Some standard content:
National Standard of the People's Republic of China
Numerical tables of statistical distribution
t distribution
Tables for statistical distributionst-distribution
UDC 311.13(083.5)
GB 4086.3--83
This standard includes two numerical tables of distributions commonly used in statistics. Their names, table intervals and precision are as follows: 1 Distribution function table
t distribution quantile table
=1(1)20(2)30(5)50,60
t =0(0.1)4(0.2)5(0.5)7
b= 0.5(0.05)0.95.0.975. 0.98.0.99. 0.995.0.9975, 0.998, 0.999, 0.9995, 0.9998, 0.9999V=1(1)30(2)40(5)70(10)120150，180，240,cxThe quantiles in the table correspond to the lower probability. Although 5 to 6 decimal places are given in the table, the number of decimal places to be used in use depends on the actual problem. If the requirements cannot be met in the application, please refer to the processing method in the appendix. Issued by the National Bureau of Standards on December 21, 1983
6 decimal places
5 decimal places
19841001 implementation
838586
858537
864200
.89081
.897584
900674
.931862
951410 | 88570
.824158
. 857766
871996
.884708
936717
.942414
981619
983998
987684
290344
992304
994085
8:996303||tt ||868274
969976
981889
990448
.996255
.903048
.97587
8:999542
959761
994317
The table gives the values of the distribution function P() for the degree and t. Example: For v=8 and t=3.9, P(t;V)=0.997728. 0.774999
992467
965531
998857
999838
999906
887047
902486|| tt||8:994584
.996638
.998847
999140
999355
999810
940015
948974
956689 | | tt | .919121
941402
950342
989672
.997515
998249
577585
615345
.720167|| tt | | 8: 951484 | |tt||615540
8:972094
8:988997
997126
639119
892701
998551
.99902
7 21269
893388
909072
973472
981888
987747
998021
998388
999289
999742
753010
968614
990330
996560
997740
8:999214
999363
This table gives the values of the t distribution function P(t;) for the degree and. P(t; w)=#.752302
Example: For 14 and t=0.7,
753302
834334
979038
8:998668
:997376
997885
998628
9 98896
999111||tt| |999285
299699
100001
999945
578139
894999
8:936498
946827
.539304
578797||tt| |1989130
997052|| tt | tt||999996
817919| | tt |
t distribution function table
p(t,v) =
539375
688980
.722682
835908
926085
964690
0:980697
989816
991877
99478 2
996703
997387
.998988
.999375
.999509
999698
1999815
(1++2/v) -(v+1)/2
-αVVB(1/2, v/2)
539413
.836357
.957778
.976789
992148
285035
99 8080
999564
539444
616719
836736
879520
981553
990471
995245
.997057
1998199
.998596
.999766
8999952
GB 4086.3--83
500000
723334| |tt||755149
8:939589
.977568
981887
.998297
998680
.999790||tt| |.500000
539495
7234 94
755340
.988597
990942
0994358
578681
654206
689898
723814
0994696
.99984
The table gives the value of the t-distribution function P(t; \) for the degrees of freedom v and t. Example: For Fv=45 and t=3, P(t;v)=0.997805. 40
500000
724242
998740
999052
999883
999983
000000
539628
724392
756414
.996378
.998806
999895
999972
8:756684
.882572
. 939571
942574
.961556
.974983
980029
990242
992415
994140
996567
997396
998901
: 999185
999677
8:999765
999912
(distribution quantile Table
tp(v):f
Crazy 530
8:80000
(1 +t2/v)\(#+*) 2
VUR(1/2, V/2)
0:0008
e:0008
8: 1233
.26317
8:25692
.25658
:25419
GB4086.3-83
8:3396|| tt||8:38997
1389 78
.54348
.54153
133144
8·52967
:88485
.67801
This table For the degree of self-exit and the probability of the lower side, the quantile (V) of the distribution of 1 is given: for = 100 and -0.90, () = 1.29007. When p0.5, tp() tp(). Example: ta.t(100) and two-sided probability (the corresponding quantile is: α? (: 132
.90570
th.u(100)=-1.29007
：18957
111916
1:07928
.05932
.05752
t. (v)||tt ||.47588
.39682
.34503
.34061
.32124
.31143
.30695||tt| |129582
.29103
.28509
.91999
.94318
for +m- 100 li a- 0.2, t: α2 (v )- (3u(100) -1.29007.Distribution quantile table
tp(u):
10:7:8:01
.30265
(1+t2/)(# +1)/2
wB(1/2,/2)
.20099
.06390
.05954
.92439||tt ||2:00030
.99444
1:9996
.98667
12047 0
.09936
.09578
.08088
.07631
:50832
：4998
GB4086. 3-83
2:67729
.90602
.80703
318.30884
.06433
：81778
This table gives the quantile t(v) of the t-distribution for degrees of freedom and lower probability p. Example: For y=20 and p=0.999, tp()=3.55181. 4.
.43698
3:45019
.22041
When p 0.5 When, tp(v)-tr-p(v). Example: to.ol(20)-- to.9gy(20)=-3.55181. .01500
.29053
159794| |tt||3183.0988
79-7000
.67757
2:81839
3:88218
Quantile corresponding to the two-sided probability α The number is t::a/2(V). Example: For = 20 and α 0.002, t1-a/z ()-to.99g (20) = 3.5518113.
GB4086.3-83
Appendix A
Calculation Method
(reference)
Here are the calculation methods for the two numerical tables of this standard. When the numerical tables listed in this standard cannot meet the requirements, these algorithms and the procedures in Appendix B can be used for reference. Calculate, or use interpolation method for rough calculation. A.1 Definition and Notation of t Distribution
The density function of t distribution with degrees of freedom is
f(t;v)=
Where: B(a,b)=
The distribution function is
B(,）
=1,2,3,.
l a-1( 1-)b-1d.
P(t, v)
The quantile tp() corresponding to the lower probability p satisfies the following equation: (1+
f (t,v) dt.| |tt||P(tp(v),v)= p.
Note: The limiting distribution of t distribution when the degrees of freedom v→α is normal distribution. A.28 Relationship between distribution and t distribution| |tt||A.2.1 Definition and Notation of B Distribution
The density function of the beta distribution with parameter α and b is B(d,6)a+(1)--, 0≤<, a>0 , b>0.fe(a,a, b)=
The distribution function is
-29-1 (1- )b\1d.
Ir(a,b) = {
. The quantile p(a, b) corresponding to the lower probability p of B(a, b)
satisfies the equation rxp(a, b)
A.2.2 The relationship between β distribution and t distribution
B(a,b)
Comparing t distribution with β distribution, we get the distribution function of t distribution P(t;v)=1
Wu Zhong ： 3
al(1-α)b-Id=p.
Ir(a,b),
quantile is
where:|| tt||l 2(1 - p)
A.3 Calculation method
GB4086.3—83
tp(v)= sign(p-
A .3.1 References to other distributions
p* (a,b)
a. According to the above, the calculation of t distribution can be converted into the calculation of β distribution. Therefore, the calculation method of β distribution is actually given here. b. This appendix involves the calculation of the quantile up of normal distribution. Please refer to Appendix A of GB4086.1-83 "Normal Distribution of Statistical Distribution Numerical Tables".
A.3.2B The calculation of distribution density function and distribution function uses the recursive formula obtained by the method of integration by parts I(a+1,b)= Is(a,b)-1Ur(a,b),
IUr(a,b),
I, (a, 6+1)= I(a, b)+-
Us (a+1,b)=Us(a,b) α+b
tU (a,b+1)=U+(a,b)-+tb
(1-α).
In the formula: U(a,b)=
-α\(1-α)b
Here we only consider the case where α and b are both integer multiples. In this case, the recursive initial value is [1（）
）={-
I（,)=,
I（1,）=V-,
I（１，1）=α，
U(,)=(1-),
U( ）(),
U（1,）=
lUr（，１）=α(1-z).
A.3.38 Calculation of distribution quantile
If parameters a, b are one or 1, the following formula can be used to directly calculate
GB4086.3-83
αp(
1+ tan2((1- p)),
1)=2,
p(1,)=1(1-p),
l cp(1,1)= p.
In other cases, the substitution method is used for calculation (see Appendix A.3 of GB4086.1), and the initial value is given by the following method: rp(a,b)=
a. If one of the two parameters α and b is equal to, then define
t = tp*+
where u* is the normal quantile corresponding to the lower probability p*, p* = -
F=2 (when b=
-Yi(up*),
v=26,F=
(when a=
22（u3+u),
(5u5+ 16u3 +3u),
Y2(u)=
Y4(u)=
(3u? +19u5+17u3-15u),
(79u9 +776u? + 1482u5-1920u3- 945u),Ys(u)=
(27ull+339u9+930u?-1782u5-765u3+17955u). 213.32.5
b. If none of the two parameters c and 6 are equal, then define F
where: *=1p, c=
(1-c)(1-d)+up*(ic)2d+(1-d)2c-cdus.(1- d)2- dug*
B.1 Description
GB4086.3—83
Appendix B
Calculation Programs
(Reference)
This list gives four FORTRAN programs that are actually used in this standard. They are BFTAFD: used to calculate the density function and distribution function of β distribution, BETAXD: used to calculate the quantile of β distribution, TFD: used to calculate the distribution function of β distribution, and TXD: used to calculate the quantile of t distribution.
The program uses the calculation method of Appendix A, that is, the t distribution is converted into β distribution calculation, so the calculation programs of two distributions are necessary. In addition, when they are running, they need to call two FORTRAN programs NORFD and NORXD. Please refer to Appendix B of GB4086.J: Although the number tables printed in this standard only take 5 to 6 decimal places, the calculation accuracy of the program is usually up to 10·10 Programs
SUBROUTINE BETAFD(X,N2A,N2B,P,DENS)INTEGER N2A,N2B
DOUBLE PRECISION TURN
N2A=2*A. (A-1)TH POWER OF X IN BETA DISTRIBUTIONBETF0001
BETF0002
BETF0003
BETF0004
BETF0005
BETF0006
BEIF0007
BETF0008
BETF0009
BETF0010||tt| |BETF0011
BETF0012
BETF0013
BETFO014
N2B=2*B. (B-1)TH POWER OF (1-X) IN BETA DISTRIBUTION BETF0015BETF0016
BETA POINT
RE1F0017
LOWER PROBABILITYbZxz.net
DENSITY
**REQUIREDROUTINES **
** ALGORITHM **
THE RECURRENCE FORMULA
I(X,A+1,B) = I(X,A,B) - (U(AB)/A).ICX.A ,B+1) = I(X,A,B) + (U(A,B)/B),U(A+1,B) = U(AB)*X*(A + B)/AU(A,B+1) = U(A,B)*(1-X)*(A+B)/B,WHERE
I(XA,B) IS DISTRIBUTION FUNCTION OF BETA DISTRIBUTION.U(A,B) = X**A*(1-X)**B/BETA(A,B),BETA(A,B) IS BETA FUNCTION.
INITIAL VALUES
I(X,1/2,1/2) = 1 - (2/PAI)*ATAN(SQRI((1-X)/X)),I(X,1/2,1) = SQRI(X).
I(X,1. 1/2) = 1 - SQRT(1-X).
I(X,1,1) = X,
BETF0018
BETF0019
BETF0020
BETF0021
BETF0022
REIF0023
BETF0024
BETF0025
BETF0026
BETF0027
BETF0028
BFIF0029
BE IF0030
BETF0031
BETF0032
BI.TF0033
BEIF0034
BETF005S
BETFO036
BE1F0037
BETF0038
GB4086.3--83|| tt||U(1/2.1/2)=SQRT(X(1-X))/PAI，U(1/2,1) = SQRt(X)*(1-X)/2.
U(1.1/2) = X+SQRT(1-X)/2,
U<1,1) =000 - ||IF《M.EQ. 2)GOTO 20
IF (N .EQ. 2) G0 T0 10
N2AISODDANDN2BISODD
P = 1.0D0 - 0.63661977236758134D0*DAIAN(S /DSQRT(X))DENS=0.31830988618379067D0*S*DSQRT(X)GOTO 40
CONTINUE
N2A IS EVEN AND N28 IS 000
P = 1.000 - s
DENS = 0.500*S*X
GD TO 40
CONTINUE
IF (N .EQ. 2) GO TO 30
N2A IS ODD AND N2B IS EVEN
P = DSQRT(X)
DENS = 0.5DO*(1.0DO X)*P
GO TO 40
CONTINUE
N2A IS EVEN ANU N2B IS EVEN
DENS = X*(1.ODO X)
40 CONTINUE
SO CONTINUE|| tt||RECURRENCE
DO 60 I = N,N2A,2
IF (N2A .LE. I) GO TO 70
P = P - 2.ODO/FI*DENS
DENS = DENS*X *(FM + FI)/FI
60 CONTINUE
7O CONTINUE
DO 80 I = M.N2B,2
IF (N2B .LE. I) GO TO 90
P = P + 2.0D0/FM*BENS
DENS - DENS*(1.ODO - X)*(FM + FI)/FM80 CONTINUE
90 CONTINUE| |tt||DENS = DENS/(X*(1.ODO - ( BETF0045
BETF0046
BETF0047
BETF0048
BETF0049
BETF0050
BETF0051
BETF0052
BETF0053| |t t||BETF0054
BETF0055
BETF0056
BEIF0057
BETF0058
BETF0059
BETF0060
BETF0061||tt| |BETF0062
BETF0063
BETF0Q64
BETF006S
BETF0066
BETF0067
BETF0068
BETF0069
BE TF0070
BETF0071
BETF0072
BETF0073
BETF0074
BEIF0075
BETF0076
BETF0077
BETF0078| |tt||BETF0079
BETF0080
BETF0081
BETF0082
BETF0083
BETF0084
BETF0085
BETF00 86
BETF0087
BETF0088
BETF0089
BETF0090
BETF0091
BETF0092
BETFQ093
BETF0094| |tt||BETF0095
BETF0096
BETF0097
BETF0098
BETF0099
BETF0100
GB4086.3—
SUBROUTINE BETAXD(P,N2A.N28,EPS,X,IER)INTEGER N2A,N28,IER
DOUBLE PRECISION P,EPS.X
** PURPOSE **
PERCENTAGE POINT OF BETA DISTRIBUTION (INCOMPLETE BETAFUNCTION RATIO) WITH HALF INTEGER PARAMETERS** ARGUMENTS **
DN NETRY
N2A=2*A, (A-1)TH POWER OF × IN BETA DISTRIBUTIONN2A||tt| |ON RETURN
BETX0001
BETX0002
BETX0003
BETX0004
BEIX0005
BETX0006
BETX0007
BETX0008| |tt||BEIX0009
BETX0010
BETX0011
BE1X0012
BE1x0013
BETX0014
N2B=2*B, (B-1 )TH POWER OF (1-X) IN BETA DISTRIBUTION BETX0015LOWER PROBABILITY
REQUIRED PRECISION
PERCENTAGE POINT
ERROR PARAMETER
IER.NE.0 INDICATES P.LE.O OR P.GE.1** REQUIRED ROUTINES **
BETAFD| |tt||PERCENTAGE POINT OF NORMAL DISTRIBUTIONDISTRIBUTION FUNCTION OF NORMAL DISTRIBUTIODISTRIBUTION FUNCTION OF BETA OISTRIBUTION**ALGORITHM **
WHEN N2A AND N2B ARE EQUAL TO 1 OR 2, THE PERCENTAGE POINTX(P,N2A/2.N28/ 2) IS CALCULATED BY FORMULAX(P,1/2,1/2) = 1/(1 + (TAN((PA1/2)*(1-P)))**2),X(P.1/2,2/2) = P**2,
X(P,2/ 2,1/2) = 1 - (1-P)**2,X(P,2/2,2/2) = P,
FOR THE OTHER UALUES OF N2A AND N2B,HITOTUMATU-S ITERATIONMETHOD IS USED, INITIAL VALUE IS CALCULATED BY YAMAUTI-SFORMULA AND PAULSON-TAKEUCHI-S FORMULA. SEE Z. YAMAUTI.STATISTICAL TABLES AND FORMULAS WITH COMPUTER APPLICATION,JSA-1972, B(2.10)-P.27, B (2.31)-P.33.INTEGER N,MN
DOUBLE PRECISION PAIS.PP,U.DENS,DDENS,R,RDOT.ABCDF1.F2DATA PAI5/1.57079632679489662D0/IER=O
IF (P .GT. 0.ODO .AND. P .LT. 1.ODO) GO TO 10IER = 129
IF (P .LE. 0.0D0) X = 0.0D0
IF (P .GE. 1.0D0) N2A .GT, 2 .OR. N2B .GT. 2) GO TO 30CASE N2A=1.2 AND N2B=1,2
IF (N2A .EQ. 1 ,AND. N2B -EQ. 1) X = 1.0 D0/(1.0D0 +(DSIN(PAI5*(1.ODO-P))/DCOS(PAI5*(1.ODO-P)))**2)IF (N2A .EQ. 1 .AND. N?B .EQ. 2) X = P**2BETX0016
BETX0017
BETX0018
BETX0019
BETX0020
BEIX0021
BEIX0022
BETX0023||tt| |BE1X0024
BETX0025
BETX0026
BETX0 027
BETX0028
BETX0029
BETX0030
BETX0031
BETX0032
8E1X0033
BETX0034
BETX0035| |tt||BETX0036
BETX0037
BEIX0038
B ETX0039
BETX0040
BETX0041
BETX0042
BETX0043
BETX0044
BEiX0045
BETX0046
BETX0047| |tt||BETX0048
BETX0049
BETX0050||t t||BETX0051
BETX0052
BEIX0053
BEIX0054
BETX0055
BETX0056
BE1X0057
BEIX0058||tt| |BEIXOS9
BETX0060
BETXOCG1
BETX0062
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