statistical computation of tolerance limits
TRANSCRIPT
NASATechnicalMemorandum
NASA TM- 108424
CsP
(NASA-TM-10842_) STATISTICALCOMPUTATION OF TOLERANCE LIMITS
(NASA) 45 p
N94-15535
Unclas
G3/65 0190807
STATISTICAL COMPUTATION OF TOLERANCE LIMITS
By J.T. Wheeler
Structures and Dynamics Laboratory
Science and Engineering Directorate
October 1993
NASANational Aeronautics and
Space Administration
George C. Marshall Space Flight Center
MSFC - Form 3190 (Rev. May 1983)
TABLE OF CONTENTS
INTRODUCTION .........................................................................................................................
MATHEMATICAL PROCEDURES ............................................................................................
One-Sided Tolerance Limits ...............................................................................................
Method A ...............................................................................................................
Method B (New Theory) .......................................................................................Two-Sided Tolerance Limits .............................................................................................
Method A ...............................................................................................................
Method B (New Theory) ........................................................................................
NUMERICAL RESULTS .............................................................................................................
NONPARAMETRIC TOLERANCE LIMITS ..............................................................................
CONCLUSIONS ...........................................................................................................................
APPENDIX A: Derivation of Chi Distribution .............................................................................
APPENDIX B 1: One-Sided Kp ....................................................................................................
B2: Two-Sided Kp ....................................................................................................
APPENDIX CI: Simulation Code for One-Sided k .....................................................................
C2: Simulation Code for Two-Sided k .....................................................................
REFERENCES ...........................................................................................................................
Page
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pRF.CE!)_IG PAGE BLANK NOT FILMED
Figure
1.
2.
3.
LIST OF ILLUSTRATIONS
Title
Normal distributions for one- and two-sided tolerance limits .....................................
Area of integration for one-sided tolerance limits .......................................................
Area of integration for two-sided lower limits .............................................................
Page
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6
8
L
w
ivL
LIST OF TABLES
Table
1.
2,
.
.
.
,
,
,
.
12.
Title
One-sided k; Pr(T v <_k I'_ I Kp l'g) = 7' ;
One-sided k; Pr(Tv <-k l'_l Kp l'ff) = 7';
One-sided k; Pr(Tv <- k :ff l Kp l-if) = 7';
One-sided k; Pr(Tv <- k l'ff l Kp g'if) = 7";
One-sided k; Pr(T v <_k l-ff I Kp l'if) = 7";
Two-sided k; Pr(T v _k 1-ff I Kp gh') = 7';
Two-sided k; Pr(T v <_k l'_ l Kp g-if) = 7";
Two-sided k; Pr(T v <_k ,/'dik e g'_) = 7';
Two-sided k; Pr(T v < k ,/-ff IKp g-if) = 7";
Two-sided k; Pr(Tv <-k l'_ l Kp g'if) = 7';
7' = 0.50 .....................................................
7' = 0.75 .....................................................
7' = 0.90 .....................................................
y = 0.95 .....................................................
7'- 0.99 .....................................................
7' = 0.50 .....................................................
7' = 0.75 .....................................................
7'= 0.90 .....................................................
7'= 0.95 .....................................................
7' = 0.99 .....................................................
Comparison of approximate and exact results for two-sided tolerance factor kfor n = 2 and 10 .........................................................................................................
Nonparametric tolerance limits ...................................................................................
Page
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TECHNICAL MEMORANDUM
STATISTICAL COMPUTATION OF TOLERANCE LIMITS
INTRODUCTION
The need for improved, accurate computation of tolerance factors is accomplished by a new
theory associated with the numerical integration of the equations to yield exact solutions. The computer
codes have been developed specifically to integrate the equations of some statistical properties for a
normal (or Gaussian) distribution by application to the particular one- and two-sided tolerance limits.
For the normal variate X with known mean # and known standard deviation o', a specified pro-
portion of the normal population (1) is contained above the lower tolerance limit, L = p-Kp o', or below
the upper tolerance limit, L = I.t+Kpcr, for the one-sided case, and (2) falls within the lower and upper
tolerance limits for the two-sided case. Kp is the deviate corresponding to the proportion for the inverse
normal probability distribution.
However, in most situations, the mean p and standard deviation o"frequently are unknown.
Therefore, the objective of the tolerance-limit analysis is to obtain the numerical solutions to the prob-
lems involving the probability distribution for the sampled population with the normal distribution with
unknown mean p and unknown standard deviation o'. Furthermore, it results in the calculation of the
tolerance limits based on the mean x and the standard deviation s of the random sample. Thus, by the
statistical theory, the tolerance limits can be calculated out by the following equations:
L = F-ks
U = X+ks,
(1)
where x is an estimate of/.t and s is an estimate of ¢rof the normal distribution computed from a sample
of size n. The quantity k is the tolerance factor. Since x and s are considered random variables, the toler-
ance limit statement is applied to the given probability. Now, to ensure stability of the numerical solu-
tions, it is necessary to determine k such that the probability is 7, that at least a proportion p of the popu-
lation is above F-ks or below X+ks for the one-sided case and lying between the limits for the two-sided
case.
The statistical tolerance interval :_+ks covers the prespecified proportion p of the sampled distri-
bution with 100 ypercent confidence for the cases where a sample x!, x2, x3 ..... xn represents the random
sample from the normal distribution with mean p and standard deviation 0". The ordinary statistical
methods assume the following equations:
X ± _ xi, (2)--tl i= 1
as a sample mean and
s=_/nl_-l_(Xi-_}2i= ' (3)
as a sample standard deviation.
In the one- and two-sided cases, tolerance limits can be derived using noncentral t-distribution.
Mathematically, the problem is to find k such that
Pr{Pr(X<_._+ks) > p} = 7, (4)
where X has the normal distribution with mean/1 and standard deviation o'. 7is specified probability.
Some tables for the tolerance factor k have been generated using inputs of sample size n, proportion, andprobability. The values of k pertain to percentage points of the noncentral t-distribution. Specifically,
Pr{noncentral t _<k/-gl g = Kp,/-ff} = y, (5)
where the noncentral t-distribution has v degrees of freedom.
Normal distributions for one- and two-sided tolerance limits are depicted in figure 1.
Wald and Wolfowitz, in their technical paper, had developed an approximate formula, which
shows that k is approximated by
k = ru, (6)
where r is determined from the normal distribution by
.(1 + r)__(1- r), (7)
with
and u is defined by
• (t)=_ **ex --_- dz,
v (8)u= Z v) '
where Z _v) is the upper 10Og, percentage point of the chi-square distribution with v= n-1 degrees of
freedom. For v = n-l, the approximation converges to the true limits as n approaches infinity.
2
i'÷ksOne-sidedtolerancelimits
Theprobability is (1-o0 thatat least100p percent of the population lies below (or above)the tolerance limit F = _e+k _s (or F -- :g-k :).
Two-sided tolerance limits
is(1- _)that at least 100 p percent of the population is containedThe probability
between the tolerance limits F_ = :r+k2s and F2 = 7t-k2s.
Figure 1. Normal distributions for one- and two-sided tolerance limits.
3
MATHEMATICAL PROCEDURES
The mathematical procedures are given for determination of exact tolerance limits for the one-
and two-sided cases.
One-Sided Tolerance Limits
Method A
For the noncentral t-distribution, let w denote a random variable having a normal distribution
with mean zero and variance one; let v denote a random variable that has a chi-square distribution with v
degrees of freedom; let the quantity 6 be any constant; and let w and v be stochastically independent.
Then T(fi) = (w+_)/_ has a noncentral t-distribution with noncentrality parameter 8 and v degrees of
freedom.
Thus, a standardized normal variate is given
W 2
1 ---U
g(w) = _ e ,-_,<w<_ (9)
and an independent chi-square variate based on v degrees of freedom is
V
h(v) = 1 (_'_) -_v v e 0<v<o* (10)
Then the joint distribution ofw and v is the product of equations (9) and (10):
Q(w,v) = g(w)h(v) ,
1Q(w,v) =e . 1 v v e -_ . (11)
The new random variables are introduced, namely,
t= w+S u = ,17 . (12)
Applying the change of variables technique, the new distribution is obtained:
f(t,u) = ¢p(w,v) IJI , (13)
where J is the Jacobian determinant of the transformation,
4
Ow Ow
igt i3uJ=
onv onv
Ot t)u
Consider the transformation w = tul,/V- S, v = u 2
_vm= 0
_t
_Vm=2u ._u
The Jacobian is obtained
(14)
Therefore,
1 --_" ,F(_D2_ e _)f(t,u) = _ e v
(15)
The noncentral t-distribution is obtained by integrating over u from 0 to oo
¢tu 8_2to u 2
fo -_TV- 1" v-1 ---2 Uf(t) 1 e u e p du .
-- X_ 12
(16)
Equation (16), the density function, is integrated with respect to t from minus infinity to t to give thecumulative distribution function:
F(t;v,_5) = Pr{Tv < t} - fo G[ tU - eS) uk,/Vv-i G'(u)du , (17)
where
a'(x)- i e-_ ,
and
G(x) = fx__**G'(t)dt .
Thedistributionhasthefollowing properties:
Pr{Tv(8 ) < to} = 1-Pr{Tv(-¢5 ) < to} ,
Pr{Tv(8) < O} = G(-5).
Equation (17) has been formulated in one of the computer codes to evaluate the one-sided k fac-
tors, but the listing of the code for this method is not included.
Method B (New Theo_)
For the one-sided tolerance limits, the probability equation involves the form
Pr((-_)<-Kp,(-_)>Kp)=y . (18)
The joint probability is calculated about the standard normally distributed random variable
z =/ff (X-/.t)/cr and the random variable w = slot. One can approximate or, the true standard deviation, by
s, the sample standard deviation. The joint probability density is integrated over the region defined by
z =1_ kw-1-_ K e . (19)
Equation (19) is designated as the shaded area of the strip as shown in figure 2.
w
0toJ-K;)
N
Figure 2. Area of integration for one-sided tolerance limits.
Kp/k is located at the vertex on the w-axis and -Kp _ is at w = 0 On the z-axis. The shaded area is
integrated to give the summation of the elements. W is the chi-square distribution with v degrees of free-
dom, and z is a standard normal variable. In the one-sided case, the joint distribution of w and z is
derived in the following form:
I = 2 ff _ w(V-1) e 2
Let w = x/ dV. Then dw = dx/ dV .
F
6
Substitutingtheaboverelationsintoequation(20)resultsin
where
e-_7"
l=2f: ± dx (21)
7.2
f:**e-Tdz¢ (Y) = 2,_-Y "
After those manipulations, equation (21) can be expressed in the final form
X 2
f-1--r ¢[:n (kw-Xp)],_ (22)= __ x (v-l) eI
&
Equation (22) is programmed in the program code which is listed in appendix C 1.
Two-Sided Tolerance Limits
Method A
From method A described for the one-sided tolerance limits, equation (17) is readily imple-
mented as a summation for a set of lower and upper limits of the integrals to represent the two-sided
tolerance limits. In terms of the tolerance interval, a proportion of a distribution is constructed as a frac-
tion from which the sample is drawn. Let wl and w2 constitute a two-sided normal distribution with zero
means and unit variances. Also, let v be distributed independent of the w' s and let v obtain a square root
of a chi-square distribution with v degrees of freedom. Consequently, it follows that the expressions
(wl+S1)/v and (w2+_)/v have noncentral t-distributions with v degrees of freedom and noncentrality
parameters 81 and 62.
A two-sided noncentral t-distribution is, therefore, defined as the joint distribution of (wl+gl)/V
and (w2+62)lv. A random sample Xl, x2, x3 ..... xn from a normal distribution is assumed. Values of k fac-
tor are assigned from which tolerance limits x-ks (lower limit) and x+ks (upper limit) are computed,
specifying with probability that no more than the proportion Pl of the normal population is below F-ks
and no more than the proportion P2 is above :_+ks.
Accordingly, equation (17) is written in the following form for a set of lower (0,R) and upper
(R,oo) limits for two-sided noncentral t cumulative distribution function:
and
F(t,g;O,R) =
F(t,g;R,_) =
foR G[ tU _ g] u V-l G'(u) du
g] duf: .
(23)
(24)
Thus,summationof equations(23)and(24)yields
Pr {Tv < tl_} = F(t,g;O,R) = F(t,g;R,oo) . (25)
Method B (New Theory)
Dealing with a problem of solving the tolerance factor k based on the given probability and pro-
portion after the iterative process for the two-sided tolerance limits has the following form,
where
O(k) = f/ tp(z)Q[w(z)]dz ,
4/_)_ e -V_-dx
(26)
(27)
is the chi distribution and
_ 1 .d¢p(z) - _ e- 2 ,
(28)
is the standard normal probability de0sity (zero mean and unit standard deviation). Derivation of the chi
distribution is described in appendix A.
In equation (27), the lower limit of the integral, w(z), needs to be determined. The probability is
such that the quantities (L-/l)/cr and (U-la)/trexceed Kp with the proportion P being at least p, namely,
: Prob {G[(U-lz)la]-G[(L-lz)la] > p } . (29)
Equation (29) is evaluated by integrating the joint density of z and w over the shaded area in figure 3.
/,Iv
Kp/kJ
v Z
Figure 3. Area of integration for two-sided lower limits.
E
8I::
7
Kflk is represented at the vertex on w-axis for the lower limit (parabolic curve). The area is inte-grated with respect to z, thus summing the area elements into a vertical strip extending from the bound-ing curve of the lower limit upwards. For any value of z, a strip is defined along the line
Z = g-ff Kp -/'ff kw and another line z =/'_ Kp + 1"ffkw. So, let
(U-l.t)ltr = Kp=---_ + kw , (3O)
and
(L-It)1 cr = Kp = --_ - kw .(31)
Kp is defined by
,and the values of Kp for one- and two-sided cases are tabulated in appendices B1 and B2.
Using equations (30) and (31), the function for the lower limit becomes
f(z,w) = G[(U-I.t )l tr]-G[(L-l.t )l tr]-p = 0 ,
or
:<z,w)=C ÷kw)-C-kw)-p=O.
Equation (32) is, for some function f(z,w), the total or exact differential
(32)
Of Ofdf = -_zdz + _-_dw , (33)
ofv,._ and _w are the partial derivatives of f with respect to z and w, respectively. Then equationwhere
(32) may be written df= O. Rearranging equation (33) to obtain a differential equation
_fdw Oz
3w
3f__ _1 [e-(_ +k_)_/2- e-{_- _} _I2]. _ ] ,
=k[ e-(_+_)2r_+e-(_-_)_a]J"
THUS,
__ 0fSubstituting these values of and _ into equation (34) gives a final expression
(34)
(35)
(36)
9
dw = I___L_[e-(_+_2a-e-{_ -_)_t2] (37)
The hyperbolic tangent function of u is defined by the formula
tanh u = e-"-e-" (38)e-U+e-U .
Substitution of equation (38) yields the differential expression for the lower limit
dw l__ tanh (.__n kw) (39)dz - kl"ff
The methods for obtaining numerical solutions to differential equations and for computing the
integrals by numerical methods in the computer codes reduce equation (39) to
w = w(z) . (40)
The values of w(z) are applied to the lower limit of the integral of equation (27) for computation.
The computer code for this method is listed in appendix C2.
NUMERICAL RESULTS
A method for one-sided tolerance limits and another method for two-sided tolerance limits have
been devised and enhanced, using a new theory, for the exact computation of tolerance factors k. Some
tables of tolerance factors k have been generated to illustrate the exact results.
The k data are provided, for one-sided case, with the following parameters of all combinations:
Tables 1 through 3
7= 0.50, 0.75, 0.90
p = 0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.8413, 0.85, 0.90, 0.95, 0.975, 0.9773, 0.99, 0.9987,0.999, 0.9999, 0.999968, 0.99999
n = 2(1)50
Tables 4 and 5
)'= 0.95, 0.99
p = 0.50, 0.75, 0.90, 0.95, 0.99, 0.999, 0.9999, 0.99999
n = 2(1)10
and for two-sided case:
10
r_
_=
Tables 6 through 8
V= 0.50, 0.75, 0.90
p = 0.50, 0.55, 0.60, 0.65, 0.6827, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 0.9545, 0.975, 0.99, 0.9973,
0.999, 0.9999, 0.999937, 0.99999
n = 2(1)50
T= 0.95, 0.99
p = 0.50, 0.75, 0.90, 0.95, 0.99, 0.999, 0.9999, 0.99999
n =2(1)10.
Methods A and B for the one- and two-sided cases have involved several different algorithms
needed for numerically integrating, differentiating, and iteratively root-solving the equations to obtainthe exact solutions for the tolerance factors k. In method A, the Pegasus method with an estimated order
of convergence superior to a secant method and the more accurate 20-point Gaussian quadrature proce-
dure have been employed along with some other algorithms for numerical solutions.
A procedure for method B uses the functions of normal density, normal distribution and inversenormal distribution, calculation of factorial, secant method, Simpson method for numerical integration,
and the third-order Runge-Kutta method for numerical solutions to differential equations. The Runge-
Kutta method requires three derivative evaluations per step, which has an accumulated truncation error
proportional to (Ax) 3. Those Pegasus and secant methods require input of good initial estimates in order
to solve a root-finding problem f(x) = 0.
A comparison of approximate and exact results for the two-sided tolerance factor k for n = 2 and
10 is given in table 11 with confidence level ),and proportion p.
Based on table 11, the approximate data for n = 2 differ from the exact data by at most 3.18 per-
cent. But as n increases, the error percentage decreases to less than 1 percent, as the n = 10 data can
verify.
11
Table 1. One-sided k.
P r ( Tv < k ,/-ff l K p ,/-ff) = )', )'=0.50
P P P P P P P P P P0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.8413 0.85 0.90tl
234
56789
10111213
14151617
181920212223242526272829303132333435363738
394041424344454647484950
0.000010.000000.000_
0.000000.000000.000000.000000.000000.000000.000000.000030.00300O.O0(g)O0.00(_0.000000.000000.000000.000000.000000.000000.000000.000_0.000000.000000.000000.000000.00(00.000000.000000.000000.00(0
0.000000.000000.000000.000000.000000.000000.00(00.00000
0.000000.00(00.000000.000000.000000.000000.000000.000000.000000.00000
0.157900.141940.136490.133760.132130.131040.130270.129690.129240.128890.128600.128350.128150.127970.127820.127680.127570.127460.127370.127280.127210.127140.127080.127020.126970.12692
0.126870.126830.126790.126760.12672
0.126690.126660.126630.126600.126580.126550.126530.126510.126490.126470.126450.126440.12642
0.126400.126390.126380.126360.12635
0.320390.286730.275460.269850.266500.264290.262710.261530.260620.259880.259290.258790.258380.258020.257710.257440.257200.256980.256790.256620.256470.256330.256200.256080.255980.255880.255780.255700.255620.255540.255470.255410.255350.255290.255240.255180.255140.255090.255050.255010.254970.254930.254890.254860.254830.254800.254770.254740.25471
0.49221
0.437430.419590.410810.405590.402140.399690.397860.396440.395310.394390.393620.392980.392420.391940.391520.391150".39082
0.390530.390270.390030.389810.389620.389430.389270.389110.388970.388840.388720.388600.388500.388400.388300.388210.388130.388050.387980.387910.387840.387770.387710.387660.387600.387550.387500.387410.387400.387360.38732
0.679000.59798
0.572460.560020.552680.547840.544410.541860.539880.538300.537020.53595
0.535050.534280.533610.533030.532520.53206
0.531650.53129
0.530960.530660.530380.530130.529900.529690.529490.529310.529140.528980.528830.528690.528560.528440.528320.528210.528110.528010.527920.527830.527740.527660.527590.527520.527450.527380.527320.527260.52720
0.887370.77325
0.738480.721740.711920.70548
0.700920.69753
0.694910.692820.691120.689710._852
0.687500.686620.685860.685180.684580.684040.683560.683120.682730.682360.682030.681730.681450.681190.680950.680720.680520.68032
0.680140.679960.679800.679650.679510.67937
0.679240.679120.679000.678890.678790.678690.678590.678500.678420.678330.678250.67817
1.127010.970660.924540.902600.889810.881450.875550.871170.867790.865100.862910.861100.859570.85826
0.857130.856140.855270.854500.853810.853200.852640.852130.851660.851240.850850.850490.850160.849850.849560.849300.84904
0.848810.848590.848380.848 190.848000.84783
0.847660.847510.847360.847220.847080.846960.846830.846720.846610.846500.846400.84630
1.360221.159621.101911.074721.058951.048661.041421.036061.031921.028631.025961.023741.021871.020281.018901.017701.016641.015701.014861.014101.013421.012801.012241.011721.01125
1.010811.010411.010031.009681.009361.009051.00876
1.008501.008241.008011.007781.007571.007371.007181.007001.006831.00666
1.006511.006361.006221.006081.005951.005821.00571
1.414471.203251.142791.114351.097871.087131.079581.073981.069661.066241.063451.061141.059191.057531.056091.054841.053731.052751.051881.05109
1.05038
1.04974
1.049151.048611.04812
1.047661.047241.046851.046481.046141.045831.045531.045251.044991.044741.044511.044281.044081.043881.043691.043511.043341.04318
1.043021.042881.042741.042601.042471.04234
1.784301.498541.418931.381871.360511.346641.336901.329701.324151.319751.316181.313211.310721.308591.306751.305141.303731.302481.301361.300351.299441.298621.297871.297181.296551.295971.295431.294931.294461.294031.293621.293241.292891.292551.292241.291941.291661.291391.291141.290901.290671.290451.290241.290041.289861.289681.289511.289341.28919
12
Table 1. One-sidedk (continued).
Pr(Tv<k,rfflKp1"ff)=_', y= 0.50
234
56789
1011121314151617181920212223242526272829303132333435363738
394041424344
454647484950
P0.95
2.338641.938351.829451.779221.750411.731741.718661.70900
1.701571.695691.690901.68694
1.683611.680761.678301.676161.674271.672601.671111.669771.668561.66746
1.666461.665541.664701.663921.663211.662541.661921.661341.660801.660301.659821.659371.658951.658561.65818
1.657821.657491.657171.656861.656571.65630
1.656041.655781.655541.655321.655091.65488
P
0.975
2.819722.320572.186202.124462.089112.066232.050232.03841
2.029322.022122.016282.011442.00736
2.003882.000881.998261.995961.993921.99210
1.990461.988991.987641.986421.985301.984281.983331.982451.981 641.980881.980181.979521.978901.978321.97778
1.977261.976781.976321.975881.975481.975081.974711.974361.974021.973701.973401.973101.972821.972561.97229
P0.9773
2.880622.369062.231482.168282.132102.108692.092312.080222.07092
2.063562.057572.052622.04845
2.044892.041822.039142.036792.034702.032842.031162.029652.02828
2.027032.02588
2.024832.023862.022972.022142.021362.020642.019972.019332.018742.018182.017662.017162.016692.016252.015832.015432.015052.014692.014342.014022.013702.013402.013112.012852.01259
P0.99
3.376052.764542.600882.525832.482902.455142.435722.421392.410382.401652.394562.388702.38376
2.379552.375912.372742.369952.367482.365272.363292.361502.35987
2.358392.35704
2.355792.354642.353582.352602.351682.350832.350032.349282.348582.347922.347302.346712.346162.345632.345142.344662.344212.343782.343382.342992.342622.342262.341932.341602.34128
P0.9987
4.391243.57922
3.362613.263383.206663.16998
3.144343.125413.110873.099343.089993.082243.07572
3.070163.06536
3.061173.057493.054223.051313.048693.046333.044193.042233.040443.03880
3.037283.035883.034583.033383.032253.031193.030213.029283.028413.027593.026813.026083.025393.024733.024103.023513.022953.022413.021903.021413.020943.020493.020073.01966
P0.999
4.526643.688173.46454
3.362113.303543.265683.23922
3.219673.204663.192763.183103.175103.168373.162633.157673.153353.149553.146183.143173.140473.138033.135823.133803.131953.130263.12869
3.127243.125903.124663.123493.122403.121383.120433.119543.118693.117883.117123.116413.115733.115093.114473.113893.113333.112803.112303.111823.111353.110913.11057
P
0.9999
5.468274.447034.174784.050083.978803.932713.900473.87669
3.858403.843923.832173.82246
3.814173.807233.801223.795873.791313.787173.783523.780223.777313.774563.772063.769883.767833.765873.76408
3.762513.760973.759543.758223.756993.755853.754743.753673.752703.751813.750953.750103.749303.748563.747873.747193.746533.745913.745333.744783.744243.74368
P0.999968
5.888114.785834.492024.357434.280474.230724.195924.170244.150504.134864.122164.11165
4.102804.095254.088734.083064.078054.07362
4.069674.066114.063044.060014.057364.054944.052734.050704.048844.047144.045594.044204.042974.041914.041044.040404.040014.03993
4.035464.034554.033694.032894.032164.03149
4.030904.030384.029964.029664.029484.02946
4.02963
P0.99999
6.283495.10513
4.790974.647084.564694.511814.474294.447034.425774.409264.395324.384594.37461
4.366814.360024.353424.348444.343894.339414.335514.332484.32903
4.326294.323734.321494.318924.316784.315234.313584.311664.309994.308794.307614.30619
4.304804.303754.302884.301894.300774.299784.299024.29832
4.297514.296664.295914.295304.294724.294084.29322
13
Table 2. One-sided k.
Pr(Tv < k l'ff l Kp,/-ff) = y, y=0.75
23456789
1011121314
151617
18192021222324252627282930313233343536
3738394041
424344i45!4647i48149 _
50
P0.50
0,70712
0.471400.382450.331250.296670.27121
0.251430.235460.222220.211000.201340.192890.185430.178780.172800.167380.162440.157920.153760.149910.146330.143000.139890.136970.13423
0.131640.129200.126900.124710.122630.12065
0.118760.116960.115240.113590.11202
0.110500.10905
0.107650.106300.10501
0.103760.102550.10138
0.100260.099170.098110.097090.09610
P0.55
0.941300.639290.53421
0.475840.437220.40917
0.387570.370280,356020.343990.333660.324680.316760.309720.303400.297680.292490.287730.283360.279330.275580.272100.268850.265800.262940.260250.257710.255310.253030.250870.248820.246860.244990.243210.241500.239870.238300.236800.235350.233960.232620.231330.230080,228880.227720.226590.225510.224450.22343
P0.60
1.202910.818290.693750.626820.583380.552230.528490.509600.494120.481120.470000.460360.451890.444370.437640.431560.426050,421010.416390.412120.408170.404500.401080.397870.394870.392040.389370.386850.384470.382210.380060.378010.376060.374190.372410.370700.369070.367500.365990.364540.363150.361800.360500.359250.358040,356870.355740.354650,35359
P
0.65
1.497941.011910.864080.786960.737810.703010.676710.655940.639000.624840.612780.602350.593220.585130.577900.571400.565500.560120.555180.550640.546440.542540.538900.535500.532320.529320.526500.523840.521320.518930.516660.514500.512450.510480.508610.506810,505090.503440.501860.500330.498870.497460.496090.494780.493510.492290.491 I00.489960.48884
P0.70
1.834801.225081.049410.960200.904280.865130.835800.812780.794110.778580.765390.754020.744090.735320.727500.72047
0.714110.708310.703010.698130.693620.689440.685550.681910.678510.675310.672310.669470.666780.664240.661820.659530.657340.655250.653260.651360.649530.647780.646100.644490.642940.641440.640000.638610,637270.635980.634720.633510.63234
P
0.75
2.224761.464341.255311.151661.087661.043331.010380.984680.963940.946750.932210.91971
0.908820.899220.890690.883020.87610
0.869810.86406
0.858770.853890.849370.845160.84124
0.837570.834130.830890.827830.824950.822210.81962
0.817160.814810.81258
0.810440.80840
0.806450.80458
0.802780.80105
0.799400.797800.796260.79478079334
0.791960.790630.789330.78808
P0.80
2.686051.740271.490741.369601.295811.245201.207851,178891,155621.136421.120231.106361.09429
1.083691.074271.065831.05822
1.051311.045001.039211.033881.028941.024341.020071.016071.01232
1.008791.005471.002330.999360.99655
0.993880.991340.988910.986600.984390.982280.980250.978310.97644
0.974650.972930.971270.969670.96812
0.966630.965190.963800.96245
P0.8413
3.144132.009401.718921.580101.496411.439441.397631.365361.339521.318271.300401.285111.271851.260211.249891.240661.23234
1,224801.217921.211621.205811.200441.195451.190801.186461.18240
1.178581.174981.171581.168371.165331.162441.159691.157081.1 54581.152201.149921.147731.145641.14363
1.141701.139841.138051.136331.13466
1.133061.131511.130011.12856
P
0.85
3.251692.07212
1.771941.628931.542901.484411.441551.408491.382051.360301.342031.326411.312861.300981.290441.281021.272541.264841.257831.251401.245481.240001.234921.230191.225771.221621.217731.214071.210611.207341.204241.201301.198511.195841.193301.190881.188561.186331.184201.182161.180191.178301.176481.174731.173041.171411.169831.168311.16684
14 E
Table 2. One-sided k (continued).
Pr(Tv<k,/'fflKp1-_}=?', y= 0.75
2
34
56789
1011121314151617181920212223!24425 _26'27'28'29
3031323334353637383940
41424344
454647484950
P0.90
3.992642.501232.13373
1.961611.859231.790211.739941.701381.670671.645501.624421.606441.590891.577271.565221.554461.544791.536031.528061.520761.514041.507831.502081.496731.491731.487051.482661.478531.474631.470951.467461.464151.461011.45801
1.455161.452431.449831.447341.444951.44265
1.440451.438341.436301.434341.432441.43062
1.428861.427151.42551
P0.95
5.121343.151742.680522.46331
2.335522.250062.188232.141032.103602.073052.047532.025832.007111.990741.97629
1.963421.951861.941411.931901.923211.915221.907851.901031.894681.888761.883231.878031.87315
1.868551.864201.860091.856191.852491.848961.845611.842401.839341.836411.833601.830911.828331.825851.823461.821161.818941.816801.814741.812751.81082
P P0.975 0.9773
6.11237 6.23838
3.72465 3.797713.16171 3.223082.90438 2.960622.75387 2.807212.65367 2.705112.58143 2.631532.52645 2.575552.48296 2.531282.44752 2.495222.41799 2.465162.39291 2.439652.37130 2.417672.35244 2.398492.33581 2.381572.32100 2.366522.30772 2.353012.29572 2.340812.28482 2.329732.27486 2.319602.26571 2.310302.25727 2.301732.24946 2.293792.24221 2.286422.23544 2.279552.22912 2.273122.22319 2.267102.21762 2.261442.21237 2.256102.20742 2.251072.20273 2.246312.19828 2.241792.19406 2.237512.19005 2.233432.18623 2.229542.18259 2.225842.17910 2.222312.17577 2.218932.17258 2.21569
2.16952 2.212582.16659 2.209592.16377 2.206722.16105 2.203972.15844 2.201322.15592 2.198772.15350 2.196312.15116 2.193932.14889 2.191632.14670 2.18941
P0.99
7.268964.395983.725803.421283.243953.126303.041712.977482.926762.885512.851182.822072.797012.775162.75591
2.738792.723442.70959
2.697012.685522.674982.665262.656262.647922.64014
P0.9987
9.388525.63814
4.770464.378554.151344.001143.893463.811903.747633.695463.652103.615403.583833.556373.532173.510653.491423.474053.458243.443883.430753.418563.407273.396893.38722
P0.999
9.672515.804964.910834.507194.273274.118684.007883.923983.857883.804243.759643.721943.68947
3.661223.636373.614263.594483.576653.560453.545663.532203.519663.508043.49739
3_48744
P
0.9999
11.65011
6.969745.891545.405845.125054.939714.807224.706884.627964.563964.510594.465934.427254.393514.364274.33766
4.314214.293284.273764.25605
4.240674.225534.211514.198914.18764
P0.999968
12.533347.491076.330365.80826
5.506475.307495.165165.057364.972754.904064.847034.798864.75630
4.721384.690844.661474.636284.613844.593054.574224.557364.541564.526124.513064.50072
2.632862.62605
2.619642.613622.607932.602542.597442.592612.588002.583622.579442.57544
2.571632.567982.564492.561112.557882.554762.551782.548912.546142.543462.540862.53836
3.378113.369573.361593.354133.347023.340093.333843.328033.322453.316903.311463.306303.30176
3.297423.293273.288903.284773.280693.277043.273653.270393.267163.263713.26051
3.478113.469323.461133.453483.446243.439273.432603.426583.420823.415303.40978
3.404653.399683.39519
3.390833.386553.382273.378223.37440
3.370853.367433.364043.360613.35739
4.17639 4.488624.16542 4.476984.15560 4.466864.14713 4.457564.13884 4.449164.13017 4.439404.12131 4.430054.11486 4.423264.10847 4.416314.10270 4.409564.09523 4.401924.08847 4.39463
4.08243 4.388734.07813 4.383694.07344 4.378544.06854 4.372604.06285 4.367504.05732 4.361474.05268 4.357254.04902 4.352584.04564 4.348744.04185 4.343924.03745 4.339884.03308 4.33533
P0.99999
13.365887.982856.744536.188125.866425.654445.502825.388155.298175.224975.164265.113035.069325.030614.997344.96687
4.940034.91651
4.894204.874114.856494.839844.822734.809174.79647
4.783414.770514.759974.750414.742254.731124.720354.712644.706804.699944.690584.682824.676724.67177
4.666934.660264.654944.64668
4.643664.638094.635884.629584.625414.61956
15
Table 3. One-sided k.
P r ( Tv < k ,f-ff l K p /-ff} = 7 , ?'=0.90
2
3456789
1011121314
1516171819202122i
23]2425!26
2728!29!
303132333435363738394041424344454647
48 i491
50
P P P P P P P P P0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.8413 0.85
i ""
2.17634
1.088660.818870.685670.602530.544180.500250.465610.437350.41373
0.393590.376150.36085
0.347280.335150.324210.314280.305210.296890.289210.282110.275500.269330.263570.258160.253070.248270.243730.239430.235360.231480.227790.224280.220920.217700.214620.211680.208840.206120.203510.200990.198560.196220.193960.191770.189660.187610.18563
0.18372
2.72056
1.334361.012150.858230.764110.698980.650460.612500.581730.556140.534420.515680.49930
0.484810.471880.460250.449710.440110.431310.423210.415710.40875
0.402270.39621
0.390530.385200.380170.375420.370930.366670.362620.358770.355100.351600.348250.34505
0.341980.339040.336210.333490.330870.32835
0.325920.323570.321310.319120.317000.314950.31297
3.343081.602531.21950
1.041660.934870.861920.808080.766260.732580.704700.681130.660860.643200.62762
0.613760.601310.590050.57981
0.570440.561830.553870.546490.539620.533210.527210.521570.51627
0.511260.506530.502040.497790.493740.489880.486210.482700.479340.476120.473030.470070.467220.464490.461850.459310.456860.454490.452210.449990.447860.44578
4.056621.897971.444611.239231.117861.035890.975920.929650.892580.862040.836320.814280.795120.778270.763300.749890.737790.726800.716760.707530.699020.691150.683820.676990.670600.664610.658970.653660.648630.643880.639370.635080.631000.627110.623400.619850.616450.613200.610070.607070.604180.601410.598730.596150.593660.591260.588930.586680.58450
4.880672.228011.69301t.455761.317541.225121.158031.10659
1.065591.031951.003720.97960
0.958690.940350.924090.909550.896450.884570.873730.863790.854630.846160.838290.830960.824110.817690.811650.805960.800600.795520.790700.786130.781780.777640.773680.769910.766290.762830.759510.756320.753250.750300.747460.744720.742080.739530.737070.734690.73238
5.842562.602831.972261.697811.539891.435271.359861.302351.25673
1.219441.188251.16168
1.138711.118601.100811.084921.070641.057701.045911.035111.025181.016001.007480.999550.992150.985210.978700.972570.966780.961310.956130.951120.946540.942090.937840.933790.929910.926200.922630.919220.915940.912780.909740.906810.903980.901260.898620.89608
0.89362
6.987863.03934
2.294831.976091.794721.675541.590181.525431.474271.432611.397861.368351.34288
1.320641.301001.283491.267771.253551.240611.228771.217891.20785
1.198541.189881.181801.174241.167141.160471.154181.148231.142601.137261.132181.127351.122741.118351.114151.110131.106271.102571.099021.095611.092321.089161.086111.083161.080321.077571.07492
8.137223.468342.61005
2.247082.042271.908521.813191.741161.684441.638371.600051.567561.539581.51518
1.493661.474511.457331.441811.427701.414811.402961.39204
1.381921.372521.363751.355551.347861.340631.333821.327381.321291.315511.31003
1.304811.299841.29509
1.290561.286221.282061.278081.27425
1.270571.267031.263621.260341.257171.254111.251151.24830
8.403373.568682.683602.310212.099871.962691.865001.791251.733211.686101.646931.613731.585161.560241.538281.518741.501221.485391.471001457861.445781.434651.424341.414761.405831.397481.38964
1.382281.375341.368791.362591.356711.351131.345811.340751.335921.331311.326901.322671.318611.314721.310981.307381.303911.300571.297351.294241.291231.28832
16
Table3. One-sidedk (continued).
Pr(Tv<_k_lKv_)= )', )'=0.90
tt
P0.90
10.252634.258323.187962.742442.493772.332722.218662.132942.06574
2.011351.966261.928141.895401.866901.841831.819551.799601.781601.765261.750351.736671.724071.71241
1.701581.691491.682071.673241.664941.65712
1.649741.642771.636161.629881.623921.618241.612821.607641.602691.597951.593411.589041.584851.58082
1.576951.573211.56961
1.566131.562771.55952
P0.95
13.089705.311323.956453.399743.091792.893722.754212.649822.56830
2.502542.448182.402332.363042.328912.298932.272342.248562.227142.207712.190012.173782.158842.145042.132232.120312.109182.098752.088962.07975
2.071072.062852.055082.047702.040692.034012.027652.021582.015772.010212.004881.999771.994871.990151.985611.981241.97702
1.972961.969031.96523
P0.975
15.586246.243824.637053.981383.620493.389253.226893.105733.011302.935292.872542.819702.774472.735232.700802.670292.643032.618492.596262.576002.557452.540392.524622.510012.4964 12.483712.471832.460682.450202.440312.430962.422122.413722.405752.398172.390942.384042.377442.371132.365092.359292.353722.348372.343222.338272.33349
2.328882.324432.32013
P0.9773
15.904196.363004.724084.055743.688063.452573.287283.163963.067872.990532.926702.872962.826972.787072.752062.721042.693332.668392.645802.625212.606362.589022.573012.558162.544342.531442.519372.508052.497392.487352.477862.468872.460352.452252.444552.437212.430202.423502.417102.41096
2.405072.399422.393982.388762.383732.378872.37419
2.369672.36531
P0.99
18.500097.34060
5.438354.666074.242623.972113.782633.641523.531733.443493.370743.309553.257233.211883.172123.136923.105493.077213.051613.028303.006962.987342.969222.952432.936812.922242.908602.895822.883792.872452.861742.851602.841992.832872.824182.815902.808002.800462.793242.786332.779702.773332.767222.761342.755682.750222.744942.73986
2.73494
P
0.9987
23.862969.380236.928525.93969
5.399635.055744.815484.636964.498354.387104.295544.218614.152954.09597
4.046294.00213
3.962803.927483.895473.86635
3.839763.815303.792643.771773.752373.734183.717183.701323.686403.672273.658923.646363.634463.623123.612363.602033.592173.582933.573963.565373.557153.549293.541723.534433.527403.520653.514153.50788
3.50182
P0.999
24.581389.653157.129246.111215.555465.201674.954544.770974.62847
4.514094.420004.340934.273464.214944.163904.11850
4.078094.041824.008923.978983.95178
3.926613.903313.881903.862013.843313.825833.80956
3.794273.779753.766033.753143.740943.72930
3.718273.707633.697623.688033.678963.670363.661563.653383.645753.638263.631153.624143.617453.610883.60480
P0.9999
29.5865111.611238.533127.311016.645516.222445.927305.708415.538615.40210
5.290225.195875.116105.046424.985874.932024.883754.841584.802104.766654.734414.70445
4.676364.651234.628024.606014.584654.56575
4.548254.531414.513844.497984.485024.47207
4.458544.444784.431564.422114.411524.401704.390754.380584.371804.363844.355714.347064.339424.331444.32221
P0.999968
31.82294
12.570669.16226
7.849927.134086.680026.363686.128825.947795.800335.680765.578575.493935.418935.354255.296825.245155.199635.157405.118875.085225.05404
5.020574.995564.973684.947854.925134.905134.887064.86871
4.848774.833594.817854.80727
4.790514.774684.763294.751024.739754.727834.718364.707424.698094.688794.68171
4.673484.661844.652384.64034
P0.99999
33.9315913.546749.756688.35726
7.596417.111846.774106.52611
6.332016.175526.049435.940735.850735.77042
5.701915.641495.585915.53804
5.492935.449595.416445.383955.349505.321925.296775.270715.244275.225005.205465.187855.164415.149145.133135.117945.101195.086055.073835.060455.049625.036325.025285.014935.002744.995464.98802
4.980604.968834.955294.94138
17
Pn 0.50
2 4.464593 1.685864 1.17668
5 0.953396 0.822647 0.734458 0.669849 0.61985
10 0.57968
Table 4. One-sided k.
Pr(Tv <k1"_lKvg'ff)= Y, y= 0.95
P
0.75
11.763573.806192.617612,149691.895031.732251.617821.532201.46524
P0.90
20.581686,155264.16193
3.406633.006262.755432.581912.453762.35464
P0.95
26.258747.655955.143894.202683.707683.399473.187293,031242,91096
P
0.99
37.0945410,552737.042355.741105.061974.641724.353864.143023.98112
P
0,999
49.2743713.857159,21413
7.501876.611796.062835,687545.413355.20320
P0.9999
59.3012516.5979611.01895
8,965997.900517.244126.796236.469266.21886
P0.99999
68.0071518.9855512.5929810.24328
9.024898.274907.763487.390327.10472
Table 5. One-sided k.
Pr{Tv<_kg'51Kpg'5} = y, y=0.99
n
23456789
10
P0,50
22.509104,021052,270351,675691.373731,187821.059940.965490,89222
P0.75
58.937658.728194.715193.454112.848092.490722.253372.083141.95433
P0.90
103.0070713.99520
7.379955.361754.411083.859133.49721
3.240413.04791
P0.95
131.44022
17.370209.083436.578335,405544.727864,285253.972263.73832
P0.99
185.6286823.8960412.387338,939077.334566.41193
5.811775.388905.07374
P0.999
246.6043231.3476516.1757011.649329.549928.345737.564167.014406.60539
P0.9999
296.7162937,5327619.3274213.9065311.395469.956919.024038.368427.88102
P0.99999
340,2216142.9218522.0779215.8773313.00732
11.3640710.299079.550928.99499
18m
=
Table 6. Two-sided k.
Pr(Tv<kl'_lKpl'_=),, r= 0.50
23456789
10111213141516171819202122
232412526272829303132333435363738394041
424344
454647484950
P P0.50 0.55
1.24272 1.384110.94201 1.05228
0.85225 0.953080.80826 0.904360.78196 0.875180.76439 0.85568
0.75181 0.841690.74235 0.831160.73496 0.822930.72903 0.816330.72417 0.810910.72011 0.806390.71666 0.802540.71370 0.799240.71114 0.796380.70888 0.793860.70689 0.791640.70512 0.789660.70353 0.787880.70210 0.786290.70080 0.784840.69962 0.783530.69854 0.782320.69756 0.781220.69664 0.780200.69580 0.779260.69502 0.778380.69430 0.777570.69362 0.776820.69299 0.776120.69240 0.775460.69185 0.774840.69133 0.774250.69084 0.773710.69037 0.773190.68994 0.772700.68952 0.772240.68913 0.771800.68876 0.771380.68841 0.770990.68807 0.770610.68775 0.770250.68744 0.769910.68715 0.769580.68687 0.76927
0.68660 0.768970.68635 0.768690.68610 0.76841
0.68587 0.76815
P0.60
1.533241.169051.060091.006470.974310.952780.937320.925680.916580.90927
0.903270.898250.893990.890330.887150.88436
0.881900.879700.87773
0.875960.874350.872890.871550.87032
0.869190.868150.867180.866280.865440.864650.863920.863230.862590.861980.861400.860860.860340.859860.859390.858950.858540.858140.857760.857390.857050.856710.856400.856090.85580
P0.65
1.69250
1.294221.175021.116281.080981.057331.040331.027511.017481.009421.002810.997270.992570.988520.985010.981930.979200.976780.974600.972640.970860.969240.967760.966400.965150.963990.962920.961920.961000.960130.959320.958550.957840.957160.956530.955920.955360.954820.954300.953820.953350.952910.952490.952090.951700.951330.950980.950640.95032
P0.6827
1.803671.381841.255611.193351.155901.130781.11272
1.099081.088421.079841.072801.066901.061891.057581.053841.050561.047651.045061.042741.040651.038751.037021.035441.033991.032661.031421.030281.029211.028221.027301.026431.025621.024851.02413
1.023451.022811.022201.021621.021071.020551.020061.019591.019141.018711.018301.01790
1.017521.017161.01682
P0.70
1.865291.430491.300401.236211.197581.171651.153011.138931.127911.119051.11177
1.105671.100491.096041.092171.08877
1.085771.083091.08069
1.078521.076561.074771.073141.071 641.07025
1.068981.067791.066691.065661.06471
1.063811.062971.062171.061431.060721.060061.059431.058831.058261.057721.057211.056721.056261.055811.055391.054981.054591.054221.05386
P0.75
2.056701.581911.44000
1.369911.327661.29928
1.278841.263391.251291.241561.233551.226841.221141.216241.211971.208221.204911.201961.199311.196921.194751.192781.190981.189321.187791.186381.185071.183861.182721.181661.180671.179741.178861.178041.177261.176521.175831.175161.174541.173941.173381.172831.172321.171821.171351.170901.170471.170061.16966
P0.80
2.274971.755101.599991.523321.477061.445931.423501.40652
1.393211.382501.373671.366281.359991.354581.349871.345731.342071.338811.335881.333241.330841.328661.326661.324821.323131.321571.320121.318771.317511.316341.315241.314211.313241.312321.311461.310641.309871.309141.308441.30778
1.307151.306551.305981.305431.30491
1.304411.303931.303471.30303
P0.85
2.535081.96210
1.791581.707271.656351.622051.597291.578541.563831.55197
1.542191.534001.527021.521021.515791.51119
1.507121.503491.500241.497301.49463
1.492201.489971.487931.486041.48430
1.482681.481181.479781.478471.477241.476091.475001.473981.473021.472111.471241.470431.469651.468911.468211.467541.466901.466281.465701.46514
1.464611.464091.46360
19
Table 6. Two-sided k (continued).
Pr(Tv<k_lKp_}=_', _'= 0.50
tl
23456789
10111213141516171819
202122
232425262728293031323334353637383940414243444546 I
47484950
P P P P P P P P P P
0.90 0.95 0.9545 0.975 0.99 0.9973 0.999 0.9999 0.999937 0.99999
2.869352.228912.039011.945171.888461.850211.822571.801611.785141.771851.760891.751701.743861,737111.731231.726051.721471.71738
1.713711.710391.707381.704641,702121.699811,697681.695711.693891.692181,690601.689121.687731.686421.685191.684041.682941.681911,680931,680011.679131.678291.677491.676731.67600
1.675311.674651.674011.673401,672821.67226
3.375602.634232.415742.307962.242822.198842.167022.142862.123862.108502.095822.085162.076082.068242.061402.055382.050052.045282.041002.037132.033622.030412.027482.024772.022282.019982.017842.015842.013982.012252.010622.009082.007642.006292.005002.003792.002642.001552.000511.999531.998591.997691.996841.99602
1.995241.994491.993781.993091.99243
3.439422.685422.463382.353882.287692.243012.210682.186132.166822.151212.138312,127482.118242,110262.103312.097192.091762.086922.082562.078622.075042.071782.068792.066042.063512.061162.058982.056952.055062.05329
2.051632.050072.048602.047222.045912.044682.04350
2.042392.041342.040332.039382.038462.037602,03676
2.035972.035212.03448
2.033782.03310
3.822222,992762.749612.629922.557622.508812.473472.446622.425482.408382.394252.382372.372232.363472.355832.349102.343132,337792.332992.328652.324712.321112,317812.314782,311982.309382.306982.304732.302642.30068
2,298842.297122.295502.293962.292522.291152.289852.28862
2.287452.286342.285282.284262.283302.28238
2.281492.280652.279842.27906
2.27831
4.347623.415353.143693.010362.929912.875622.836312.806422.78288
2.763822.748062.734792.723472.71368
2.705132.697602.690912.684932.679542.674682.670252.666212.662502.659092.655942.653022.650312.647782.645422.643212.641142.63920
2.637362.63563
2.634002.632452.630982.62959
2.628272.627012.625812.624672.623582.622532.621532.620572.619652.618772.61793
5.007593.947093.640183.490 t 13.399753.338823.294723.261193.23476
3.213373.195663.180753.168013,156993.147363,138873.131323.124573.118493.112993.107983.103413.099213.095343.091773.088463.085393.082523.079843.077333.074983.07277
3.070683.068713.066853.065093.063423.06184
3.060333.058893.057523.056223.054973.053783.052643.051543.050493.04949
3.04852
5.456374.309103.978473.817233.720283.654973.607713.571783.54348
3.520553.501573.485583.471923.460103.449773.440663.432553.425303.418773.412853.407473.402553.398033.393873.390033.38646
3.383153.380063.377173.374473.371933.369543.367293.365173.363163.361263.359463.357743.356113.354563.353083,351673.350323,349033.347803.346613.345483.344393.34334
6,376975.052424.67364
4.489824.379644,305574.252034.211374.17934
4.153394.131924.113834.098364.084974.073264.062934.053744.045514.038104.031384.025264.019674.014534.009794.005424.001363.997583.994053.990763.987673.984773.982053.979473.977043.974753.972573.970513.968543.966683.964903.963203.961583.960043.958563.957143.955783.954473.953223.95202
6.544065.187434,79995
4.612084.499534.423894.369244.327744.29504
4.268574.246664.228194.212414.198754.186804.176254.166874.158474.150904.144044.137804.132084.126844.122004.117534.113384.109524.105924.102564.099414.096444.093664.091034.088554.086204.083974.081864.079864.077954.076134.074404.072744,071164.069644.068194.066804.065474.064194.06296
7.179475.700985.280585.07736
4.955884.874364.815514.770854.735694.707234.683674.663824.646864.632174.619334.607994.597904.588864.580724.573344.566624.560474.554824.549614.544794.540324.536164.532284.528664.525254.522064.519054.516214.513534.511004.508604.506324.504154.502094.500124.498254.496464.494744.493114.491544.490034.488594.487204.48586
20
P
n 0.50
2 2.673263 1.491234 1.210745 1.08297
6 1.008997 0.96038
8 0.9258091 0.89984
10 0.87957111 0.8632612 0.8498213 0.8385414 0.8289115 0.8205916 0.8133217 0.8069018 0.8011919 0.7960720 0.7914521 0.7872522 0.7834323 0.7799224 0.7767025 0.7737226 0.7709627 0.7683928 0.7659929 0.7637530 0.7616431 0.7596732 0.7578033 0.7560534 0.7543835 0.7528136 0.7513137 0.7498938 0.7485439 0.7472540 0.7460241 0.7448442 0.7437143 0.7426444 0.7416045 0.7406146 0.7396547 0.73873
48 0.7378549 0.7370050 0.73618
Table 7. Two-sided k.
Pr(Tv<kl'_lKp1-ff)=?', y= 0,75
P0.55
2.972341.66318
1.352581.210891.128741.074691.036211.00729
0.984700.966510.951520.938920.928170.918880.910760.90359
0.897200.89148
0.886320.881630.877360.873440.869830.866500.863410.86053
0.857850.855340.852990.850780.848690.846730.844870.843100.841430.839840.838330.836880.835510.83419
0.832930.831720.830560.829450.828380.827350.826360.825410.82449
P
0.60
3.287591.844991.50287
1.346641.255951.196201.153621.121601.096561.076391.059761.045781.033851.023531.014511.006540.999450.993090.987350.982140.977390.97303
0.969020.965310.961870.958680.955690.95290
0.950280.947820.945510.943320.941250.939280.937420.935650.933970.932360.930830.929360.927960.926610.925320.924080.922900.921750.920650.919590.91856
P
0.65
3.62408
2.039601.664071.492461.392711.326931.28000
1.244681.217051.194771.176391.160941.14774
1.136331.126341.117521.109671.102631.096271.090501.085241.080411.075961.071851.068041.064501.061191.058101.055201.052471.049901.047471.045171.043001.040931.038971.037101.035321.03362
1.031991.030431.028941.027511.026141.02482
1.023551.022331.021151.02001
P
0.6827
3.858882.175681.77698
1.594701.488681.418721.368771.331161.30172
1.277981.258381.241901.227821.215641.204991.195581.187191.179671.172881.166721.16110
1.155941.151191.146801.142731.138941.13541
1.132101.129001.126081.123331.120741.11828
1.115961.113751.111651.109651.107751.105931.104191.102531.100931.099401.097931.096521.095171.093861.092601.09138
P
0.70
3.989022.251211.839691.65152
1.542051.469771.41816
1.379291.348851.324291.304021.286981.272411.259811.248781.239041.230361.222581.215551.209171.203341.198001.193081.188541.184321.180401.176741.173311.170101.167081.164241.161551.159011.156601.154311.152141.150061.14809
1.146211.144411.142681.141031.139441.137921.136461.135061.133701.132401.13114
P0.75
4.393152.48613
2.035011.828661.708511.629111.572351.52957
1.496041.468981.446621.427811.411731.39782
1.385631.374871.365281.356671.34890
1.341841.335401.329491.324041.319011.314351.31001
1.305951.302161.298601.295261.292101.289131.286311.283641.281111.278701.276401.274221.272131.270131.268221.266391.264631.262951.261331.259771.258271.256821.25542
P P
0.80 0.85
4.85395 5.403052.75463 3.075352.25863 2.526192.03173 2.275001.89952 2.12855
1.81206 2.031601.74949 1.962181.70228 1.909751.66526 1.868601.63536 1.835331.61064 1.807821.58983 1.784641.57203 1.764811.55662 1.747631.54313 1.732581.53120 1.719261.52056 1.707401.51102 1.696741.50240 1.687111.49457 1.678361.48742 1.670371.48086 1.663041.47482 1.656281.46923 1.650031.46405 1.644241.45923 1.638841.45473 1.633811.45052 1.629091.44656 1.624661.44284 1.620501.43934 1.616581.43603 1.612881.43290 1.609371.42994 1.606051.42712 1.602891.42444 1.599891.42189 1.597041.41946 1.594311.41714 1.591711.41492 1.589221.41279 1.586841.41076 1.584561.40880 1.582371.40693 1.580271.40512 1.578251.40339 1.576301.40172 1.574431.40011 1.572631.39856 1.57089
21
Table 7. Two-sided k (continued).
P r ( Tv < k g'ff l K p1"ff ) = )" , )'=0.75
n
2
3456789
10111213
14151617181920212223242526272829303132333435!3637
38394041424344
4546474849
50
|
P0.90
6.10874
3.488532.871482.589362.424812.315792.237662.178602.132202.094672.063602.037402.01498
1.995541.978491.963411.949961.937881.92696
1.917041.907971.899641.891971.884881.878291.872161.866431.861071.856041.85131
1.846851.842631.838641.834861.831271.82786
1.824601.82150
1.818541.81571
1.813001.810401.807901.805511.803211.800991.798861.79680
1.79482
P0.95
7.177614.116063.396873.068362.876732.74969
2.658562.589612.535382.491482.455102.424392.398092.375262.355242.337512.321692.307472.294612.282912.272222.26240
2.253352.24498
2.237202.229962.223202.216862.210912.205322.200042.195052.190332.185862.181612.177562.173712.170042.166522.163172.159952.156872.153922.151082.148352.145722.143192.140752.13840
P0.9545
7.312384.195303.463283.128962.933942.804642.71189
2.641702.586502.541792.504752.47348
2.446702.423442.403052.384992.368872.354392.341282.329362.318472.308462.299242.290702.282782.275402.268502.262042.255982.250282.244902.239812.235002.230442.226102.221982.218052.214312.210732.207302.204032.200882.197872.194982.192192.18951
2.186932.184442.18205
P0.975
8.120794.671103.862323.493243.277983.135243.032802.955252.894222.844772.803772.769142.73946
2.713692.691072.671042.653152.637062.622512.60926
2.597162.586032.575772.566282.557462.549242.541572.534382.527632.521272.515282.50961
2.504252.499162.494332.489732.485352.481172.477182.473362.469712.466202.462842.459612.456502.453512.450632.44785
2.44517
P0.99
9.230625.325384.411663.995163.752343.591323.475733.388173.319243.263353.216993.177823.144223.115023.089393.066673.046383.028123.011602.996552.982792.97015
2.958482.947682.937652.928302.919562.911372.903682.896442.889602.883142.877032.871232.865722.860472.855472.850702.846152.841792.837612.833612.829772.826082.822522.819112.815822.812642.80958
P0.9973
10.625096.148825.103774.628044.350884.167114.03517
3.935213.85647
3.792613.739603.694793.656333.622903.593543.567503.544223.523283.504313.487033.471223.456693.443283.430863.419313.408553.398493.389063.380203.371853.363983,356533.349483.342793.336433.330383.324613.319103.313843.308813.303983.299363.294923.290653.286553.28260
3.278793.275123.27158
P0.999
11.573546.70953
5.575425.059574.759184.560044.417084.308764.223424.154184.096704.048104.00638
3.970103.938233.909953.884683.861923.841313.822543.80535
3.789553.774963.761453.748893.737173.726223.715953.706313.697223.688643.68053
3.672853.665563.658623.65203
3.645743.639743.634003.62851
3.623243.618203.613363.608703.604223.599913.595763.591753.58789
P0.9999
13.519577.861166.544765.946935.599125.368665.203245.077904.979144.899004.832454.776154.727814.685764.648794.615994.586664.560244.536294.514484.49450
4.476134.459164.443444.428814.415174.402424.390454.379214.368614.358614.349154.340184.331684.323584.315884.308534.301524.294824.288404.282254.276354.270694.265244.260004.254964.250104.245414.24088
P0.999937
13.872848.070366.720916.108245.751855.515735.346255.21784
5.116665.034554.966364.908684.859144.816054.778174.744554.714494.687414.662874.640504.620024.60119
4.583794.567674.552684.538694.525614.513344.501814.490944.480684.470984.461784.453054.444754.436854.429314.422124.415244.408664.402344.396294.390484.384894.37952
4.374344.369354.364544.35990
P
0.99999
15.216378.866257.391266.722206.333236.075625.890765.750705.640345.55078
5.476405.41347
5.359415.312395.271045.234345.201515.171945.14513
5.120705.098325.077735.058725.04109
5.024695.009404.995094.981664.969044.957154.945924.93530
4.925234.915674.906584.89793
4.889674.881794.87425
4.867034.860124.853484.847114.840984.835094.829424.823944.818674.81357
22
Table8. Two-sidedk.
Pr{Tv < k1"ff l Kpg'_) = )', ),=0.90
2345
6789
1011121314
1516
171819202122232425262728293031323334
35363738394041424344454647484950
P P P P P P P P P0.50 0.55 0.60 0.65 0.6827 0.70 0.75 0.80 0.85
,==,
6.808262,491851.765621.472711.313561.212991.143371.09212
1.052691.021330.995740.974420.956340.940800.92729
0.915410.904870.895450.886980.879300.872320.865930.860050.854640.849620.844960.840620.836560.832750.829180.825820.822640.81964
0.816800.814110.811550.809110.806790.804570.802460.800430.798490.796630.794840.793130.791480.789890.78836
0.78688
7.566072,776041.970411,645351.468571.356771.279291.22220
1.178261.143291.114741.090941.070761.053411.03831
1.025041.013261.002730.993260.984680.976870.969720.963160.957100.951480.946270.941410.936870.932610.928620.924850.921300.91794
0.914770,911750.908880.906160.903560.901080.898710.896440.894270,892190.890190.888270,88642
0.884640.88293
0.88128
8.364783.076322.187211.82837
1.633081.509461.423721.36051
1.311811.273041.241361.214951.192551.173281.156501.141751.128671.116971.106431.096891.088211.080261.072951.066211.059971.054171.048761.043711.038971.034521.030341.026381.02265
1.019111.015751.012561.009531.006631.003871.001240.998710.996290.993970.991750.989610.98755
0.985570.983660,98182
9.217303.397622.419562.02478
1,809811,673621,579101.50935
1,455591.412761.377751.348541.323761.302441.28388
1.267551,253061.240101.228431.217861.208231.199421.191331.183861.176941.170501.164511.158911.153651.148721.144071.139691.135551.131621.127901.124361.120991,117781.114721.111791,108991.106311.103731.101261.098891.09661
1.094411.092291.09025
9.812163.622232.582212.16241
1.933751.788821.688181.61388
1.556591.510931.473601.442451.416011.393251.37344
1,356001.340531.326691.314221.302941.292651.283241.274591.266601.259211.252341,245931.239941.234321.229051.224081.219401.214971.210771.206791.203011.199411.195981.192701.189571,186571.183701.180951.178311.175771.17333
1.170981.168721.16653
10.141853.746862.672532.238882.002641.852881.748851.67204
1,612801.565571.526951.494721.467361.443811,423301,40525
1.389231.374901.362001.350311.339661.329921.32096
1.312691.305031.297911.291271.285071.279251.273791.268651.263791.259201.254861.250731.24681
1.243081.239521.236131.232891.229781.226811.223961.221221,218591.21606
1.213631.211281.20902
11.165724.134492.953722.477152.217432.052691.938181.85358
1.788281,736201.693581.658001.62779
1.601771.57910
1.559151.541441,525591,511311.498381.486591.475811.465891.456731.448251.440371.433021.426141.419701.413651.407951.402571.397481.392671.388101.383751.379621.375681.371911.368321.364881.361581.358421.355391.352471.34967
1.346971.344361.34186
12.333174.577463.275542.750152.463742.281982.155572.062111.98993
1.932331.885171.845781.812311.783481.758361.73624
1.716591.699011.683161.668811.655721.643741.632731.622561.613141.604381.596211.588571.581411.574681.568351.562371.556711.551361,546271.541441.536841.532461.528271.524271,520451.516781.513261.509891.50664
1.503521.500521.497621.49483
13.724375.106513.660463.077032.758942.556972.416432.312462.23212
2.167962.115412.071492,034162.001981.973931.94922
1.927271.907621,889911.873861.859221,845821.833501.822121.811571.801761.79262
1.784061.776041.768511.761411.754711.74837
1.742371.736681.731261.726111.721191.716501.712021.707721.703611.699671.695891.692251.68875
1.685381.682131.67900
23
Table8. Two-sidedk (continued).
2
3456789
1011
1213
1416[17118
19120121!
22
2324252627282930313233343536373839404142
4344454647484950
24
P0.90
15.51241
5.788094.157093.499273.140582.912762.754142.636732.545942.47340
2.413952.36422
2.321942.285482.253672.225652.200742.178442.158332.140092.123472.108242.094232.081282.069292.058132.047732.038002.028872.020292.012212.004581.997371.990531.984041.977881.972001.966401.961061.955951.951061.94637
1.941881.937561.933421.929431.925581.921881.91831
Pr(Tv<_kg'fflKeg-ff}=y, y= 0.90
P0.95
18.220846.823284.91267
4.142473.722533.455743.269883.132233.025712.940542.870682.812222.76247
2.719552.682092.649062.619692.593382.569652.548122.528482.510482.493922.478622.464432.451232.438912.427392.416592.40643
2.396862.387822.379272.371162.363472.356162.349192.342552.336212.330152.324352.318792.313452.30833
2.303412.298672.294112.289712.28547
P0.9545
18.562346.954015.008184.223833.796193.524493.335213.195013.086522.999762.928602.869042.818352.774622.736452.702792.672862.646042.621862.599912.579892.561552.544662.529062.514602.501142.488592.476842.465822.455462.445702.436482.427772.419502.411662.404202.397102.390332.38386
2.377682.371762.366092.360652.355422.350402.345572.340922.336442.33211
P0.975
20.610977.739015.582054.712914.239103.938053.728283.572853.452533.35628
3.277313.211183.154893.106303.063873.026452.99316_.963332.936412.911982.889692.869262.850452.833072.816952.801952.787952.774852.762562.751012.740122.729842.720112.710882.702132.693802.685872.678312.671092.664192.65758
2.651242.645172.639332.633722.628322.623122.618112.61328
P0.99
23.423618.818636.372135.386774.849734.508504.270694.094453.957963.848753.759103.68400
3.620043.56481
3.516573.474013.436133.402173.371523.343703.318303.295023.273583.253763.235373.218263.202293.187343.173313.16012
3.147683.135933.124823.114283.104283.094763.085703.077053.068793.060903.053343.046093.03914
3.032463.026043.019873.013923.008183.00265
P0.9973
26.9579010.177587.367676.236525.620205.228634.955724.753434.596734,47129
4.368304.281994.208464.144944.089444.040453.996843.957723.922413.890343.861073.834223.809483.786623.765403.745643.727203.709933.693733.678483.664113.650533.637683.625503.613923.602923.592433.58242
3.572873.563733.554983.546593.538543.530803.523373.516213.509323.502673.49626
P0.999
29.36192
11.103088.046176.815976.145815.720075.423365.203395.032994.896574.78453
4.690634.610614.541484.481064.42772
4.380234.337634.299164.264224.232324.203054.176094.151164,128024.106484.08636
4.067524.049844.033214.017524.002713.988683.97538
3.962743.950723.939273.928343.917913.90792
3.898363.889203.880403.871953.863833.856013.848473.841213.83420
P0.9999
34.2947413.004249.440898.007647.227176.731456.385996.129875.931445.772555.642045.532625.439365.358775.288315.226095.170685.120965.076045.035244.997974.963784.932274.903124.876064.850874.827334.80529
4.784604.76513
4.746764.729414.712984.697394.682594.668504.655084.642274.630034.618324.607114.59636
4.586044.576124.566594.557414.548564.540044.53181
P P0.999937 0.99999
35.19027 38.5961213.34964 14.663769.69438 10.659088.22430 9.048947.42381 8.172396.91541 7.615766.56110 7.227886.29844 6.940326.09493 6.717525.93197 6.53911
5.79810 6.392545.68588 6.269655.59023 6.164905.50756 6.074355.43528 5.995185.37146 5.925265.31461 5.862975.26360 5.807075.21752 5.756575.17566 5.710685.13742 5.668775.10234 5.630305.07000 5.594845.04009 5.562045.01233 5.531594.98647 5.503224.96232 5.476734.93970 5.451914.91846 5.428614.89848 5.406674.87963 5.385984.86181 5.366434.84495 5.347924.82895 5.330354.81375 5.313664.79929 5.297784.78551 5.282654.77236 5.268214.75980 5.25441
4.74778 5.241204.73626 5.228564.72523 5.216434.71463 5.204794.70445 5.193604.69466 5.182844.68523 5.172484.67615 5.162504.66740 5.152884.65895 5.14359
|
Pn 0.50
2 13.651953 3.58453
2.287651.81188
6 1.565937 1.415268 1.313169 1.23916
10 1.18293
Table 9. Two-sided k.
Pr(Tv <klrfflKp1-_)= 7', 7'=0.95
P0.75
22.383005.937513.818403.040962.638412.391142.223042.100842.00770
P0.90
31.092578.305995.368094.290613.732583.389533.156042.986072.85631
P0.95
36.519629.788806.341105.076894.422164.019603.745513.545903.39343
P0.99
46.9449212.647178.220686.597995.757765.241114.889234.632844.43691
P0.999
58.8443115.9200510.376918.345307.293556.646896.206435.885455.64007
P0.9999
68.7290318.6440112.173549.802378.575027.820467.306526.931966.64561
P0.99999
77.3484921.0218713.7428911.075679.695248.846648.268677.847457.52540
Table 10. Two-sided k.
Pr{Tv < k1-_ l Kp l'_} = 7', 7'=0.99
234567i89
10
P P P P P P P P0.50 0.75 0.90 0.95 0.99 0.999 0.9999 0.99999
68.319378.121934.028552.823872.269571.953781.750151.607881.50274
112.0022413.434956.706184.724203.811683.291222.955082.719772.54553
155.5778018.782979.416286.655025.383264.657604.188663.860133.61664
182.7305022.1314011.118017.869846.373555.519644.967714.580944.29420
234.8908228.5865014.4056310.220248.291637.190776.479065.980195.61021
294.4267435.9783418.1777312.9206210.497529.114208.219737.592657.12749
343.8829242.1308521.3211615.1727912.3384010.720119.673648.939938.39562
387.0087847.5018424.0671617.1411013.9478012.1244810.9453610.118609.50523
25
Table 1 i. Comparison of approximate and exact results for two-sided tolerancefactor k for n = 2 and 10.
n=2 _'
0.90 0.95 0.99
p Approx. Exact Approx. Exact Approx. Exact
0.500.75
0.90
0.95
0.99
0.999
6.80958
11.40606
15.97787
18.79974
24.16655
30.22649
6.80826
11.16572
15.51241
18.22084
23.42361
29.36192
13.64602
22.85712
32.01879
37.67366
48.42846
60.57225
13.65195
22.38300
31.09257
36.51962
46.94492
58.84431
68.27314
114.35770
160.19493
188.48716
242.29504
303.05230
68.31937112.00224
155.57780
182.73050
234.89082
294.42674
P
0.50
0.75
0.90
0.95
0.99
0.999
n=10
Approx.
1.04151
1.77504
2.53516
3.01829
3.9593O
5.04563
0.90
Exact
1.05269
1.78828
2.54594
3.02571
3.95796
5.03299
Approx.
1.16612
1.98740
2.83846
3.379414.43299
5.64929
Exact
1.182932.00770
2.85631
3.393434.43691
5.64007
0.99
Approx.
1.47161
2.50805
3.58207
4.26472
5.594327.12926
Exact
1.50274
2.54553
3.61664
4.29420
5.610217.12749
Nonparametric Tolerance Limits
The nonparametrics (distribution-free) statistical analysis is applied to the one- and two-sidedtolerance limits to determine the level of dependency of the interval on the sample size at given proba-
bility and proportion of the population. The nonparametrics analysis assumes a continuous distribution,
but it precludes any requirement for assumptions, regarding the distribution of data. The data from tables
1 through 10 cannot be applied to the nonparametrics analysis.
One-sided tolerance limits pertain to the relative effect of the proportion: at least a prOportion P
of the population is greater than X (r) or less than x(n-l-m) with probability y. In a comparison, two-sidedtolerance limits primarily concern the interval within which at least a proportion P of the population
occurs. A typical problem embraces a question of how large the sample size n should be so that at least a
proportion P of the population is between A"(r) and x(n-l-_) with probability yor more. One interpretationof this nonparametrics problem demonstrates a situation, for example, where one has a sample of nevents and determines to know how large sample size n can be obtained if one has attained a confidence
level of 90 percent that at least 80 percent of the population lies between x_ 1) and xO,), the smallest and
largest events in one's sample.
26
Therandomsamplevalues,whicharearrangedin orderof increasingmagnitude,arerepresentedasxo), x(2), x(3) ..... x_,) from a distribution having a density functionf(x) and distribution function F(x).
The order statistics X(r) and x(s) consist of the nonparametric tolerance limits. Let x be a continuous ran-
dom variable and one calculates the lower limit L and upper limit U. Accordingly, the probability thatthe value of the random variable falls between L and U is established as
(41)
where f(x) is the unknown continuous probability density function of the random variable x. L and U will
be defined to be 100 P percent tolerance limits at probability _.
Letting L = X(r) (lower tolerance limit) and U = x(s) (upper tolerance limit) take place in equation
(41), the resulting expression becomes
Pr(fX(s) f(x)dx>P}=y'[Jx(r)(42)
Defining
gt
F(t) = J_ f(x)dx ,
it follows that equation (42) is now rewritten as
Pr_.F(xs)-F(xr)] >--P} = Y. (43)
A random sample xl, x2, x3 ..... Xn of size n is obtained from a cumulative distribution function
F(x), and their distribution is defined by
dB = dF(xl) dF(x2) dF(x3) ... dF(xn) (44)
Hence, a transformation is made to perform integration for
= f xr dE(t) = F(xr) . (45)Zr
The distribution of Zr is derived from a beta distribution of the first kind given by
n!dW = (r-1)!(n-r)! zr-a(1-Zr)_-rdzr ' 0 < Zr < 1 . (46)
ThUS, considering next a distribution for the rth order statistic Xr, one puts F(xr) = Zr to obtain the follow-
ing expression:
n! {F(xr)} r-1 (1-F(xr)} nr f(xr)dx r (47)dW - (r-l)!(n-r)[
27
Sincethesamplesof sizen constitute any distribution F(x) with a continuous probability density func-
tion fix), the sampling distribution of F(xr) yields
where
dW = (S(Xr)) r-I (1-e(Xr)) n-r dF(xr)
B(r,n-r+l)
n!B(r,n-r+ 1) - (r- 1) !(n-r) ! '
from the gamma function. If Xr and xs are independent and continuous, then the joint distribution for
F(xr) and F(xs), r < s, is given by
NV-" (F(xr))r-I (F(xs)-F(xr)}S'-r-' (1-F(xs))n-'s dF(xr) dF(xs)B(r,s-r)B(s,n-s+ l )
New variables of the transformation y,z can be introduced by setting
y = F(xs)-F(x r) , z = F(xr) •
Furthermore, the Jacobian is obtained
_)y _y
_z _z
0s _r
1 -1[--" "-- 1 °
0 1
(48)
(49)
(50)
Then, by the change of variable formula,
f ,z y,z)=f,f y,z)IJI, (51)
is obtained. The bars I I denote absolute value.
The process from equation (51) yields the following result
Z r-lys-r-l(1-y--z) n-_'dydz (52)dVy 'z = B(r,s-r) B(s,n-s+l)
When the joint distribution of two random variables is given, one needs to obtain the distribution of one
of the variables alone. Thus, equation (52) is integrated out the variable z over its range (0, l-y), obtain-
ing for the marginal distribution of y
s-r-I fOy.. dy l-y
dVr's = B(r,s-r) B(s,n-s+l) zr-l(1-y-z)_-Sdz . (53)
28
Useof z = (1-y)t reduces equation (53) to
dVr_ $ Ys-r-l(1-Y)n'4+rdy fo 1B(r,s-r) B(s,n-s+ l ) tr-l(1-t)'_dt =
dvrs' = B(s-r,n-s+r+l)
y s-r-1(l-y) n-4+rdy B(r, n-s+ 1)B(r,s-r)B(s,n-s+ l )
, 0_y_l
(54)
Thus, y = F(xs)-F(xr) is distributed as a beta variate of the first kind with parameters depending only on
the difference (s-r).
Using equation (54), equation (53) becomes
Pr{y > P} = Ys-r-l(1-Y)n-_+rdYB(s-r,n-s+r+ 1) = Y "
(55)
Equation (55) is rewritten in terms of the incomplete beta function as
Pr {F(xs)-F(x _) > P} = 1-lp(s-r,n-s+r+l) = 7 • (56)
Equation (56) provides the nonparametric tolerance interval (Xr, Xs) and related parameters such as the
minimum proportion P of F(x), the probability 7, the sample size n and the order statistics' positions rand s.
Equation (56) is equivalent to another equation, namely,
7>r._, (7) (l_p)_p_ , (57)
where the sample size n depends on the solution for both types of one-sided tolerance limits and for the
two-sided tolerance limits. (7) is the binomial coefficient, defined as
(7)- °'i!(n-i) !
The data for nonparametric tolerance limits computed from equation (56) are tabulated in table
12. The numerical data have shown that the sample number of at least 230 items would be required if the
acceptance criteria without any failure are satisfied for 99-percent proportion at 90-percent confidencelevel. Should a single failure occur, then a sample of 388 items would be obtained. The sample of atleast 3,000 items is based on achievement of 23 failures.
29
Table 12. Nonparametrictolerancelimits.
0.50
0.75
0.90
0.95
0.99
Notes:
P
0.50
0.75
0.90
0.95
0.99
0.50
0.75
0.90
0.95
0.99
0.50
0.750.90
0.95
0.99
0.500.75
0.90
0.95
0.99
0.50
0.75
0.90
0.95
0.99
No Failure
r n
1 21 3
1 7
1 14
1 69
1 3
1 5
1 14
1 28
1 138
1 4
1 91 22
1 45
1 230
1 5
1 111 29
1 59
1 299
1 7
1 17
1 44
1 9O
1 459
One Failure
i-
2
22
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
2
2
2
2
4
7
17
34
168
510
27
54
269
7
15
3877
388
8
18
46
94
473
11
24
64
130
662
c_ = confidence level
P = proportionFailure to meet acceptance criteria.
Two Failures
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
33
3
3
33
3
3
n
6
11
27
54
268
7
1539
78
392
9
20
52
105
531
11
24
61
124
628
14
31
81
165
838
Three Failures
F
8
15
37
74
367
10
20
51
102
510
12
25
65
132
667
1429
76
153
773
17
37
97
198
1,001
(r-l) Failures
r n
4
44
4
4
4
4
4
4
4
4
4
4
4
4
4
4
44
4
4
44
4
4
1,500751300
151
31
1,481734
289
143
27
1,463720
280
135
24
1,453711
274
13122
1.433
695263
124
19
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,0003,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
3,000
CONCLUSIONS
The four computer codes have been written to implement the algorithms to calculate the parame-ters for the exact tolerance factor k's for derivation of the one- and two-sided statistical tolerance limits,
using the new theory. One resulting conclusion from this study has shown that the computer codes can
perform faster and more efficiently. The exact data, based on normal distributions, have demonstrated
that the approximation is somewhat satisfactory for even small sample sizes.
30
APPENDIX A
Derivation of Chi Distribution
Chi-square probability function (tail area) is constructed in the form of
1 ft?v fi- _e-½drQ[w(z)]= _ x2 '
_(_)where v denotes degrees of freedom and F(-) is the gamma function
(fo )Let t = vx 2 and dt = 2 vxdx.
Substituting the preceding equations into equation (A1) gives
v_ 1
1 fx_ (VX2)-_Q[w(z)] = ._ v_2
_(_)
v._2
_..12_.e 2vxdr .
After manipulation, equation (A2) becomes
VX 2_(_)_x v-1 e dxQ_w_l-_,_ _
For the reason of symmetry based on addition of integration area, the final expression is given by
4(_)_ _x_Q[w(z)] = x v-I e dr .
Equation (A4) is the chi distribution.
(A1)
(A2)
(A3)
(A4)
31
APPENDIX B1
One-Sided Ke:-
Standardized normal random variable exceeded with probability p
Proportion K t,
0.99999
0.99996832876
0.9999
0.999
0.998650102
0.9975
0.996
0.99
0.97724987
4.2648907939
4.0000000000
3.71901648553.0902323062
3.0000000000
2.8070337684
2.6520698079
2.3263478743
2.0000000000
0.975
0.96
0.95
0.9350.90"
0.85
0.84134477
1.9599640498
1.7506860747
1.64485362781.5141018878
1.2815515655
1.0364333895
1.0000000000
0.80
0.75
0.700.65
0.60
0.55
0.50
0.8416212336
0.6744897502
0.5244005127
0.3853204664
0.2533471031
0.1256613469
0
32
APPENDIX B2
Two-Sided Kp
Standardized normal random variable exceeded with probability p
Proportion Kp
0.99999
0.99993665752
0.9999
0.999
0.997300204
0.99
0.975
0.95449974
4.4171734135
4.0000000000
3.8905918864
3.2905267315
3.0000000000
2.5758293036
2.2414027281
2.0000000000
0.950.90
0.85
0.80
0.75
0.70
0.6826894925
1.9599640498
1.64485362781.4395314711
1.2815515655
1.1503493804
1.0364333895
1.0000000000
0.65
0.60
0.55
0.50
0.450.40
0.35
0.30
0.25
0.200.15
0.10
0.05
0
0.9345892911
0.8416212336
0.7554150264
0.6744897502
0.5977601260
0.5244005127
0.4537621902
0.3853204664
0.3186393640
0.2533471031
0.18911842630.1256613469
0.0627067779
0
33
APPENDIX C1
Simulation Code, One-Sided k
'K1-FACTORDEFDBL A-Z
INPUT"SAMPLE SIZE=";NSINPUT "PROPORTION";PINPUT "CONFIDENCE=";CLSTART=TIMERPI=3.141592654#'INVERSE NORMALQ=I-P:T=SQR(-2*LOG(Q))AO=2.30753:A1=.27061:B1=.99229:B2=.04481N U=A0+AI*T:DE= I+B 1*T+B2*T*TX=T-NU/DE =L0: Z=I/SQR(2*PI)*EXP(-X*X/2):IF X>2 GOTO L3V=25-13*X*XFOR N=I 1 TO 0 STEP-1U=(2*N+I)+(-1)^(N+I)*(N+I)*X*XJVV=U:NEXT NF=.5-Z*XNW=Q-F:GOTO L2L3:V=X+30FOR N=29 TO 1 STEP-1U=X+NNV=U :NEXT NF=Z/V :W=Q-F :GOTO L2L2:L=L+IR=X:X=X-W/Z
E=ABS(R-X)IF E>.00001 GOTO L0'END OF INVERSE NORMAL
'CALCULATION OF FACTORIALN=NS:NU=N-1MT=INT(NU/2):UT=NU-2*MTGT=IFOR I= 1 TO MT- I+UTKT=IIF UT=0 GOTO L1KT=I-.5L1 :GT=GT*KTNEXT IGT=GT* (1+UT*(SQ R(PI)- 1))G F=GT*2^(N U/2-1 )'END OF FACTORIAL
34
'SECANT METHODKP=X:J=I :K=KPK0=K:GOSUB INTEGRATION:SF0=SF
K=K*(1 +.0001 ):KI=K: GOSUB INTEGRATION:SF1 =SFBEGIN:K=K1-SFI*(K1-K0)/(SF1-SF0)IF ABS((K1-K)/K1)<.000001 GOTO RESULTJ=J+I:K0=KI:KI=K:SF0=SF1GOSUB INTEGRATION:SFI=SF:GOTO BEGINRESULT:FINISH=TIMERBEEP:BEEPPRINT K
PRINT "K";"(";J;") =";USING"##.####";KPRINT "TIME=";FINISH-START;"SECONDS"'END OF SECANT METHODWHILE INKEY$--"":WEND
'SIMPSONINTEG RATION:L1=0: L2= 10IF N>40 THEN L2=20
DL=KP*SQR(N):TP=K*SQR(N)Y=NU/2
M=2:E=0:H=(L2-L1)/2X=LI:GOSUB FUNCTIONY0=Y:X=L2:GOSUB FUNCTIONYN=Y:X=LI+H:GOSUB FUNCTION
U=Y:S=(Y0+YN+4*U)*H/3START:M=2*MD=S:H=H/2:E=E+U:U=0FOR I-1 TO M/2
X=L1 +H*(2*I-1):GOSUB FUNCTIONU=U+YNEXT I
S=(Y0+YN+4*U+2*E)*H/3IF ABS((S-D)/D)>.00001# GOTO STARTSF=S/GF-CLRETURN'END OF SIMPSON
FUNCTION: Z=TP*X/SOR(NU)-DLT0=Z:G0= 1/SQR(2*PI)*EXP(-Z*Z/2)A1 =.3193815: A2=-.3565638: A3= 1.781478:A4=- 1.821256:A5= 1.330274IF Z<0 THEN T0=-ZW= 1/(1 +.231649"T0)P1 = ((((A5*W+A4)*W+A3)*W+A2)*W+A 1)*WPH=I-G0*P1IF Z<0 THEN PH=I-PH
Y= PH*X^(N U- 1)* EXP(-X*X/2)RETURN
3S
APPENDIX C2
Simulation Code, Two-Sided k
'K2-FACTORSTART:DEFDBL A-ZPI=3.141592654#
INPUT"SAMPLE SIZE=";NINPUT"PROPORTION=";PINPUT"CONFIDENCE=";CLSTART=TIMER
NU=N-1 :X=(NU+2)/2DEF FN HTN(X)=(EXP(X)-EXP(-X))/(EXP (X)+EXP(-X))CALL GAMMACALL NORMIN
PRINT"KP=";KP'SECANT METHODK=KP:J=IK0=K:GOSUB INTEGRATION:SF0=SFK=K*(l+.0001):K1 =K:GOSU B INTEGRATION:SFI=SFBEG IN:K=K1 -SF 1*(K1 -K0)/(SF 1-S F0)IF ABS((K1-K)/K1)<.000001 GOTO RESULTJ=J+l :K0=K1 :K1 =K:SF0=SF 1GOSUB INTEGRATION:SFI=SF:GOTO BEGINRESULT:FINISH=TIMERBEEP'BEEPPRINT KPRINT" "'" "- -" " " "K, ( ,J, ) = ;USING ##.#### ;KPRINT "TIM E=";FINISH-START;"SECONDS"'END OF SECANT METHODWHILE INKEY$="":WENDGOTO START
'START SIMPSONINTEGRATION:A=0:B=5:M=32H=(B-A)/2/MFE=0:FU=0
X=A:W=KP/K:UC=K/SOR(N):HC=H/K/SQR (N)CALL FUNCTION:FA=FFOR I-1 TO 2"M-1CALL RUNGEX=A+I*HCALL FUNCTIONU=l MOD 2IF U=0 THEN FE=FE+F ELSE FU=FU+F
36
NEXT ICALL RUNGEX=B:CALL FUNCTION:FB=FS=H*(FA+2*FE+4*FU+FB)/3SF=2*S-CLRETURN
SUB RUNGE STATICSHARED HC,UC,W,H,X'RUNGE-KUTTA/3
XI=X:WI=W:U=X*W*UC:KI=HC*FNHTN(U)X=XI+H/2:W=WI+K1/2:U=X*W*UC:K2=HC*FNHTN(U)X=XI+H:W=W1-KI+2*K2:U=X*W*UC:K3=HC*FNHTN(U)W=W1+(K1+4*K2+K3)/6END SUB
SUB FUNCTION STATICSHARED PI,NU, GNU,F,W,X'CHI-SQUAREWL=NU*W*W
C=(WL/2)^(NU/2)*EXP(-WI_/2)/GNUQ=I :J=0:S=0LINE2: J=J+I:D=NU+2*JR=WL/D:Q=Q*R:S=S+QIF Q>.000001 GOTO LINE2
P=(I+S)*C'END CHI-SQUARE
F=(1-P)*EXP(-X*X/2)/SQR(2*PI)END SUB
SUB GAMMA STATICSHARED X,PI,GNUZ=X+4:IF X>4 THEN Z=X
G=SQR (2*PI/Z)*EXP(Z*LOG(Z)-Z+ 1/12/Z-1/360/Z^3+ 1/1260/Z^5 - 1/1680/'Z^7+ 1 /1188/Z^9)IF X>4 GOTO L1G=G/X/(X+I)/(X+2)/(X+3)LI:GNU=GEND SUB
SUB NORMIN STATIC
SHARED KP,P,PIQ=(1-P)/2:T=SQR(-2*LOG(Q))A0=2.30753:A1 =.27061 :B1 =.99229:B2=.04481NU=A0+A 1*T: D E= 1+ B 1*T+B2*T*TX=T-NU/DE
3?
LO: Z=I/SQR(2*PI)*EXP(-X*X/2):IF X>2 GOTO L3V=25-13*X*XFOR N=I 1 TO 0 STEP-1U=(2*N+I)+(-1)A(N+I)*(N+I)*X*X/VV=U:NEXT NF=.5-Z*X/VW=Q-F:GOTO L2L3:V=X+30FOR N=29 TO 1 STEP-1U=X+NNV=U :NEXT NF=ZN :W=Q-F :GOTO L2L2:L=L+IR=X:X=X-W/ZE=ABS(R-X)IF E>.000001 GOTO LOKP=XEND SUB
38
REFERENCES
It
.
.
,
.
.
Abramowitz, M., and Stegun, I.A.: "Handbook of Mathematical Functions." Dover Publications,Inc., New York, NY, 1965.
Conover, W.J.: "Practical Nonparametric Statistics." Second Edition, John Wiley and Sons, NewYork, NY, 1980.
Guttman, I.: "Statistical Tolerance Regions: Classical and Bayesian." Hafner Publishing Company,Darien, CN, 1970.
Kendall, M., and Stuart, A.: "The Advanced Theory of Statistics." Vols. I and II, Macmillan
Publishing Company, Inc., New York, NY, 1977.
Owen, D.B.: "A Special Case of a Bivariate Noncentral t-Distribution." Biometrika, vol. 52, Nos. 3
and 4, 1965, pp. 437-446.
Webb, S.R., and Friedman, H.A.: "Exact Two-Sided Tolerance Limits Under Normal Theory."
Rocketdyne, a Division of North America Rockwell Corporation, report NAR 54416, 1968.
39
APPROVAL
STATISTICAL COMPUTATION OF TOLERANCE LIMITS
By J.T. Wheeler
The information in this report has been reviewed for technical content. Review of any informa-
tion concerning Department of Defense or nuclear energy activities or programs has been made by the
MSFC Security Classification Officer. This report, in its entirety, has been determined to be unclassified.
J.C.
Director, Structures and Dynamics Laboratory
"_" U.S, GOVERNMENT PRINTING OFFICE 1993--533-108/80127
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1. AGENCY USE ONLY (Leave blank) | 2, REPORT DATE 3. REPORT TYPE AND DATES COVERED
I October 1993 Technical Memorandum
4. TITLE AND SUBTITLE S. FUNDING NUMBERS
Statistical Computation of Tolerance Limits
6. AUTHOR(S)
J.T. Wheeler
7. PERFORMINGORGANIZATIONNAME(S)AND ADDRESS(ES)
George C. Marshall Space Flight Center
Marshall Space Flight Center, Alabama 35812
9. SPONSORING/MONITORINGAGENCYNAME(S)AND ADDRESS(ES)
National Aeronautics and Space Administration
Washington, DC 20546
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AGENCY REPORT NUMBER
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Prepared by Structures and Dynamics Laboratory, Science and Engineenng Directorate
12a.DISTRIBUTION/AVAILABILITYSTATEMENT
Unclas sifted--Unlimited
12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200words)
Based on a new theory, two computer codes have been developed specifically to calculate theexact statistical tolerance limits for normal distributions within unknown means and variances for the
one-sided and two-sided cases for the tolerance factor, k. The quantity k is defined equivalently in terms
of the noncentral t-distribution by the probability equation. Two of the four mathematical methods
employ the theory developed for the numerical simulation. Several algorithms for numerically integrat-
ing and iteratively root-solving the working equations are written to augment the program simulation.
The program codes generate some tables of k's associated with the varying values of the proportion and
sample size for each given probability to show accuracy obtained for small sample sizes.
14. SUBJECTTERMSstatistics, probability, life-testing, process control, reliability,
nonparametrics, stochastic process, tolerance limits, numerical statistics,
quality control, tolerance factors17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION
OF REPORT OF THIS PAGE OF ABSTRACT
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