Teoria degli errori e fondamenti di statistica/G

Questo testo è stato riletto e controllato.

Maurizio Loreti - Teoria degli errori e fondamenti di statistica (2006)

G Tabelle

Informazioni sulla fonte del testo

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F.2

H

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[p. 311 modifica]

Appendice G

Tabelle

Nelle pagine seguenti sono riportati alcuni valori tabulati relativi alle distribuzioni normale, di Student, del $\chi ^{2}$ e di Fisher.

Per le tabelle della distribuzione normale, per i valori dell’ascissa compresi tra 0 e 4 sono state calcolate sia l’ordinata della funzione di Gauss standardizzata

$y=f(x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}$

che i valori $I_{1}$ ed $I_{2}$ di due differenti funzioni integrali:

I_{1}=\int _{-x}^{x}f(t)\,\mathrm {d} t

e

I_{2}=\int _{-\infty }^{x}f(t)\,\mathrm {d} t

.

Per la distribuzione di Student, facendo variare il numero di gradi di libertà $N$ (nelle righe della tabella) da 1 a 40, sono riportati i valori dell’ascissa $x$ che corrispondono a differenti aree $P$ (nelle colonne della tabella): in modo che, indicando con $S(t)$ la funzione di frequenza di Student,

$P=\int _{-\infty }^{x}S(t)\,\mathrm {d} t$ .

Per la distribuzione del $\chi ^{2}$ , poi, e sempre per diversi valori di $N$ , sono riportati i valori dell’ascissa $x$ corrispondenti ad aree determinate $P$ , così che (indicando con $C(t)$ la funzione di frequenza del $\chi ^{2}$ ) risulti

$P=\int _{0}^{x}C(t)\,\mathrm {d} t$ .

Per la distribuzione di Fisher, infine, per i soli due valori 0.95 e 0.99 del livello di confidenza $P$ , sono riportati (per differenti gradi di libertà $M$ ed [p. 312 modifica] $N$ ) le ascisse $x$ che corrispondono ad aree uguali al livello di confidenza prescelto; ossia (indicando con $F(w)$ la funzione di Fisher) tali che

P=\int _{0}^{x}F(w)\,\mathrm {d} w

.

(G.1)

Per calcolare i numeri riportati in queste tabelle si è usato un programma in linguaggio C che si serve delle costanti matematiche e delle procedure di calcolo numerico della GNU Scientific Library (GSL); chi volesse maggiori informazioni al riguardo le può trovare sul sito web della Free Software Foundation, sotto la URL http://www.gnu.org/software/gsl/.

La GSL contiene procedure per il calcolo numerico sia delle funzioni di frequenza che di quelle cumulative per tutte le funzioni considerate in questa appendice; e per tutte, meno che per la funzione di Fisher, anche procedure per invertire le distribuzioni cumulative. Per trovare l’ascissa $x$ per cui l’integrale (G.1) raggiunge un valore prefissato si è usato il pacchetto della GSL che permette di trovare gli zeri di una funzione definita dall’utente in un intervallo arbitrario. [p. 313 modifica]

$x$	$y$	$I_{1}$	$I_{2}$	$x$	$y$	$I_{1}$	$I_{2}$
0.00	0.39894	0.00000	0.50000	0.40	0.36827	0.31084	0.65542
0.01	0.39892	0.00798	0.50399	0.41	0.36678	0.31819	0.65910
0.02	0.39886	0.01596	0.50798	0.42	0.36526	0.32551	0.66276
0.03	0.39876	0.02393	0.51197	0.43	0.36371	0.33280	0.66640
0.04	0.39862	0.03191	0.51595	0.44	0.36213	0.34006	0.67003
0.05	0.39844	0.03988	0.51994	0.45	0.36053	0.34729	0.67364
0.06	0.39822	0.04784	0.52392	0.46	0.35889	0.35448	0.67724
0.07	0.39797	0.05581	0.52790	0.47	0.35723	0.36164	0.68082
0.08	0.39767	0.06376	0.53188	0.48	0.35553	0.36877	0.68439
0.09	0.39733	0.07171	0.53586	0.49	0.35381	0.37587	0.68793
0.10	0.39695	0.07966	0.53983	0.50	0.35207	0.38292	0.69146
0.11	0.39654	0.08759	0.54380	0.51	0.35029	0.38995	0.69497
0.12	0.39608	0.09552	0.54776	0.52	0.34849	0.39694	0.69847
0.13	0.39559	0.10343	0.55172	0.53	0.34667	0.40389	0.70194
0.14	0.39505	0.11134	0.55567	0.54	0.34482	0.41080	0.70540
0.15	0.39448	0.11924	0.55962	0.55	0.34294	0.41768	0.70884
0.16	0.39387	0.12712	0.56356	0.56	0.34105	0.42452	0.71226
0.17	0.39322	0.13499	0.56749	0.57	0.33912	0.43132	0.71566
0.18	0.39253	0.14285	0.57142	0.58	0.33718	0.43809	0.71904
0.19	0.39181	0.15069	0.57535	0.59	0.33521	0.44481	0.72240
0.20	0.39104	0.15852	0.57926	0.60	0.33322	0.45149	0.72575
0.21	0.39024	0.16633	0.58317	0.61	0.33121	0.45814	0.72907
0.22	0.38940	0.17413	0.58706	0.62	0.32918	0.46474	0.73237
0.23	0.38853	0.18191	0.59095	0.63	0.32713	0.47131	0.73565
0.24	0.38762	0.18967	0.59483	0.64	0.32506	0.47783	0.73891
0.25	0.38667	0.19741	0.59871	0.65	0.32297	0.48431	0.74215
0.26	0.38568	0.20514	0.60257	0.66	0.32086	0.49075	0.74537
0.27	0.38466	0.21284	0.60642	0.67	0.31874	0.49714	0.74857
0.28	0.38361	0.22052	0.61026	0.68	0.31659	0.50350	0.75175
0.29	0.38251	0.22818	0.61409	0.69	0.31443	0.50981	0.75490
0.30	0.38139	0.23582	0.61791	0.70	0.31225	0.51607	0.75804
0.31	0.38023	0.24344	0.62172	0.71	0.31006	0.52230	0.76115
0.32	0.37903	0.25103	0.62552	0.72	0.30785	0.52848	0.76424
0.33	0.37780	0.25860	0.62930	0.73	0.30563	0.53461	0.76730
0.34	0.37654	0.26614	0.63307	0.74	0.30339	0.54070	0.77035
0.35	0.37524	0.27366	0.63683	0.75	0.30114	0.54675	0.77337
0.36	0.37391	0.28115	0.64058	0.76	0.29887	0.55275	0.77637
0.37	0.37255	0.28862	0.64431	0.77	0.29659	0.55870	0.77935
0.38	0.37115	0.29605	0.64803	0.78	0.29431	0.56461	0.78230
0.39	0.36973	0.30346	0.65173	0.79	0.29200	0.57047	0.78524

Tabelle della distribuzione normale (I)

[p. 314 modifica]

$x$	$y$	$I_{1}$	$I_{2}$	$x$	$y$	$I_{1}$	$I_{2}$
0.80	0.28969	0.57629	0.78814	1.20	0.19419	0.76986	0.88493
0.81	0.28737	0.58206	0.79103	1.21	0.19186	0.77372	0.88686
0.82	0.28504	0.58778	0.79389	1.22	0.18954	0.77754	0.88877
0.83	0.28269	0.59346	0.79673	1.23	0.18724	0.78130	0.89065
0.84	0.28034	0.59909	0.79955	1.24	0.18494	0.78502	0.89251
0.85	0.27798	0.60467	0.80234	1.25	0.18265	0.78870	0.89435
0.86	0.27562	0.61021	0.80511	1.26	0.18037	0.79233	0.89617
0.87	0.27324	0.61570	0.80785	1.27	0.17810	0.79592	0.89796
0.88	0.27086	0.62114	0.81057	1.28	0.17585	0.79945	0.89973
0.89	0.26848	0.62653	0.81327	1.29	0.17360	0.80295	0.90147
0.90	0.26609	0.63188	0.81594	1.30	0.17137	0.80640	0.90320
0.91	0.26369	0.63718	0.81859	1.31	0.16915	0.80980	0.90490
0.92	0.26129	0.64243	0.82121	1.32	0.16694	0.81316	0.90658
0.93	0.25888	0.64763	0.82381	1.33	0.16474	0.81648	0.90824
0.94	0.25647	0.65278	0.82639	1.34	0.16256	0.81975	0.90988
0.95	0.25406	0.65789	0.82894	1.35	0.16038	0.82298	0.91149
0.96	0.25164	0.66294	0.83147	1.36	0.15822	0.82617	0.91309
0.97	0.24923	0.66795	0.83398	1.37	0.15608	0.82931	0.91466
0.98	0.24681	0.67291	0.83646	1.38	0.15395	0.83241	0.91621
0.99	0.24439	0.67783	0.83891	1.39	0.15183	0.83547	0.91774
1.00	0.24197	0.68269	0.84134	1.40	0.14973	0.83849	0.91924
1.01	0.23955	0.68750	0.84375	1.41	0.14764	0.84146	0.92073
1.02	0.23713	0.69227	0.84614	1.42	0.14556	0.84439	0.92220
1.03	0.23471	0.69699	0.84849	1.43	0.14350	0.84728	0.92364
1.04	0.23230	0.70166	0.85083	1.44	0.14146	0.85013	0.92507
1.05	0.22988	0.70628	0.85314	1.45	0.13943	0.85294	0.92647
1.06	0.22747	0.71086	0.85543	1.46	0.13742	0.85571	0.92785
1.07	0.22506	0.71538	0.85769	1.47	0.13542	0.85844	0.92922
1.08	0.22265	0.71986	0.85993	1.48	0.13344	0.86113	0.93056
1.09	0.22025	0.72429	0.86214	1.49	0.13147	0.86378	0.93189
1.10	0.21785	0.72867	0.86433	1.50	0.12952	0.86639	0.93319
1.11	0.21546	0.73300	0.86650	1.51	0.12758	0.86896	0.93448
1.12	0.21307	0.73729	0.86864	1.52	0.12566	0.87149	0.93574
1.13	0.21069	0.74152	0.87076	1.53	0.12376	0.87398	0.93699
1.14	0.20831	0.74571	0.87286	1.54	0.12188	0.87644	0.93822
1.15	0.20594	0.74986	0.87493	1.55	0.12001	0.87886	0.93943
1.16	0.20357	0.75395	0.87698	1.56	0.11816	0.88124	0.94062
1.17	0.20121	0.75800	0.87900	1.57	0.11632	0.88358	0.94179
1.18	0.19886	0.76200	0.88100	1.58	0.11450	0.88589	0.94295
1.19	0.19652	0.76595	0.88298	1.59	0.11270	0.88817	0.94408

Tabelle della distribuzione normale (II)

[p. 315 modifica]

$x$	$y$	$I_{1}$	$I_{2}$	$x$	$y$	$I_{1}$	$I_{2}$
1.60	0.11092	0.89040	0.94520	2.00	0.05399	0.95450	0.97725
1.61	0.10915	0.89260	0.94630	2.01	0.05292	0.95557	0.97778
1.62	0.10741	0.89477	0.94738	2.02	0.05186	0.95662	0.97831
1.63	0.10567	0.89690	0.94845	2.03	0.05082	0.95764	0.97882
1.64	0.10396	0.89899	0.94950	2.04	0.04980	0.95865	0.97932
1.65	0.10226	0.90106	0.95053	2.05	0.04879	0.95964	0.97982
1.66	0.10059	0.90309	0.95154	2.06	0.04780	0.96060	0.98030
1.67	0.09893	0.90508	0.95254	2.07	0.04682	0.96155	0.98077
1.68	0.09728	0.90704	0.95352	2.08	0.04586	0.96247	0.98124
1.69	0.09566	0.90897	0.95449	2.09	0.04491	0.96338	0.98169
1.70	0.09405	0.91087	0.95543	2.10	0.04398	0.96427	0.98214
1.71	0.09246	0.91273	0.95637	2.11	0.04307	0.96514	0.98257
1.72	0.09089	0.91457	0.95728	2.12	0.04217	0.96599	0.98300
1.73	0.08933	0.91637	0.95818	2.13	0.04128	0.96683	0.98341
1.74	0.08780	0.91814	0.95907	2.14	0.04041	0.96765	0.98382
1.75	0.08628	0.91988	0.95994	2.15	0.03955	0.96844	0.98422
1.76	0.08478	0.92159	0.96080	2.16	0.03871	0.96923	0.98461
1.77	0.08329	0.92327	0.96164	2.17	0.03788	0.96999	0.98500
1.78	0.08183	0.92492	0.96246	2.18	0.03706	0.97074	0.98537
1.79	0.08038	0.92655	0.96327	2.19	0.03626	0.97148	0.98574
1.80	0.07895	0.92814	0.96407	2.20	0.03547	0.97219	0.98610
1.81	0.07754	0.92970	0.96485	2.21	0.03470	0.97289	0.98645
1.82	0.07614	0.93124	0.96562	2.22	0.03394	0.97358	0.98679
1.83	0.07477	0.93275	0.96638	2.23	0.03319	0.97425	0.98713
1.84	0.07341	0.93423	0.96712	2.24	0.03246	0.97491	0.98745
1.85	0.07206	0.93569	0.96784	2.25	0.03174	0.97555	0.98778
1.86	0.07074	0.93711	0.96856	2.26	0.03103	0.97618	0.98809
1.87	0.06943	0.93852	0.96926	2.27	0.03034	0.97679	0.98840
1.88	0.06814	0.93989	0.96995	2.28	0.02965	0.97739	0.98870
1.89	0.06687	0.94124	0.97062	2.29	0.02898	0.97798	0.98899
1.90	0.06562	0.94257	0.97128	2.30	0.02833	0.97855	0.98928
1.91	0.06438	0.94387	0.97193	2.31	0.02768	0.97911	0.98956
1.92	0.06316	0.94514	0.97257	2.32	0.02705	0.97966	0.98983
1.93	0.06195	0.94639	0.97320	2.33	0.02643	0.98019	0.99010
1.94	0.06077	0.94762	0.97381	2.34	0.02582	0.98072	0.99036
1.95	0.05959	0.94882	0.97441	2.35	0.02522	0.98123	0.99061
1.96	0.05844	0.95000	0.97500	2.36	0.02463	0.98173	0.99086
1.97	0.05730	0.95116	0.97558	2.37	0.02406	0.98221	0.99111
1.98	0.05618	0.95230	0.97615	2.38	0.02349	0.98269	0.99134
1.99	0.05508	0.95341	0.97670	2.39	0.02294	0.98315	0.99158

Tabelle della distribuzione normale (III)

[p. 316 modifica]

$x$	$y$	$I_{1}$	$I_{2}$	$x$	$y$	$I_{1}$	$I_{2}$
2.40	0.02239	0.98360	0.99180	2.80	0.00792	0.99489	0.99744
2.41	0.02186	0.98405	0.99202	2.81	0.00770	0.99505	0.99752
2.42	0.02134	0.98448	0.99224	2.82	0.00748	0.99520	0.99760
2.43	0.02083	0.98490	0.99245	2.83	0.00727	0.99535	0.99767
2.44	0.02033	0.98531	0.99266	2.84	0.00707	0.99549	0.99774
2.45	0.01984	0.98571	0.99286	2.85	0.00687	0.99563	0.99781
2.46	0.01936	0.98611	0.99305	2.86	0.00668	0.99576	0.99788
2.47	0.01888	0.98649	0.99324	2.87	0.00649	0.99590	0.99795
2.48	0.01842	0.98686	0.99343	2.88	0.00631	0.99602	0.99801
2.49	0.01797	0.98723	0.99361	2.89	0.00613	0.99615	0.99807
2.50	0.01753	0.98758	0.99379	2.90	0.00595	0.99627	0.99813
2.51	0.01709	0.98793	0.99396	2.91	0.00578	0.99639	0.99819
2.52	0.01667	0.98826	0.99413	2.92	0.00562	0.99650	0.99825
2.53	0.01625	0.98859	0.99430	2.93	0.00545	0.99661	0.99831
2.54	0.01585	0.98891	0.99446	2.94	0.00530	0.99672	0.99836
2.55	0.01545	0.98923	0.99461	2.95	0.00514	0.99682	0.99841
2.56	0.01506	0.98953	0.99477	2.96	0.00499	0.99692	0.99846
2.57	0.01468	0.98983	0.99492	2.97	0.00485	0.99702	0.99851
2.58	0.01431	0.99012	0.99506	2.98	0.00470	0.99712	0.99856
2.59	0.01394	0.99040	0.99520	2.99	0.00457	0.99721	0.99861
2.60	0.01358	0.99068	0.99534	3.00	0.00443	0.99730	0.99865
2.61	0.01323	0.99095	0.99547	3.01	0.00430	0.99739	0.99869
2.62	0.01289	0.99121	0.99560	3.02	0.00417	0.99747	0.99874
2.63	0.01256	0.99146	0.99573	3.03	0.00405	0.99755	0.99878
2.64	0.01223	0.99171	0.99585	3.04	0.00393	0.99763	0.99882
2.65	0.01191	0.99195	0.99598	3.05	0.00381	0.99771	0.99886
2.66	0.01160	0.99219	0.99609	3.06	0.00370	0.99779	0.99889
2.67	0.01130	0.99241	0.99621	3.07	0.00358	0.99786	0.99893
2.68	0.01100	0.99264	0.99632	3.08	0.00348	0.99793	0.99896
2.69	0.01071	0.99285	0.99643	3.09	0.00337	0.99800	0.99900
2.70	0.01042	0.99307	0.99653	3.10	0.00327	0.99806	0.99903
2.71	0.01014	0.99327	0.99664	3.11	0.00317	0.99813	0.99906
2.72	0.00987	0.99347	0.99674	3.12	0.00307	0.99819	0.99910
2.73	0.00961	0.99367	0.99683	3.13	0.00298	0.99825	0.99913
2.74	0.00935	0.99386	0.99693	3.14	0.00288	0.99831	0.99916
2.75	0.00909	0.99404	0.99702	3.15	0.00279	0.99837	0.99918
2.76	0.00885	0.99422	0.99711	3.16	0.00271	0.99842	0.99921
2.77	0.00861	0.99439	0.99720	3.17	0.00262	0.99848	0.99924
2.78	0.00837	0.99456	0.99728	3.18	0.00254	0.99853	0.99926
2.79	0.00814	0.99473	0.99736	3.19	0.00246	0.99858	0.99929

Tabelle della distribuzione normale (IV)

[p. 317 modifica]

$x$	$y$	$I_{1}$	$I_{2}$	$x$	$y$	$I_{1}$	$I_{2}$
3.20	0.00238	0.99863	0.99931	3.60	0.00061	0.99968	0.99984
3.21	0.00231	0.99867	0.99934	3.61	0.00059	0.99969	0.99985
3.22	0.00224	0.99872	0.99936	3.62	0.00057	0.99971	0.99985
3.23	0.00216	0.99876	0.99938	3.63	0.00055	0.99972	0.99986
3.24	0.00210	0.99880	0.99940	3.64	0.00053	0.99973	0.99986
3.25	0.00203	0.99885	0.99942	3.65	0.00051	0.99974	0.99987
3.26	0.00196	0.99889	0.99944	3.66	0.00049	0.99975	0.99987
3.27	0.00190	0.99892	0.99946	3.67	0.00047	0.99976	0.99988
3.28	0.00184	0.99896	0.99948	3.68	0.00046	0.99977	0.99988
3.29	0.00178	0.99900	0.99950	3.69	0.00044	0.99978	0.99989
3.30	0.00172	0.99903	0.99952	3.70	0.00042	0.99978	0.99989
3.31	0.00167	0.99907	0.99953	3.71	0.00041	0.99979	0.99990
3.32	0.00161	0.99910	0.99955	3.72	0.00039	0.99980	0.99990
3.33	0.00156	0.99913	0.99957	3.73	0.00038	0.99981	0.99990
3.34	0.00151	0.99916	0.99958	3.74	0.00037	0.99982	0.99991
3.35	0.00146	0.99919	0.99960	3.75	0.00035	0.99982	0.99991
3.36	0.00141	0.99922	0.99961	3.76	0.00034	0.99983	0.99992
3.37	0.00136	0.99925	0.99962	3.77	0.00033	0.99984	0.99992
3.38	0.00132	0.99928	0.99964	3.78	0.00031	0.99984	0.99992
3.39	0.00127	0.99930	0.99965	3.79	0.00030	0.99985	0.99992
3.40	0.00123	0.99933	0.99966	3.80	0.00029	0.99986	0.99993
3.41	0.00119	0.99935	0.99968	3.81	0.00028	0.99986	0.99993
3.42	0.00115	0.99937	0.99969	3.82	0.00027	0.99987	0.99993
3.43	0.00111	0.99940	0.99970	3.83	0.00026	0.99987	0.99994
3.44	0.00107	0.99942	0.99971	3.84	0.00025	0.99988	0.99994
3.45	0.00104	0.99944	0.99972	3.85	0.00024	0.99988	0.99994
3.46	0.00100	0.99946	0.99973	3.86	0.00023	0.99989	0.99994
3.47	0.00097	0.99948	0.99974	3.87	0.00022	0.99989	0.99995
3.48	0.00094	0.99950	0.99975	3.88	0.00021	0.99990	0.99995
3.49	0.00090	0.99952	0.99976	3.89	0.00021	0.99990	0.99995
3.50	0.00087	0.99953	0.99977	3.90	0.00020	0.99990	0.99995
3.51	0.00084	0.99955	0.99978	3.91	0.00019	0.99991	0.99995
3.52	0.00081	0.99957	0.99978	3.92	0.00018	0.99991	0.99996
3.53	0.00079	0.99958	0.99979	3.93	0.00018	0.99992	0.99996
3.54	0.00076	0.99960	0.99980	3.94	0.00017	0.99992	0.99996
3.55	0.00073	0.99961	0.99981	3.95	0.00016	0.99992	0.99996
3.56	0.00071	0.99963	0.99981	3.96	0.00016	0.99993	0.99996
3.57	0.00068	0.99964	0.99982	3.97	0.00015	0.99993	0.99996
3.58	0.00066	0.99966	0.99983	3.98	0.00014	0.99993	0.99997
3.59	0.00063	0.99967	0.99983	3.99	0.00014	0.99993	0.99997

Tabelle della distribuzione normale (V)

[p. 318 modifica]

Probabilità (in percentuale)

$N$	99.9	99.8	99.5	99.0	98.0	95.0	90.0	80.0	75.0	60.0
1	318.309	159.153	63.657	31.821	15.895	6.314	3.078	1.376	1.000	0.325
2	22.327	15.764	9.925	6.965	4.849	2.920	1.886	1.061	0.816	0.289
3	10.215	8.053	5.841	4.541	3.482	2.353	1.638	0.978	0.765	0.277
4	7.173	5.951	4.604	3.747	2.999	2.132	1.533	0.941	0.741	0.271
5	5.893	5.030	4.032	3.365	2.757	2.015	1.476	0.920	0.727	0.267
6	5.208	4.524	3.707	3.143	2.612	1.943	1.440	0.906	0.718	0.265
7	4.785	4.207	3.499	2.998	2.517	1.895	1.415	0.896	0.711	0.263
8	4.501	3.991	3.355	2.896	2.449	1.860	1.397	0.889	0.706	0.262
9	4.297	3.835	3.250	2.821	2.398	1.833	1.383	0.883	0.703	0.261
10	4.144	3.716	3.169	2.764	2.359	1.812	1.372	0.879	0.700	0.260
11	4.025	3.624	3.106	2.718	2.328	1.796	1.363	0.876	0.697	0.260
12	3.930	3.550	3.055	2.681	2.303	1.782	1.356	0.873	0.695	0.259
13	3.852	3.489	3.012	2.650	2.282	1.771	1.350	0.870	0.694	0.259
14	3.787	3.438	2.977	2.624	2.264	1.761	1.345	0.868	0.692	0.258
15	3.733	3.395	2.947	2.602	2.249	1.753	1.341	0.866	0.691	0.258
16	3.686	3.358	2.921	2.583	2.235	1.746	1.337	0.865	0.690	0.258
17	3.646	3.326	2.898	2.567	2.224	1.740	1.333	0.863	0.689	0.257
18	3.610	3.298	2.878	2.552	2.214	1.734	1.330	0.862	0.688	0.257
19	3.579	3.273	2.861	2.539	2.205	1.729	1.328	0.861	0.688	0.257
20	3.552	3.251	2.845	2.528	2.197	1.725	1.325	0.860	0.687	0.257
21	3.527	3.231	2.831	2.518	2.189	1.721	1.323	0.859	0.686	0.257
22	3.505	3.214	2.819	2.508	2.183	1.717	1.321	0.858	0.686	0.256
23	3.485	3.198	2.807	2.500	2.177	1.714	1.319	0.858	0.685	0.256
24	3.467	3.183	2.797	2.492	2.172	1.711	1.318	0.857	0.685	0.256
25	3.450	3.170	2.787	2.485	2.167	1.708	1.316	0.856	0.684	0.256
26	3.435	3.158	2.779	2.479	2.162	1.706	1.315	0.856	0.684	0.256
27	3.421	3.147	2.771	2.473	2.158	1.703	1.314	0.855	0.684	0.256
28	3.408	3.136	2.763	2.467	2.154	1.701	1.313	0.855	0.683	0.256
29	3.396	3.127	2.756	2.462	2.150	1.699	1.311	0.854	0.683	0.256
30	3.385	3.118	2.750	2.457	2.147	1.697	1.310	0.854	0.683	0.256
31	3.375	3.109	2.744	2.453	2.144	1.696	1.309	0.853	0.682	0.256
32	3.365	3.102	2.738	2.449	2.141	1.694	1.309	0.853	0.682	0.255
33	3.356	3.094	2.733	2.445	2.138	1.692	1.308	0.853	0.682	0.255
34	3.348	3.088	2.728	2.441	2.136	1.691	1.307	0.852	0.682	0.255
35	3.340	3.081	2.724	2.438	2.133	1.690	1.306	0.852	0.682	0.255
36	3.333	3.075	2.719	2.434	2.131	1.688	1.306	0.852	0.681	0.255
37	3.326	3.070	2.715	2.431	2.129	1.687	1.305	0.851	0.681	0.255
38	3.319	3.064	2.712	2.429	2.127	1.686	1.304	0.851	0.681	0.255
39	3.313	3.059	2.708	2.426	2.125	1.685	1.304	0.851	0.681	0.255
40	3.307	3.055	2.704	2.423	2.123	1.684	1.303	0.851	0.681	0.255

Percentili della distribuzione di Student

[p. 319 modifica]

Probabilità (in percentuale)

$N$	99.9	99.5	99.0	95.0	90.0	75.0	50.0	25.0	10.0	5.0	1.0	0.5	0.1
1	10.83	7.88	6.63	3.84	2.71	1.32	0.45	0.10	0.02	0.00	0.00	0.00	0.00
2	13.82	10.60	9.21	5.99	4.61	2.77	1.39	0.58	0.21	0.10	0.02	0.01	0.00
3	16.27	12.84	11.34	7.81	6.25	4.11	2.37	1.21	0.58	0.35	0.11	0.07	0.02
4	18.47	14.86	13.28	9.49	7.78	5.39	3.36	1.92	1.06	0.71	0.30	0.21	0.09
5	20.52	16.75	15.09	11.07	9.24	6.63	4.35	2.67	1.61	1.15	0.55	0.41	0.21
6	22.46	18.55	16.81	12.59	10.64	7.84	5.35	3.45	2.20	1.64	0.87	0.68	0.38
7	24.32	20.28	18.48	14.07	12.02	9.04	6.35	4.25	2.83	2.17	1.24	0.99	0.60
8	26.12	21.95	20.09	15.51	13.36	10.22	7.34	5.07	3.49	2.73	1.65	1.34	0.86
9	27.88	23.59	21.67	16.92	14.68	11.39	8.34	5.90	4.17	3.33	2.09	1.73	1.15
10	29.59	25.19	23.21	18.31	15.99	12.55	9.34	6.74	4.87	3.94	2.56	2.16	1.48
11	31.26	26.76	24.72	19.68	17.28	13.70	10.34	7.58	5.58	4.57	3.05	2.60	1.83
12	32.91	28.30	26.22	21.03	18.55	14.85	11.34	8.44	6.30	5.23	3.57	3.07	2.21
13	34.53	29.82	27.69	22.36	19.81	15.98	12.34	9.30	7.04	5.89	4.11	3.57	2.62
14	36.12	31.32	29.14	23.68	21.06	17.12	13.34	10.17	7.79	6.57	4.66	4.07	3.04
15	37.70	32.80	30.58	25.00	22.31	18.25	14.34	11.04	8.55	7.26	5.23	4.60	3.48
16	39.25	34.27	32.00	26.30	23.54	19.37	15.34	11.91	9.31	7.96	5.81	5.14	3.94
17	40.79	35.72	33.41	27.59	24.77	20.49	16.34	12.79	10.09	8.67	6.41	5.70	4.42
18	42.31	37.16	34.81	28.87	25.99	21.60	17.34	13.68	10.86	9.39	7.01	6.26	4.90
19	43.82	38.58	36.19	30.14	27.20	22.72	18.34	14.56	11.65	10.12	7.63	6.84	5.41
20	45.31	40.00	37.57	31.41	28.41	23.83	19.34	15.45	12.44	10.85	8.26	7.43	5.92
21	46.80	41.40	38.93	32.67	29.62	24.93	20.34	16.34	13.24	11.59	8.90	8.03	6.45
22	48.27	42.80	40.29	33.92	30.81	26.04	21.34	17.24	14.04	12.34	9.54	8.64	6.98
23	49.73	44.18	41.64	35.17	32.01	27.14	22.34	18.14	14.85	13.09	10.20	9.26	7.53
24	51.18	45.56	42.98	36.42	33.20	28.24	23.34	19.04	15.66	13.85	10.86	9.89	8.08
25	52.62	46.93	44.31	37.65	34.38	29.34	24.34	19.94	16.47	14.61	11.52	10.52	8.65
26	54.05	48.29	45.64	38.89	35.56	30.43	25.34	20.84	17.29	15.38	12.20	11.16	9.22
27	55.48	49.64	46.96	40.11	36.74	31.53	26.34	21.75	18.11	16.15	12.88	11.81	9.80
28	56.89	50.99	48.28	41.34	37.92	32.62	27.34	22.66	18.94	16.93	13.56	12.46	10.39
29	58.30	52.34	49.59	42.56	39.09	33.71	28.34	23.57	19.77	17.71	14.26	13.12	10.99
30	59.70	53.67	50.89	43.77	40.26	34.80	29.34	24.48	20.60	18.49	14.95	13.79	11.59

Percentili della distribuzione del $\chi ^{2}$

[p. 320 modifica]

$M$

$N$	1	2	3	4	5	6	8	12	24	36	48	$\infty$
1	161.45	199.50	215.71	224.58	230.16	233.99	238.88	243.91	249.05	250.79	251.67	254.31
2	18.51	19.00	19.16	19.25	19.30	19.33	19.37	19.41	19.45	19.47	19.47	19.50
3	10.13	9.55	9.28	9.12	9.01	8.94	8.85	8.74	8.64	8.60	8.58	8.53
4	7.71	6.94	6.59	6.39	6.26	6.16	6.04	5.91	5.77	5.73	5.70	5.63
5	6.61	5.79	5.41	5.19	5.05	4.95	4.82	4.68	4.53	4.47	4.45	4.37
6	5.99	5.14	4.76	4.53	4.39	4.28	4.15	4.00	3.84	3.79	3.76	3.67
7	5.59	4.74	4.35	4.12	3.97	3.87	3.73	3.57	3.41	3.35	3.32	3.23
8	5.32	4.46	4.07	3.84	3.69	3.58	3.44	3.28	3.12	3.06	3.02	2.93
9	5.12	4.26	3.86	3.63	3.48	3.37	3.23	3.07	2.90	2.84	2.81	2.71
10	4.96	4.10	3.71	3.48	3.33	3.22	3.07	2.91	2.74	2.67	2.64	2.54
11	4.84	3.98	3.59	3.36	3.20	3.09	2.95	2.79	2.61	2.54	2.51	2.40
12	4.75	3.89	3.49	3.26	3.11	3.00	2.85	2.69	2.51	2.44	2.41	2.30
13	4.67	3.81	3.41	3.18	3.03	2.92	2.77	2.60	2.42	2.35	2.32	2.21
14	4.60	3.74	3.34	3.11	2.96	2.85	2.70	2.53	2.35	2.28	2.24	2.13
15	4.54	3.68	3.29	3.06	2.90	2.79	2.64	2.48	2.29	2.22	2.18	2.07
16	4.49	3.63	3.24	3.01	2.85	2.74	2.59	2.42	2.24	2.17	2.13	2.01
17	4.45	3.59	3.20	2.96	2.81	2.70	2.55	2.38	2.19	2.12	2.08	1.96
18	4.41	3.55	3.16	2.93	2.77	2.66	2.51	2.34	2.15	2.08	2.04	1.92
19	4.38	3.52	3.13	2.90	2.74	2.63	2.48	2.31	2.11	2.04	2.00	1.88
20	4.35	3.49	3.10	2.87	2.71	2.60	2.45	2.28	2.08	2.01	1.97	1.84
21	4.32	3.47	3.07	2.84	2.68	2.57	2.42	2.25	2.05	1.98	1.94	1.81
22	4.30	3.44	3.05	2.82	2.66	2.55	2.40	2.23	2.03	1.95	1.91	1.78
23	4.28	3.42	3.03	2.80	2.64	2.53	2.37	2.20	2.01	1.93	1.89	1.76
24	4.26	3.40	3.01	2.78	2.62	2.51	2.36	2.18	1.98	1.91	1.87	1.73
25	4.24	3.39	2.99	2.76	2.60	2.49	2.34	2.16	1.96	1.89	1.85	1.71
30	4.17	3.32	2.92	2.69	2.53	2.42	2.27	2.09	1.89	1.81	1.77	1.62
40	4.08	3.23	2.84	2.61	2.45	2.34	2.18	2.00	1.79	1.71	1.67	1.51
60	4.00	3.15	2.76	2.53	2.37	2.25	2.10	1.92	1.70	1.61	1.57	1.39
120	3.92	3.07	2.68	2.45	2.29	2.18	2.02	1.83	1.61	1.52	1.46	1.25
$\infty$	3.84	3.00	2.60	2.37	2.21	2.10	1.94	1.75	1.52	1.42	1.36	1.00

Percentili della distribuzione di Fisher per $P=0.95$

[p. 321 modifica]

$M$

$N$	1	2	3	4	5	6	8	12	24	36	48	$\infty$
1	4052.02	4999.47	5403.35	5624.58	5763.65	5858.99	5981.32	6106.44	6234.68	6278.10	6299.92	6365.88
2	98.50	99.00	99.17	99.25	99.30	99.33	99.37	99.42	99.46	99.47	99.48	99.50
3	34.12	30.82	29.46	28.71	28.24	27.91	27.49	27.05	26.60	26.44	26.36	26.13
4	21.20	18.00	16.69	15.98	15.52	15.21	14.80	14.37	13.93	13.78	13.70	13.46
5	16.26	13.27	12.06	11.39	10.97	10.67	10.29	9.89	9.47	9.32	9.25	9.02
6	13.74	10.92	9.78	9.15	8.75	8.47	8.10	7.72	7.31	7.17	7.10	6.88
7	12.25	9.55	8.45	7.85	7.46	7.19	6.84	6.47	6.07	5.94	5.87	5.65
8	11.26	8.65	7.59	7.01	6.63	6.37	6.03	5.67	5.28	5.14	5.07	4.86
9	10.56	8.02	6.99	6.42	6.06	5.80	5.47	5.11	4.73	4.59	4.53	4.31
10	10.04	7.56	6.55	5.99	5.64	5.39	5.06	4.71	4.33	4.19	4.12	3.91
11	9.65	7.21	6.22	5.67	5.32	5.07	4.74	4.40	4.02	3.89	3.82	3.60
12	9.33	6.93	5.95	5.41	5.06	4.82	4.50	4.16	3.78	3.65	3.58	3.36
13	9.07	6.70	5.74	5.21	4.86	4.62	4.30	3.96	3.59	3.45	3.38	3.17
14	8.86	6.51	5.56	5.04	4.69	4.46	4.14	3.80	3.43	3.29	3.22	3.00
15	8.68	6.36	5.42	4.89	4.56	4.32	4.00	3.67	3.29	3.16	3.09	2.87
16	8.53	6.23	5.29	4.77	4.44	4.20	3.89	3.55	3.18	3.05	2.98	2.75
17	8.40	6.11	5.18	4.67	4.34	4.10	3.79	3.46	3.08	2.95	2.88	2.65
18	8.29	6.01	5.09	4.58	4.25	4.01	3.71	3.37	3.00	2.86	2.79	2.57
19	8.18	5.93	5.01	4.50	4.17	3.94	3.63	3.30	2.92	2.79	2.72	2.49
20	8.10	5.85	4.94	4.43	4.10	3.87	3.56	3.23	2.86	2.72	2.65	2.42
21	8.02	5.78	4.87	4.37	4.04	3.81	3.51	3.17	2.80	2.66	2.59	2.36
22	7.95	5.72	4.82	4.31	3.99	3.76	3.45	3.12	2.75	2.61	2.54	2.31
23	7.88	5.66	4.76	4.26	3.94	3.71	3.41	3.07	2.70	2.56	2.49	2.26
24	7.82	5.61	4.72	4.22	3.90	3.67	3.36	3.03	2.66	2.52	2.45	2.21
25	7.77	5.57	4.68	4.18	3.85	3.63	3.32	2.99	2.62	2.48	2.41	2.17
30	7.56	5.39	4.51	4.02	3.70	3.47	3.17	2.84	2.47	2.33	2.25	2.01
40	7.31	5.18	4.31	3.83	3.51	3.29	2.99	2.66	2.29	2.14	2.07	1.80
60	7.08	4.98	4.13	3.65	3.34	3.12	2.82	2.50	2.12	1.97	1.89	1.60
120	6.85	4.79	3.95	3.48	3.17	2.96	2.66	2.34	1.95	1.80	1.71	1.38
$\infty$	6.63	4.61	3.78	3.32	3.02	2.80	2.51	2.18	1.79	1.63	1.54	1.00

Percentili della distribuzione di Fisher per $P=0.99$