normal distribution is tabulated for the normal distribution, which is easy to normalize , but with an accuracy of two decimal places and the corresponding four decimal places . For the Student distribution, on the other hand, only the quantiles etc. are given, but with a variable number of degrees of freedom . Details on this and on the use of the individual tables can be found in the corresponding sections. ${\displaystyle x_{p}}$ ${\displaystyle p}$ ${\displaystyle p=0{,}9;0{,}95;0{,}99;0{,}995}$

Important quantile tables

Some important quantile tables are listed below. The selection of the tabulated values ​​follows

Normal distribution

The quantile table of the normal distribution , more precisely the standard normal distribution , can be found in the article Standard normal distribution table . The use of the table is also explained there and some examples are given.

Chi-square distribution

p-quantiles of the chi-square distribution with n degrees of freedom

	p = 0.005	0.010	0.020	0.025	0.050	0.100	0.250	0.500	0.750	0.900	0.950	0.975	0.980	0.990	0.995
n = 1	3,927e-5	1,571e-4	6.285e-4	9,820e-4	3,932e-3	1,579e-2	1,015e-1	4,549e-1	1.323	2.706	3.841	5.024	5.412	6.635	7,879
2	1.003e-2	2.010e-2	4.041e-2	5.064e-2	1,026e-1	2,107e-1	5,754e-1	1.386	2,773	4.605	5,991	7.378	7.824	9.210	10.60
3	7,172e-2	1.1480e-1	1,848e-1	2,158e-1	3,518e-1	5,844e-1	1,213	2,366	4.108	6.251	7.815	9.348	9,837	11.34	12.84
4th	2.070e-1	2,971e-1	4,294e-1	4,844e-1	7,107e-1	1.064	1.923	3.357	5.385	7.779	9.488	11.14	11.67	13.28	14.86
5	4,117e-1	5,543e-1	7,519e-1	8,312e-1	1,145	1.6100	2.675	4,351	6.626	9.236	11.07	12.83	13.39	15.09	16.75
6th	6,757e-1	8,721e-1	1.134	1,237	1.635	2.204	3.455	5.348	7,841	10.64	12.59	14.45	15.03	16.81	18.55
7th	0.9893	1,239	1.564	1.6900	2.167	2.833	4.255	6.346	9,037	12.02	14.07	16.01	16.62	18.48	20.28
8th	1,344	1.646	2.032	2.180	2.733	3,490	5.071	7.344	10.22	13.36	15.51	17.53	18.17	20.09	21.95
9	1.735	2.088	2.532	2,700	3.325	4,168	5,899	8.343	11.39	14.68	16.92	19.02	19.68	21.67	23.59
10	2.156	2.558	3.059	3.247	3,940	4.865	6.737	9,342	12.55	15.99	18.31	20.48	21.16	23.21	25.19
11	2.603	3.053	3.609	3.816	4,575	5.578	7.584	10.34	13.70	17.28	19.68	21.92	22.62	24.72	26.76
12	3.074	3,571	4.178	4.404	5.226	6.304	8,438	11.34	14.85	18.55	21.03	23.34	24.05	26.22	28.30
13	3.565	4.107	4.765	5.009	5,892	7,042	9.299	12.34	15.98	19.81	22.36	24.74	25.47	27.69	29.82
14th	4.075	4,660	5.368	5.629	6.571	7.790	10.17	13.34	17.12	21.06	23.68	26.12	26.87	29.14	31.32
15th	4.601	5.229	5.985	6.262	7.261	8.547	11.04	14.34	18.25	22.31	25.00	27.49	28.26	30.58	32.80
16	5.142	5.812	6.614	6,908	7,962	9.312	11.91	15.34	19.37	23.54	26.30	28.85	29.63	32.00	34.27
17th	5.697	6.408	7.255	7.564	8,672	10.09	12.79	16.34	20.49	24.77	27.59	30.19	31.00	33.41	35.72
18th	6.265	7.015	7,906	8.231	9,390	10.86	13.68	17.34	21.60	25.99	28.87	31.53	32.35	34.81	37.16
19th	6.844	7.633	8,567	8,907	10.12	11.65	14.56	18.34	22.72	27.20	30.14	32.85	33.69	36.19	38.58
20th	7.434	8.260	9.237	9,591	10.85	12.44	15.45	19.34	23.83	28.41	31.41	34.17	35.02	37.57	40.00
21st	8.034	8,897	9.915	10.28	11.59	13.24	16.34	20.34	24.93	29.62	32.67	35.48	36.34	38.93	41.40
22nd	8,643	9,542	10.60	10.98	12.34	14.04	17.24	21.34	26.04	30.81	33.92	36.78	37.66	40.29	42.80
23	9.260	10.20	11.29	11.69	13.09	14.85	18.14	22.34	27.14	32.01	35.17	38.08	38.97	41.64	44.18
24	9,886	10.86	11.99	12.40	13.85	15.66	19.04	23.34	28.24	33.20	36.42	39.36	40.27	42.98	45.56
25th	10.52	11.52	12.70	13.12	14.61	16.47	19.94	24.34	29.34	34.38	37.65	40.65	41.57	44.31	46.93
26th	11.16	12.20	13.41	13.84	15.38	17.29	20.84	25.34	30.43	35.56	38.89	41.92	42.86	45.64	48.29
27	11.81	12.88	14.13	14.57	16.15	18.11	21.75	26.34	31.53	36.74	40.11	43.19	44.14	46.96	49.64
28	12.46	13.56	14.85	15.31	16.93	18.94	22.66	27.34	32.62	37.92	41.34	44.46	45.42	48.28	50.99
29	13.12	14.26	15.57	16.05	17.71	19.77	23.57	28.34	33.71	39.09	42.56	45.72	46.69	49.59	52.34
30th	13.79	14.95	16.31	16.79	18.49	20.60	24.48	29.34	34.80	40.26	43.77	46.98	47.96	50.89	53.67
35	17.19	18.51	20.03	20.57	22.47	24.80	29.05	34.34	40.22	46.06	49.80	53.20	54.24	57.34	60.27
40	20.71	22.16	23.84	24.43	26.51	29.05	33.66	39.34	45.62	51.81	55.76	59.34	60.44	63.69	66.77
45	24.31	25.90	27.72	28.37	30.61	33.35	38.29	44.34	50.98	57.51	61.66	65.41	66.56	69.96	73.17
50	27.99	29.71	31.66	32.36	34.76	37.69	42.94	49.33	56.33	63.17	67.50	71.42	72.61	76.15	79.49
55	31.73	33.57	35.66	36.40	38.96	42.06	47.61	54.33	61.66	68.80	73.31	77.38	78.62	82.29	85.75
60	35.53	37.48	39.70	40.48	43.19	46.46	52.29	59.33	66.98	74.40	79.08	83.30	84.58	88.38	91.95
70	43.28	45.44	47.89	48.76	51.74	55.33	61.70	69.33	77.58	85.53	90.53	95.02	96.39	100.4	104.2
80	51.17	53.54	56.21	57.15	60.39	64.28	71.14	79.33	88.13	96.58	101.9	106.6	108.1	112.3	116.3
90	59.20	61.75	64.63	65.65	69.13	73.29	80.62	89.33	98.65	107.6	113.1	118.1	119.6	124.1	128.3
100	67.33	70.06	73.14	74.22	77.93	82.36	90.13	99.33	109.1	118.5	124.3	129.6	131.1	135.8	140.2
150	109.1	112.7	116.6	118.0	122.7	128.3	138.0	149.3	161.3	172.6	179.6	185.8	187.7	193.2	198.4
200	152.2	156.4	161.1	162.7	168.3	174.8	186.2	199.3	213.1	226.0	234.0	241.1	243.2	249.4	255.3
250	196.2	200.9	206.2	208.1	214.4	221.8	234.6	249.3	264.7	279.1	287.9	295.7	298.0	304.9	311.3
300	240.7	246.0	251.9	253.9	260.9	269.1	283.1	299.3	316.1	331.8	341.4	349.9	352.4	359.9	366.8
400	330.9	337.2	344.1	346.5	354.6	364.2	380.6	399.3	418.7	436.6	447.6	457.3	460.2	468.7	476.6
600	514.5	522.4	531.0	534.0	544.2	556.1	576.3	599.3	623.0	644.8	658.1	669.8	673.3	683.5	693.0
800	700.7	709.9	720.0	723.5	735.4	749.2	772.7	799.3	826.6	851.7	866.9	880.3	884.3	896.0	906.8
1000	888.6	898.9	910.3	914.3	927.6	943.1	969.5	999.3	1029	1057	1074	1089	1094,	1107,	1118

Student's t-distribution

p-quantiles of Student's t-distribution with n degrees of freedom

	p = 0.9	0.95	0.96	0.975	0.98	0.99	0.995	0.999	0.9995
n = 1	3.078	6.314	7.916	12.71	15.89	31.82	63.66	318.3	636.6
2	1,886	2.920	3.320	4.303	4,849	6,965	9.925	22.33	31.60
3	1.638	2.353	2.605	3.182	3,482	4,541	5,841	10.21	12.92
4th	1.533	2.132	2.333	2,776	2,999	3,747	4.604	7.173	8.610
5	1.476	2.015	2.191	2.571	2.757	3.365	4.032	5,893	6.869
6th	1,440	1,943	2.104	2,447	2,612	3.143	3.707	5.208	5,959
7th	1.415	1,895	2.046	2,365	2.517	2.998	3,499	4,785	5.408
8th	1.397	1,860	2.004	2.306	2,449	2,896	3.355	4,501	5.041
9	1.383	1,833	1,973	2.262	2,398	2.821	3,250	4,297	4,781
10	1.372	1,812	1,948	2.228	2,359	2.764	3.169	4.144	4,587
11	1.363	1,796	1.928	2,201	2.328	2.718	3.106	4.025	4,437
12	1.356	1,782	1.912	2.179	2.303	2,681	3.055	3,930	4,318
13	1,350	1,771	1,899	2.160	2.282	2,650	3.012	3.852	4.221
14th	1,345	1.761	1,887	2.145	2.264	2.624	2,977	3,787	4,140
15th	1.341	1.753	1,878	2.131	2.249	2.602	2.947	3.733	4.073
16	1.337	1,746	1,869	2.120	2.235	2.583	2.921	3,686	4.015
17th	1.333	1,740	1,862	2.110	2.224	2.567	2,898	3,646	3.965
18th	1.330	1.734	1,855	2.101	2.214	2.552	2,878	3,610	3,922
19th	1.328	1.729	1,850	2.093	2.205	2.539	2.861	3,579	3.883
20th	1.325	1.725	1,844	2.086	2.197	2.528	2.845	3,552	3,850
21st	1.323	1.721	1,840	2.080	2.189	2.518	2.831	3.527	3.819
22nd	1.321	1.717	1,835	2.074	2.183	2.508	2.819	3.505	3.792
23	1,319	1.714	1,832	2.069	2.177	2,500	2.807	3.485	3,768
24	1.318	1.711	1,828	2.064	2.172	2,492	2.797	3.467	3.745
25th	1,316	1.708	1,825	2.060	2.167	2.485	2.787	3,450	3.725
26th	1,315	1.706	1,822	2.056	2.162	2,479	2,779	3.435	3.707
27	1,314	1.703	1,819	2.052	2.158	2.473	2.771	3.421	3,690
28	1,313	1.701	1.817	2.048	2.154	2,467	2.763	3.408	3,674
29	1,311	1,699	1,814	2.045	2.150	2.462	2.756	3.396	3,659
30th	1,310	1.697	1,812	2.042	2.147	2.457	2.750	3.385	3,646
35	1.306	1.690	1.803	2.030	2.133	2,438	2.724	3.340	3,591
40	1.303	1.684	1,796	2.021	2.123	2,423	2.704	3.307	3.551
45	1.301	1.679	1.791	2.014	2.115	2,412	2,690	3.281	3.520
50	1,299	1.676	1,787	2.009	2.109	2.403	2.678	3.261	3,496
60	1.296	1.671	1,781	2,000	2.099	2,390	2,660	3.232	3,460
70	1.294	1.667	1,776	1.994	2.093	2.381	2.648	3.211	3.435
80	1.292	1.664	1,773	1,990	2.088	2,374	2,639	3.195	3.416
90	1.291	1.662	1,771	1.987	2.084	2,368	2.632	3.183	3.402
100	1.290	1.660	1,769	1.984	2.081	2,364	2.626	3.174	3.390
150	1.287	1.655	1.763	1,976	2.072	2,351	2.609	3.145	3.357
200	1.286	1.653	1.760	1,972	2.067	2,345	2.601	3.131	3.340
250	1.285	1.651	1.758	1,969	2.065	2,341	2,596	3.123	3.330
300	1.284	1,650	1.757	1,968	2.063	2,339	2,592	3.118	3.323
400	1.284	1.649	1,755	1,966	2.060	2,336	2,588	3.111	3.315
500	1.283	1.648	1.754	1.965	2.059	2,334	2.586	3.107	3.310
600	1.283	1.647	1.754	1.964	2.058	2.333	2.584	3.104	3.307
800	1.283	1.647	1.753	1.963	2.057	2,331	2,582	3,100	3.303
1000	1.282	1.646	1.752	1,962	2.056	2.330	2.581	3.098	3,300
100,000	1.282	1.645	1.751	1,960	2.054	2,326	2.576	3.090	3.291

Fisher distribution

0.95 quantiles of the Fisher distribution with n degrees of freedom in the denominator and m degrees of freedom in the numerator

	m = 1	2	3	4th	5	6th	7th	8th	9	10	11	12	14th	16	18th	20th	22nd	24	30th	40	50	60	100	100,000
n = 1	161.4	199.5	215.7	224.6	230.2	234.0	236.8	238.9	240.5	241.9	243.0	243.9	244.7	245.4	245.9	246.5	246.9	247.3	247.7	248.0	248.3	248.6	248.8	249.1
2	18.51	19.00	19.16	19.25	19.30	19.33	19.35	19.37	19.38	19.40	19.40	19.41	19.42	19.42	19.43	19.43	19.44	19.44	19.44	19.45	19.45	19.45	19.45	19.45
3	10.13	9,552	9.277	9.117	9,013	8,941	8,887	8,845	8.812	8,786	8,763	8.745	8.729	8.715	8.703	8,692	8,683	8.675	8.667	8,660	8,654	8.648	8,643	8,639
4th	7.709	6,944	6,591	6.388	6.256	6.163	6.094	6,041	5.999	5.964	5.936	5.912	5,891	5,873	5,858	5.844	5.832	5.821	5.811	5.803	5.795	5.787	5.781	5.774
5	6.608	5.786	5.409	5.192	5.050	4,950	4,876	4.818	4,772	4,735	4.704	4.678	4.655	4,636	4,619	4.604	4,590	4,579	4,568	4,558	4,549	4,541	4,534	4,527
6th	5,987	5.143	4.757	4,534	4.387	4.284	4.207	4,147	4.099	4.060	4.027	4,000	3,976	3.956	3,938	3,922	3.908	3.896	3.884	3,874	3.865	3.856	3.849	3.841
7th	5,591	4,737	4,347	4,120	3,972	3,866	3,787	3.726	3,677	3,637	3.603	3.575	3,550	3.529	3.511	3,494	3,480	3.467	3.455	3.445	3.435	3.426	3.418	3.410
8th	5.318	4.459	4.066	3.838	3,687	3,581	3,500	3.438	3.388	3.347	3.313	3.284	3.259	3.237	3.218	3.202	3.187	3.173	3.161	3.150	3.140	3.131	3.123	3.115
9	5.117	4.256	3.863	3,633	3,482	3.374	3.293	3.230	3.179	3.137	3.102	3.073	3.048	3.025	3.006	2.989	2,974	2.960	2.948	2.936	2.926	2.917	2.908	2,900
10	4.965	4.103	3.708	3.478	3.326	3.217	3.135	3.072	3.020	2,978	2.943	2.913	2,887	2.865	2.845	2.828	2.812	2,798	2.785	2,774	2.764	2.754	2.745	2.737
11	4,844	3,982	3,587	3.357	3.204	3.095	3.012	2.948	2,896	2.854	2.818	2,788	2.761	2.739	2.719	2.701	2.685	2,671	2.658	2,646	2.636	2.626	2.617	2.609
12	4,747	3,885	3,490	3.259	3.106	2.996	2.913	2.849	2.796	2.753	2.717	2.687	2,660	2.637	2.617	2,599	2.583	2.568	2.555	2.544	2.533	2.523	2.514	2.505
13	4,667	3.806	3.411	3.179	3.025	2.915	2.832	2.767	2.714	2,671	2.635	2.604	2.577	2.554	2.533	2.515	2,499	2.484	2.471	2,459	2,448	2,438	2,429	2,420
14th	4,600	3.739	3,344	3.112	2.958	2.848	2.764	2,699	2,646	2.602	2.565	2.534	2.507	2.484	2.463	2,445	2,428	2,413	2,400	2.388	2,377	2.367	2.357	2,349
15th	4,543	3,682	3.287	3.056	2.901	2.790	2.707	2.641	2,588	2.544	2.507	2,475	2,448	2,424	2.403	2.385	2,368	2.353	2,340	2.328	2,316	2.306	2.297	2,288
16	4.494	3,634	3.239	3.007	2.852	2.741	2.657	2.591	2.538	2,494	2,456	2,425	2.397	2.373	2,352	2.333	2,317	2.302	2,288	2.276	2.264	2.254	2.244	2.235
17th	4.451	3,592	3.197	2.965	2.810	2,699	2.614	2.548	2,494	2,450	2,413	2.381	2.353	2,329	2.308	2,289	2.272	2.257	2.243	2.230	2.219	2.208	2.199	2.190
18th	4,414	3.555	3.160	2.928	2,773	2,661	2.577	2.510	2,456	2,412	2,374	2,342	2,314	2.290	2.269	2.250	2.233	2.217	2.203	2.191	2.179	2.168	2.159	2.150
19th	4.381	3.522	3.127	2,895	2.740	2.628	2.544	2,477	2,423	2.378	2,340	2.308	2.280	2.256	2.234	2.215	2.198	2.182	2.168	2.155	2.144	2.133	2.123	2.114
20th	4,351	3.493	3.098	2,866	2.711	2,599	2.514	2,447	2,393	2,348	2,310	2.278	2.250	2.225	2.203	2.184	2.167	2.151	2.137	2.124	2.112	2.102	2.092	2.082
21st	4,325	3.467	3.072	2.840	2.685	2.573	2.488	2,420	2,366	2,321	2.283	2.250	2.222	2.197	2.176	2.156	2.139	2.123	2.109	2.096	2.084	2.073	2.063	2.054
22nd	4.301	3.443	3.049	2.817	2,661	2.549	2,464	2.397	2,342	2.297	2.259	2.226	2.198	2.173	2.151	2.131	2.114	2.098	2.084	2.071	2.059	2.048	2.038	2.028
23	4,279	3.422	3.028	2.796	2,640	2.528	2,442	2,375	2,320	2.275	2.236	2.204	2.175	2.150	2.128	2.109	2.091	2.075	2.061	2.048	2.036	2.025	2.014	2.005
24	4,260	3.403	3.009	2,776	2.621	2.508	2,423	2.355	2,300	2.255	2.216	2.183	2.155	2.130	2.108	2.088	2.070	2.054	2.040	2.027	2.015	2.003	1.993	1.984
25th	4,242	3.385	2.991	2.759	2.603	2,490	2.405	2,337	2.282	2.236	2.198	2.165	2.136	2.111	2.089	2.069	2.051	2.035	2.021	2.007	1.995	1.984	1,974	1.964
26th	4.225	3.369	2.975	2.743	2.587	2.474	2.388	2,321	2.265	2.220	2.181	2.148	2.119	2.094	2.072	2.052	2.034	2.018	2.003	1,990	1,978	1,966	1.956	1,946
27	4,210	3.354	2.960	2.728	2.572	2,459	2.373	2.305	2.250	2.204	2.166	2.132	2.103	2.078	2.056	2.036	2.018	2.002	1.987	1,974	1.961	1,950	1,940	1.930
28	4,196	3.340	2.947	2.714	2.558	2,445	2,359	2.291	2.236	2.190	2.151	2.118	2.089	2.064	2.041	2.021	2.003	1.987	1,972	1.959	1,946	1.935	1.924	1.915
29	4.183	3.328	2.934	2.701	2.545	2,432	2,346	2.278	2.223	2.177	2.138	2.104	2.075	2.050	2.027	2.007	1.989	1,973	1.958	1.945	1,932	1,921	1.910	1.901
30th	4.171	3.316	2.922	2,690	2.534	2,421	2,334	2.266	2.211	2.165	2.126	2.092	2.063	2.037	2.015	1.995	1,976	1,960	1.945	1,932	1.919	1.908	1.897	1,887
31	4.160	3.305	2.911	2,679	2.523	2.409	2,323	2.255	2.199	2.153	2.114	2.080	2.051	2.026	2.003	1.983	1.965	1,948	1.933	1,920	1.907	1,896	1,885	1,875
32	4.149	3.295	2.901	2,668	2.512	2,399	2,313	2.244	2.189	2.142	2.103	2.070	2.040	2.015	1.992	1,972	1.953	1.937	1,922	1.908	1,896	1,884	1,873	1,864
33	4,139	3.285	2,892	2.659	2.503	2,389	2.303	2.235	2.179	2.133	2.093	2.060	2.030	2.004	1,982	1.961	1,943	1.926	1.911	1,898	1,885	1,873	1,863	1.853
34	4.130	3.276	2.883	2,650	2,494	2,380	2,294	2.225	2.170	2.123	2.084	2.050	2.021	1.995	1,972	1.952	1.933	1.917	1.902	1,888	1,875	1,863	1.853	1,843
35	4.121	3.267	2,874	2.641	2.485	2,372	2.285	2.217	2.161	2.114	2.075	2.041	2.012	1.986	1.963	1,942	1.924	1.907	1,892	1,878	1,866	1,854	1,843	1,833
40	4.113	3.259	2,866	2.634	2,477	2,364	2.277	2.209	2.153	2,106	2.067	2.033	2.003	1,977	1.954	1.934	1.915	1,899	1,883	1,870	1.857	1,845	1,834	1,824
45	4.105	3.252	2.859	2.626	2,470	2.356	2.270	2,201	2.145	2.098	2.059	2.025	1.995	1,969	1,946	1.926	1.907	1,890	1,875	1,861	1,848	1,837	1,826	1,816
50	4.098	3.245	2.852	2,619	2.463	2,349	2.262	2.194	2.138	2.091	2.051	2.017	1,988	1,962	1,939	1.918	1,899	1,883	1,867	1.853	1,841	1,829	1,818	1.808
55	4.091	3.238	2.845	2,612	2,456	2,342	2.255	2.187	2.131	2.084	2.044	2.010	1.981	1.954	1.931	1.911	1,892	1,875	1,860	1,846	1,833	1,821	1,810	1,800
60	4.085	3.232	2.839	2.606	2,449	2,336	2.249	2.180	2.124	2.077	2.038	2.003	1,974	1,948	1.924	1.904	1,885	1,868	1.853	1,839	1,826	1,814	1.803	1.793
70	4.079	3.226	2.833	2,600	2,443	2.330	2.243	2.174	2.118	2.071	2.031	1.997	1,967	1.941	1.918	1.897	1,879	1,862	1,846	1,832	1,819	1.807	1,796	1,786
80	4.073	3.220	2.827	2.594	2,438	2,324	2.237	2.168	2.112	2.065	2.025	1.991	1.961	1.935	1.912	1,891	1,872	1,855	1,840	1,826	1,813	1.801	1.790	1,780
90	4.067	3.214	2.822	2.589	2,432	2,318	2.232	2.163	2,106	2.059	2.020	1,985	1,955	1.929	1.906	1,885	1,866	1,849	1,834	1,820	1.807	1,795	1,784	1,773
100	4.062	3.209	2.816	2.584	2,427	2,313	2.226	2.157	2.101	2.054	2.014	1,980	1,950	1.924	1,900	1,879	1,861	1,844	1,828	1,814	1.801	1,789	1,778	1.767
120	4.057	3.204	2.812	2.579	2,422	2.308	2.221	2.152	2.096	2.049	2.009	1,974	1.945	1.918	1,895	1,874	1,855	1,838	1,823	1.808	1,795	1,783	1,772	1,762
150	4.052	3,200	2.807	2.574	2,417	2.304	2.216	2.147	2.091	2.044	2.004	1,969	1,940	1.913	1,890	1,869	1,850	1,833	1.817	1.803	1.790	1,778	1.767	1.756
200	4.047	3.195	2.802	2,570	2,413	2,299	2.212	2.143	2.086	2.039	1.999	1.965	1.935	1.908	1,885	1,864	1,845	1,828	1,812	1,798	1,785	1,773	1,762	1.751
300	4.043	3.191	2,798	2.565	2.409	2.295	2.207	2.138	2.082	2.035	1.995	1,960	1.930	1.904	1,880	1,859	1,840	1,823	1.807	1.793	1,780	1,768	1.757	1,746
400	4.038	3.187	2,794	2.561	2.404	2.290	2.203	2.134	2.077	2.030	1,990	1.956	1.926	1,899	1,876	1,855	1,836	1,819	1.803	1,789	1.775	1.763	1.752	1,742
500	4.034	3.183	2.790	2.557	2,400	2.286	2.199	2.130	2.073	2.026	1.986	1.952	1,921	1,895	1,871	1,850	1,831	1,814	1,798	1,784	1,771	1.759	1.748	1.737
100,000	4.030	3.179	2.786	2.553	2.397	2.283	2.195	2.126	2.069	2.022	1,982	1,947	1.917	1,891	1,867	1,846	1,827	1,810	1,794	1,780	1.767	1.754	1.743	1.733

0.99 quantiles of the Fisher distribution with n degrees of freedom in the denominator and m degrees of freedom in the numerator

	m = 1	2	3	4th	5	6th	7th	8th	9	10	11	12	14th	16	18th	20th	22nd	24	30th	40	50	60	100	100,000
n = 1	4052,	4999	5403	5624	5763	5859	5928	5981	6022	6055	6083	6106	6125	6142	6157	6170	6181	6191	6200	6208	6216	6222	6229	6234
2	98.50	99.00	99.17	99.25	99.30	99.33	99.36	99.37	99.39	99.40	99.41	99.42	99.42	99.43	99.43	99.44	99.44	99.44	99.45	99.45	99.45	99.45	99.46	99.46
3	34.12	30.82	29.46	28.71	28.24	27.91	27.67	27.49	27.35	27.23	27.13	27.05	26.98	26.92	26.87	26.83	26.79	26.75	26.72	26.69	26.66	26.64	26.62	26.60
4th	21.20	18.00	16.69	15.98	15.52	15.21	14.98	14.80	14.66	14.55	14.45	14.37	14.31	14.25	14.20	14.15	14.11	14.08	14.05	14.02	13.99	13.97	13.95	13.93
5	16.26	13.27	12.06	11.39	10.97	10.67	10.46	10.29	10.16	10.05	9.963	9,888	9.825	9.770	9.722	9,680	9,643	9.610	9,580	9,553	9,528	9.506	9.485	9.466
6th	13.75	10.92	9.780	9.148	8,746	8,466	8.260	8.102	7.976	7,874	7.790	7.718	7.657	7.605	7,559	7.519	7.483	7.451	7.422	7.396	7.372	7.351	7.331	7.313
7th	12.25	9.547	8,451	7,847	7.460	7.191	6,993	6.840	6,719	6.620	6.538	6.469	6.410	6.359	6.314	6.275	6.240	6.209	6,181	6.155	6.132	6.111	6,092	6,074
8th	11.26	8,649	7.591	7.006	6.632	6.371	6.178	6.029	5.911	5.814	5.734	5.667	5.609	5.559	5.515	5.477	5.442	5.412	5.384	5.359	5.336	5.316	5.297	5.279
9	10.56	8.022	6,992	6.422	6.057	5.802	5.613	5.467	5.351	5.257	5.178	5.111	5.055	5.005	4,962	4.924	4,890	4,860	4.833	4.808	4,786	4.765	4,746	4,729
10	10.04	7,559	6.552	5,994	5.636	5.386	5,200	5.057	4,942	4,849	4,772	4.706	4,650	4.601	4,558	4,520	4,487	4.457	4,430	4.405	4,383	4.363	4,344	4.327
11	9,646	7.206	6.217	5.668	5.316	5.069	4,886	4,744	4,632	4,539	4.462	4.397	4,342	4,293	4.251	4.213	4,180	4.150	4.123	4.099	4.077	4.057	4.038	4.021
12	9.330	6,927	5,953	5.412	5.064	4.821	4,640	4,499	4.388	4,296	4,220	4.155	4,100	4.052	4.010	3,972	3,939	3.909	3.883	3.858	3.836	3.816	3,798	3,780
13	9,074	6.701	5.739	5.205	4.862	4,620	4,441	4.302	4,191	4,100	4.025	3,960	3.905	3.857	3.815	3,778	3.745	3.716	3,689	3,665	3,643	3,622	3.604	3,587
14th	8,862	6.515	5.564	5.035	4,695	4.456	4.278	4,140	4.030	3,939	3.864	3,800	3.745	3,698	3.656	3,619	3,586	3.556	3.529	3.505	3.483	3.463	3.444	3.427
15th	8,683	6.359	5.417	4.893	4,556	4,318	4,142	4.004	3,895	3.805	3.730	3,666	3,612	3.564	3.522	3.485	3.452	3.423	3.396	3.372	3.350	3.330	3.311	3.294
16	8.531	6.226	5.292	4,773	4,437	4.202	4.026	3.890	3,780	3,691	3,616	3.553	3,498	3.451	3.409	3.372	3,339	3.310	3.283	3.259	3.237	3.216	3,198	3.181
17th	8,400	6.112	5.185	4.669	4,336	4.102	3,927	3.791	3,682	3,593	3.519	3.455	3.401	3.353	3.312	3.275	3.242	3.212	3.186	3.162	3.139	3.119	3.101	3.084
18th	8.285	6.013	5.092	4,579	4,248	4.015	3.841	3.705	3,597	3.508	3.434	3.371	3.316	3.269	3.227	3.190	3.158	3.128	3.101	3.077	3.055	3.035	3.016	2,999
19th	8.185	5.926	5.010	4,500	4.171	3,939	3.765	3.631	3.523	3.434	3.360	3.297	3.242	3.195	3.153	3.116	3.084	3.054	3.027	3.003	2.981	2,961	2,942	2.925
20th	8.096	5.849	4.938	4,431	4.103	3,871	3,699	3.564	3.457	3.368	3.294	3.231	3.177	3.130	3.088	3.051	3.018	2.989	2,962	2.938	2.916	2,895	2,877	2.859
21st	8.017	5.780	4,874	4.369	4.042	3.812	3,640	3.506	3,398	3.310	3.236	3.173	3.119	3.072	3.030	2.993	2.960	2.931	2.904	2,880	2.857	2.837	2.818	2.801
22nd	7.945	5.719	4.817	4,313	3,988	3.758	3,587	3.453	3.346	3.258	3.184	3.121	3.067	3.019	2,978	2.941	2.908	2,879	2.852	2.827	2.805	2.785	2.766	2.749
23	7,881	5.664	4.765	4.264	3,939	3.710	3,539	3.406	3,299	3.211	3.137	3.074	3.020	2,973	2.931	2,894	2.861	2.832	2.805	2.781	2.758	2.738	2.719	2.702
24	7,823	5.614	4.718	4,218	3,895	3,667	3,496	3.363	3.256	3.168	3.094	3.032	2,977	2.930	2,889	2.852	2.819	2,789	2.762	2.738	2.716	2,695	2,676	2.659
25th	7.770	5,568	4,675	4,177	3,855	3,627	3.457	3.324	3.217	3.129	3.056	2.993	2.939	2,892	2,850	2.813	2,780	2.751	2.724	2,699	2,677	2.657	2.638	2,620
26th	7.721	5.526	4,637	4,140	3.818	3,591	3.421	3.288	3.182	3.094	3.021	2.958	2.904	2.857	2.815	2,778	2.745	2.715	2,688	2.664	2,642	2.621	2.602	2.585
27	7.677	5,488	4.601	4.106	3,785	3.558	3.388	3.256	3.149	3.062	2,988	2.926	2.871	2.824	2,783	2.746	2.713	2,683	2.656	2.632	2.609	2.589	2,570	2.552
28	7.636	5.453	4,568	4.074	3.754	3.528	3.358	3.226	3.120	3.032	2.959	2,896	2.842	2,795	2.753	2.716	2,683	2.653	2.626	2.602	2.579	2.559	2.540	2.522
29	7,598	5.420	4,538	4.045	3.725	3,499	3.330	3,198	3.092	3.005	2.931	2,868	2.814	2.767	2.726	2,689	2.656	2.626	2,599	2.574	2.552	2.531	2.512	2,495
30th	7,562	5.390	4,510	4.018	3,699	3.473	3.304	3.173	3.067	2.979	2.906	2.843	2,789	2.742	2,700	2,663	2.630	2,600	2.573	2.549	2.526	2.506	2.487	2.469
31	7.530	5,362	4,484	3.993	3,675	3,449	3.281	3.149	3.043	2.955	2,882	2.820	2.765	2.718	2,677	2,640	2.606	2.577	2,550	2.525	2.502	2,482	2.463	2,445
32	7.499	5.336	4.459	3,969	3,652	3.427	3.258	3.127	3.021	2.934	2.860	2,798	2.744	2,696	2.655	2.618	2.584	2.555	2.527	2.503	2,480	2,460	2,441	2,423
33	7.471	5.312	4,437	3,948	3,630	3.406	3.238	3.106	3,000	2.913	2.840	2,777	2.723	2,676	2.634	2,597	2.564	2.534	2.507	2,482	2,460	2,439	2,420	2.402
34	7.444	5.289	4,416	3,927	3,611	3.386	3.218	3.087	2.981	2,894	2.821	2.758	2.704	2.657	2,615	2.578	2.545	2.515	2.488	2.463	2,440	2,420	2,400	2,383
35	7.419	5.268	4,396	3.908	3,592	3.368	3,200	3.069	2.963	2,876	2.803	2.740	2,686	2,639	2,597	2.560	2.527	2.497	2,470	2,445	2,422	2.401	2.382	2,364
40	7.396	5.248	4.377	3.890	3,574	3.351	3.183	3.052	2.946	2.859	2.786	2.723	2,669	2.622	2,580	2.543	2.510	2,480	2.453	2,428	2.405	2.384	2,365	2,347
45	7.373	5.229	4,360	3.873	3.558	3.334	3.167	3.036	2.930	2.843	2.770	2.707	2.653	2.606	2.564	2.527	2,494	2,464	2,437	2,412	2,389	2,368	2,349	2,331
50	7.353	5.211	4,343	3.858	3,542	3.319	3.152	3.021	2.915	2.828	2.755	2,692	2.638	2.591	2.549	2.512	2,479	2,449	2,421	2.397	2,374	2.353	2,334	2,316
55	7.333	5.194	4.327	3.843	3.528	3.305	3.137	3.006	2.901	2.814	2.741	2.678	2.624	2.577	2.535	2,498	2,465	2,434	2.407	2.382	2,360	2,339	2,319	2.302
60	7.314	5.179	4,313	3.828	3.514	3.291	3.124	2.993	2,888	2.801	2.727	2.665	2,611	2.563	2.522	2.484	2,451	2,421	2,394	2.369	2,346	2,325	2.306	2,288
70	7.296	5.163	4,299	3.815	3,501	3.278	3.111	2.980	2.875	2,788	2.715	2.652	2,598	2.551	2.509	2,472	2,438	2.408	2.381	2.356	2.333	2,312	2,293	2.275
80	7.280	5.149	4,285	3.802	3.488	3.266	3,099	2.968	2.863	2,776	2.703	2,640	2.586	2.539	2.497	2,460	2,426	2,396	2.369	2,344	2,321	2,300	2.281	2.263
90	7.264	5.136	4.273	3.790	3.476	3.254	3.087	2.957	2.851	2.764	2.691	2.629	2.575	2.527	2.485	2,448	2,415	2.385	2.357	2,332	2,310	2,289	2.269	2.251
100	7.248	5.123	4,261	3,778	3.465	3.243	3.076	2.946	2.840	2.754	2,680	2.618	2.564	2.516	2,475	2,437	2.404	2,374	2,346	2,321	2,299	2.278	2.258	2.240
120	7.234	5.110	4,249	3.767	3.454	3.232	3.066	2.935	2.830	2.743	2,670	2.608	2.553	2.506	2,464	2,427	2,393	2.363	2,336	2,311	2,288	2.267	2.248	2.230
150	7.220	5.099	4,238	3.757	3.444	3.222	3.056	2.925	2.820	2.733	2,660	2,598	2.544	2,496	2,454	2,417	2.384	2.353	2,326	2.301	2.278	2.257	2.238	2.220
200	7.207	5.087	4,228	3,747	3.434	3.213	3.046	2.916	2.811	2.724	2,651	2,588	2.534	2.487	2,445	2.408	2,374	2,344	2,316	2.291	2.268	2.247	2.228	2.210
300	7.194	5.077	4,218	3.737	3.425	3.204	3.037	2.907	2.802	2.715	2,642	2.579	2.525	2.478	2,436	2,399	2,365	2,335	2.307	2.282	2.259	2.238	2.219	2,201
400	7.182	5.066	4.208	3.728	3.416	3.195	3.028	2,898	2.793	2.706	2.633	2.571	2.517	2.469	2,427	2,390	2.356	2,326	2,299	2.274	2.251	2.229	2.210	2.192
500	7.171	5.057	4,199	3.720	3.408	3.186	3.020	2,890	2.785	2,698	2.625	2.562	2.508	2,461	2,419	2.382	2,348	2,318	2.290	2.265	2.242	2.221	2.202	2.183
100,000	7.159	5.047	4,191	3.711	3,400	3.178	3.012	2,882	2,777	2,690	2.617	2.555	2,500	2.453	2,411	2,374	2,340	2,310	2.282	2.257	2.234	2.213	2.194	2.175

Individual evidence

1. ^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 383-388 , doi : 10.1515 / 9783110215274 .
2. ↑ Ulrich Krengel : Introduction to Probability Theory and Statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 , p. 246-250 , doi : 10.1007 / 978-3-663-09885-0 .
3. ↑ David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 3-540-21676-6 , pp. 577-579 , doi : 10.1007 / b137972 .