statistical methods appropriate for evaluation of fuze explosive
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Controlling Office; Naval Ordnance Lab.,White Oak, MD.
AUTHORITYOCA; Oct 31, 1983; Cmdr, Naval OrdnanceLab., Oct 10, 1985
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• i,., .g( LA S IF R E D :"-
DEFENSE !)OCUMENTATION CENTERFOR
S31ENTIFIC AND TECHNICAL INFORMATION
CAMERON STATION ALEXANDRIA. VIRGINIA
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NAVORD REPORT 2m
i -.(,Al- METHOL ; AF R1OPRIATE FOR EVALUATION OF FUZE
L XP1LOSVL"1£ FRA 'SAFETY AND RELIABILITY
.9 13 OCTOBER 1953
11
U. •, NAVAL ORDNANCE LABORATORYWHITE OAK, MARYLAND
• U.S. NAVAL OIRD.. -kNCE LABORATORY
WF,1 OAKSILVER SPRING. MARYLAND
To all holders of NAVORD Report 2101 Change 1insert change; write on cover 'Change I inser ed' 20 Sep 1954Approved by Commander, U.S. NOL - T-) 2 pages
By di ectic..
NAVORD Report 2101, "Statistical Methods Appropriate for Evaluationof FuzedExlofoive-Train Safety and Reliability",
is change a o 0ow8:
Item 1. Page i, linc 21, Replace "references (a) and (b)"-.t .•%- If11 , . .... . .Wit- "references (b) and (c)".
Item 2. Insert Page vii attached.
Insert this change sheet between the cover and the title page of your copy.
NAVORD Report 2101
REFERENCES
a. NAVORD Report No. 29-44, "Notes on Statistical MethodsApplied to Torpedo Test Results".
b. NOL S-5892, :'A Statistical Analysis for a New Procedurein Sensitivity Experiments". 0SRD AMP Report No. 101.1RSRG-P No. 40.
c. "Probit Analysis-- A Statistical Treatment of the SigmoidResponse Curve", by D. J. Finney, Cambridge UniversityPress, 1947 (2nd Ed. 1952).
d. NOLM 9910, "Comparison of the Probit Method and theBruceton Up-and--Down Method as Applied to SensitivityData."
e. Deleted.
f. NAMTC Technical Report No. 75, "A Study of Methods forAchieving Reliability of Guided Missiles". 10 July 1950.
g. NAMTC Technical Report No. 84, "General Specification forthe Safety Margins Required for Guided Missile Components."10 July 1951.
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NAVORD Report 2101
STATISTICAL MT-7, 2 '?PROPRTATE FOR EVALUATION OF FUZEEXPLOSIVE-`AIN SAiL'. •iKD RELIABILITY
Prepared by:
H. P. Culling
ABSTRACT: The statistical problems of evaluating fuze explo-sive trains for safety and reliabi1lty are presented. Th•Ie
Bruceton and Probit procedures of testing are illustrated byexamples of actual tests and the interpretation of the resultsis discussed.
Appendix A presents a step-by-step description of thecalculations necessary to determine the "50% reliable distance",the standard deviation, errors of estimate, and "degree ofreliability" by the Bruceton Method of Analysis. Tables andgraphs necessary for the calculations are appended.
Appendix B presents the Probit Method calculations neces-sary to determine the Probit Line Equation from which estimatesof the "50% unsafe distance", standard deviation, errors ofestimate, and "degree of safety" may be obtained. A test for"goodness of fit" of the Probit Line is presented. Tablesnecessary for the analysis are appended.
Appendices A and B thus provide a working handbook forthe two methods of analysis, which are both apropriate foreither safety or reliability tests.
The methods presented and tables appended have beenextracted from references (a) and (b) and adapted to theproblems of estimating fuze safety and reliability. Physicalconsiderations governing the applicability of the test methodsdescribed to various configurations of explosive trains, and
U. S. NAVAL ORDNANCE LABORATORY
White Oak, Maryland
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NAVORD Report 2101
the use of the results in the evaluation of ordnance itemsas a whole, ara nhe bubject of a separate report and arenot considered herein.
Many similar problems in ordnance evaluation and testingin which the data must be of a "go-no go" nature may pro-fitably be investigated by the methods presented in this report.For example, the minimum and average firing energy charaz-teristics of electric primers are often so determined.
Since the test points are distributed over the cpnter ofthe distribution rather than being concentrated at the ndpoints (where all or none of a small sample may be expected
n -t4^in and which usually correspond to nominal part-
dimensions, firing voltages or whatever), these methods arerelatively efficient in affording a good basis for estimatingthe effects of later design or manufacturing changes or
proposals. The value of the data from tests conducted inaccordance with these plans is therefore less likely to belost in case of such changes.
The applicability of these methods is dependent upon anunderstanding, based upon experience with large numbers ofprevious tests or demonstration of the validity on a theoret-••-c basis, of the nature of the physical processes involved.:n part .1ular, since these methods are aimed at giving anoverall estimate of the population from a very limited numberof samples, the analysis cannot, in itself, check the degreeto which the assumptions of normal distribution and linearity(or other assumed relationship) between the two variables areJustified. The greatest value of the test methods is in -
quickly confirming the suitability of a design or pointingout a marginal condition requiring correction or furtherinvestigation by the use of more samples.
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NAVORD Report 2101 13 October 1953
This report has been prepared under Task NOL-B2b-41-1, forthe following purposes:
1. To illustrate the statistical philosophy of evaluatingfuze-explosive trains for safety and reliability,
2. To present the application of two methods of analysiswhich deal with "go-no go" results as a function of a con-trolled variable, and
3. To provide a wo. Ing handbook for using these methods inthe design, analysis and presentation of explosive-trainsafety and reliability evaluations.
These procedures have been developed by the Trains and Mecha-nisms Branch, Mechanical Evaluation Division, in the courseof the evaluation of ordnance items under various Bureau ofOrdnance tasks. Tables appearing in this report have beentaken or adapted froi., the references cited.
The Naval Ordnance Laboratory is indebted to Professor SirRonald A. Fisher, Cambridge, to Dr. Frank Yates, Rothamsted,and to Messrs. Oliver and Boyd Ltd., Edinburgh, for per-mission to reprint Table No. IX from their book "StatisticalTables for Biological, Agricultural, and Medical Research".
JOHN T. HAYWARDCaptain, USNCommander
R. +HG OWERBy direction
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C ONTENTSPage
Introduction. . . .......... . . . . . . . . . .. . .. . 1Inadequacy of Confidence Limit Estimates. . . . . .. . 1Method of Testing Between Armed and Unarmed Positions . 2Methods of Analysis and Calculated Parameters . . . ... 2Specific Applications for Each Method . . . . . . . . . 3Corparison • Reliability or Safety Probabilities . . . 4A Measure •X the Degree of Reliability or Safety. . . . 4Lower Limit for the Degree of Reliability or Safety . . 6Degree of Rel.iability Versus Degree of Safety . . . . . 9Conclusions .. . . ... . . ......... . . . 10
Appendix A
The 3ruceton Method of Sensitivity Testing
Introduction. . . . ........ . ................... . . . 1Test Procedure .......... . . . .. 1Calculation of the 50%-Reliable Distance: .... . 2Calculation of the Standard Deviation . . . .... ... 3Calculation of Percentage-Reliable Curve . . . . . . .. 4Sampling Error, (ýa), for the 50%-Reliable Estimate . . 5Sampling Error, (ka), of the Standard Deviation . . . . 5Confidence Intervals Determined from aR and a .5..
Conclusions Concerning the Degree of Reliability ... 6The Previous Methods Applied to Safety Tests. . . .... 7
Appendix B
The Probit Method of Sensitivity Testing
Introduction ...... ................. . . . . . 1Test Procedure for the Probit Method: .I. 1Description of a "Probit" . . . . ...... 2Analysis of Test Data by Provisional Probit Line . . . 3The Method of Calculating the Probit Line . . . . . . . 5
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CONTENTS (CONTINUED)
Page
Calculation of the 50%-Unsafe Distance, (YU), and
Standard Deviation, (U). .... 8
The Degree of Safety, (Ds), and Comparison'Betweenthe Provisional Estkmates and Calculated Estimates.. 8
Sampling Errors of the 50%-Unsafe Distance and Slope,b, of the Calculated Probit Line . . . . . . . . .. 9
A Test for "Goodness of Fit" ..
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NAVORD Report 2101
ILLUSTRATIONS
Tables 1 - 7 Tabulation of Safety and Reliability TestResults
Graphs I - 7 Graph of Estimated Percentage-Reliable andPercentageoUnkafe Curves
APPENDIX A
Table I Table of s for Obtaining the Sample StandardDeviation
Table 2 Values of t for N Degrees of Freedom(N values of I to 120)
Graph 1 Curve of M versus s for Estimating theStandard Deviation (for M > 0.40)
Graph 2 Curve of M versus s for Estimating theStandard Deviation (for M < 0.40)
Graph 3 Curve of G versus s for Estimating theStandard Error of the Mean (for s > 0.5)
Graph 4 Curve of Hi versus a for Estimating theStandard Error of a (s > 0.5)
Graph 5 Curve for Estimating the Standard Errorof the Mean and a (s < 0.5)
APPENDIX B
Table 1 Transformation of Percentagea to ProbitsTable 2 Tne&Weightlng Coefficient (w)Table 3 Working Probits Given for Provisional
Probit, Y, and Empirical Percent, pGraph 1 Graph of Prob!ts of Percentage Unsafe
versus Distance
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STATISTICAL METHODS APPROPRIATE FOR EVALUATION OF FUZEEXPLOSIVE-TRAIN SAFETY AND RELIABILITY
INTRODUCTION
I. Present-day emphasis on highly reliable and safe weaponscalls for increasingly accurate methods of estimating the pro-babilities of reliable and safe functioning of components.The usefulness of statistical procedures in determiningreliability is discussed in references (f) and (g). Thegeneral procedure, however, is based on a large sample ofquantitative data which is obtained for a given test con-dition. This report illustrates the statistical methodswhich are applicable to "go-no go" data (such as the firingor non-firing of an explosive element) obtained at variouslevels of a test condition.
2. Organizations engaged in the evaluation of fuzes todetermine their suitability for release to production havethe problem of determining the reliability and safety of fuzeexplosive trains. This report, which is limited to thestatistical aspects of the problem, has the followingobjectives:
a. To present the statistical philosophy of evaluatingfuze explosive trains for safety and reliability,
b. To illustrate the application of two methods ofanalysis (the Bruceton and Probit) which deal with "go-no go"results as a function of a controlled variable, ani
c. To provide a working handbook for using these methodsin the design, analysis and presentation of explosive-trainsafety and reliability evaluations.
Inadequacy of Confidence Limit Estimates
3. In order to meet the explosive-safety requirement, anexplosive train usually contains an out-of-line safety devicewhich rotates or slides from the unarmed to the armed position.The safety or reliability of the train may be estimated roughlyby testing in the unarmed and armed positions, respectively.
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But if the sample size is limited to, say, 20 for each test,the statistical interpretation of the results cannot establishthe high assurance of' 9afety or reliability necessary forrelease. For example, given a sample of 20, all of whichfire reliably in the armed position, one may expect fromfuture samples a reliability as low as 83% (or 17% failures).If the sample is increased to 100, one may still expect asmany as 4% failures. These estimates of percentage failuresare based on the lower 95% confidence limit (reference (a)).Since an explosive train is probably more reliable thanindicated by such± a test, a more accurate estimate is necessary.
Method of '. ang Be~weei Ar?--- and Unarmed Positions
4. If the safety device were placed half way between thearmed and unarmed positions and the explosive train stillfunctioned reliably, one would expect the reliability of theV&raL 4- armed position to be considerably better than shownin the previous example. Then, if more samples were testedat various distances of partial arming and in such a mannerthat an array of high to low percentage-functioning distanceswas obtained, a graph could be plotted of percentage reliableversus distance from armed position. By methods of statis-tical extrapolation, a percent-reliable curve could be calcu-lated from which an estimate of the 99%-reliable distancecould be read. If the 99%-reliable distance occurs when thesafety device is still considerably out of line, it isprobable that the explosive train will function consistentlyin the armed position.
5. The safety test could be run in a similar fashion, wherean array of high to low percent-unsafe conditions versusdistance was obtained. The calculated percent-unsafe curvewould give the estimated 1%-unsafe distance.
Methods of Analysis and Calculated Parameters
6. Since the estimates of the 99%-reliable distance and1%-unsafe distance are the bases of conclusions concerningreliability and safety of the train, the method of extra-polation for obtaining these distances should be as accurateas possible. The method should also be applicable for small
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NAVORD Report 2101
samples. These two restrictions prohibit the use of severalestablished methods which employ relatively simple test pro-cedures. Two established methods which meet these require-ments are the Bruceton Method of Sensitivity Testing (reference(b)) and Probit Analysis (reference (c)). Both methods maybe employed to estimate the 50%-reliable distance, XR, or the50%-unsafe distance, XU, and the respective standard deviations,oRW and a, Each Xminus Its appropriate a gives an estimateo the 8 distance. If a is added to X, an estimate of the15% distance is obtained. By using the appropriate constantsobtained from a table of normal eurve areas, one may estimateany percentage distance. Therefore, the estimates of YR and1U and OR and aU are the only measurements necessary todetermine a measure of reliability or safety. In appendicesA and B, detailed descriptions of the calculation of thesemeasurements for the Bruceton Method and the Probit Methodare presented.
Sp~ecif 1 Applications for Each Method
7. Although both of the methods provide a means of estimatingthe same measures (XR, XU, OR, OU), there are specific reasonsfor the application of one instead of the other, depending onthe purpose of the test.
a. The Bruceton Method of testing centers the testing inthe region of the 50% distanct, while the Probit Method oftesting includes the high and low percentage points. Thus,the Bruceton Method is more accurate for estimates of 7, andYU, while the Probit Method is more accurate for estima~es ofoR and aU.
b. The Bruceton Method requires fewer samples to deter-ri-ne XR or XU and a rough estimate of a and au, so it may beused for smahl samples (not less than 20). It should be heldin mind that the Bruceton estimates of OR and au tend to beslightly less than the Probit estimates; and, in connectionwith safety or reliability tests, such estimates are overlyoptimistic.
c. The Bruceton Analysis may not be used except when theBruceton scheme has been followed, using equally spaced testdistances and the up-and-doan procedure of testing (described
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in appendix A). On the other hand, the Probit Analysis isadaptable to data of -wnequal sample sizes at each test distanceand the test distances need not be equally spaced.
d. The Bruceton Method of Analysis is based on theassumption that the percentage functioning is related to týtest distances by means of the normal error function, whilethe Probit method may be analyzed by other appropriate trans-formations, if necessary.
Thus, the Probit Method may be considered more adaptable tomost test data. except when the sample sizes are too small,or when XR or AU are the most important measures. In some
cases the Bruce4Eon Method of Analysis is preferred becausethe computations involved are relatively simple, but thisfact alone should not determine the choice of method. Seereference (d) for a more detailed discussion of the twomethods.
Comparison of Reliability or Safety Probabilities
8. The purpose of an explosive-train reliability or safetytest is to determine Just how reliable or safe the train isin the armed or unarmed positions, respectively. It ispossible by means of the proposed methods to determine theprobability of each; but, in most cases, this probability isin the range of 99% to 99.099999%. Obviously, small samplesdo not warrant so many significant figures, yet the trainsrepresented by many nines after the decimal point are cer-tainly more reliable than those with only one or two. Anotherdisadvantage of the probability figure is that it does notindicate the amount of variation due to sample differences inthe explosive components and mechanical parts (or inconsistencyin manufacture of the design). In order to determine adifference between two high probabilities, a measure whichgives the ratio of X to a should be used.
A Measure of the Degree of Reliability or Safety
9. Two measures termed the "degree of reliability", DR andthe "degree of safety", DS, which are not subject to thediscrepancies of the probability values, may be calculated.
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DR is defined as XR in terms of CR, or
DR = __R
aR
where Y. is measured from the armed position for reliability.DS is obtained in a similar manner by
D xU
where YU is measured from unarmed position. The values forX and a are obtained from thv Bruceton or Probit estimates ofthe reliability curve and safety curve, respectively. A com-parison of two reliabi-lity functions which have identical XR'sbut different values of ca may best illustrate the usefulnessof this measure.
too-
m 75--J
I 5-
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CL 0
a 2-0*- I I
0 I 2 3 4 5 6 7 8DISTANCE FROM ARMED POSITION (32 IVDS OF AN INCH)
The above graph shows XR for I and II is at 2.5/32 of an inch.
The reliability curve I has a aR equal to 0.5/32 inches, andcurve II has a oa equal to 1/32 inches. The reliability pro-bability (P) in •he armed position for curve I equals 99.99997%while curve II has a P = 99,-4%. In a practical sense, thesetwo values of P do not indicate any significant difference whenestimated from small samples. But by means of the degreemeasure, we find that curve I has a DR = 5.0 while curve Igives a DR = 2.5. By comparing the two DR values, one canconclude that DR for the train of curve I is twice the valueof DR for the train of curve II, while the probability figures
might lead to the conclusion that the two trains are almostsimilar.
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10. A more difficult comparison between probability valuesoccurs when t'wo cur'Tes have slightly different values of XRand UR, but •Le probability values occur in the 99 to 100;
region. Here again the DR of each will indicate which is themore reliable.
Lower Limit for the Degree of Reliability or Safety
11. Consideration must be given to the degree of reliabilityor safety that is necessary to insure consistent operation ofthe explosive train. A lower limit for DR and DS has beendesignated as 5.0 to insure better than 99.5% operation. Thislower limit was calculated to allow for the errors of estimatesfrom small samples of not less than 20. If the exact 50% cis-tance, YE, and the exact standard deviation, aE, were known,99.5% operation would be assured with a degree equal to 2.6.(In other words, 99-5% reliability is insured if X1/oE = 2.6where 2.6 is the normal-deviate constant corresponaing to99.5ý.) But the small sample estimates of X and a are subjectto sampling errors, so these must be allowed for in additionto the degree of 2.6.
12. The error in X (sample estimate) may be calculated by:
aR = a/j-n, where n is the number of results thatresponded, but not the total sample size, and a"is the standard deviation estimated from thesample.
if we consider samples of 20 or more with at least 9 favorableresults, then aX will be less than or equal to al;9or 0/3.The 95%6lower limit for X can be found by 1 - l.8 6 ox, or
_ 1.8 6 a which equals X - 0.6a. Therefore, to insure that3 _
the exact XE has been included in the measure of D, an allowanceof 0.6a must be subtracted from the sample X.
This 95% lower limit for X may be explained by thefollowing short discussion. Confidence limits maybe Gomputed for a mean by the expressions,
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NAVORD Report 2101
Xttag or ±t 1 (since al ),
where the value for t may be found in table 2,appendix A using n-- 1 degrees of freedom andthat probability or "risk" of excluding I whichwe are willing to allow. Thus, if n = 9 (as inthe example above) and we wished to be 90% sureof includiag the real X (i.e., we desired toknow the 90% confidence limits), we should enterthe t table with 9 - 1 or 8 degrees of freedomand a probability level of .100% - 90% or 10%.This would give a t value of 1.86, and so the90% confidence limits for n , 9 are
Y ± 1.86a or Y± 0.6a.
The other~way of looking at these 90% confi-dence limits is that the probability is 10%that the real X has been excluded. This 10%probability is the sum of the 5% probabilitythat the true Z lies above the greater 90%confidence limit, X + 0.6a, and the 5% pro-bability that the true X lies below thesmaller 90% confidence limit, X - 0.6a. Inother words, the probability is 95% that thetrue Y lies above the smaller 90% confidencelimit, 7 - 0.6a, which is the value givenabove for the 95% lower limit for Y.
13. The sampling error of a may be determined by obtainingan upper limit of a which will include the true aE with 95%confidence. A relation between aE and the sample a is givenby the F-ratio, defined as
F 2/a2.F=E
Tables are available in most statistical texts which givevalues of F for specific confidence levels and for varioussample sizes used for estimating a. The 95% upper limit ofa is calculated using the F-value given at the 5% levelunder N co (representing the size of the population) andN2 = 8 (representing n - I which was used to obtain a). Thisvalue of F is 2.9 which gives the 95% upper limit of a as:
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NAVORD Report 2101
2._9 or 1.7a a=
Therefore, the error in a is allowed for by estimatilng Eas 1.70.
14. It was pre--to,,ly sta-!d that 99.5% reliability wasinsured if
E/E 2.6.
By substituting the errors for the sample estimates in theabove relation, it becomes,
S- 0.6a2J1.-oa - 2.6 or Y/a = 5.0.
Thus D, obtained from a sample of 20 or more, must be greaterthan 5.0, if o operation is desired. (It souldA be notedthat the samples tested must be representative of the produc-
tion lots for these figures to be meaningful.)
Looking back on this problem, it can be statedthat if DR (which is the ratio of X to a) is5.0 or greater, then even if the sample esti-mates of a and X were both so overly optimisticthat they might be exceeded only 5% of the timefrom a sample as small as 9 the ratio 7 •/OE (thetrue X to the true a) should be'equal to orgreater than 2.6. It was stated above that avalue of 2.6 would insure at least 99.5%operation (or safety). The requirement impliedhere is that the ratio of the pessimistic 95%limits for a and X be greater than 2.6. Thevalue of 5.0 was merely found to be the necessarylower limit to the ratio of the estimates of 7to a from a sample as small as 9 to be certainthat the 2.6 requirement for the ratio of thepessimistic limits for I and a was met. It canbe seen that while a D of 5.0 always implies99.5% operation, the fRilure of a set of datato yield a DR of 5.0 does not mean that the 2.6value for the ratio of the peLsimistic 95%limits for Y and a (and hence 99.5% operation)
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NAVORD Report 2101
has not been met, since the sample size may!±-ye been larger than 9 and this would resultin the pessimistic 95% limits for 7 and abeing closer to the sample estimates of Iand a. The ratio of these pessimistic 95%limits for X and a should be computed when-ever DR or Ds is found to be less than 5.0,since i.f this ratio is equal to or greaterthan 2.6, the statement that the item is 99.5%safe or operable may still be made.
Degree of Reliability Versus Degree of Safety
15. Since an explosive train must be both reliable and safe,these two opposing requirements must be balanced. In otherwords, DR must not exceed D to such an extent that the safetyIs endangered, and conversely. In a theoretically ideal traii,the di.fference between DR and D5 would be minimized. In
practice, the difference may be large so long as both valuesexceed the lower limit of 5.0. Tables 1 through 7 present
actual test results determined by the Bruceton and ProbitMethods. Graphs 1 through 7 are the respective graphs ofpercent-reliable and percent-unsafe conditions estimated fromthe data given in the tables. The calculated DR and Ds foreach of the samples are as follows:
D E DS oDegree of Degree of
Reliability Safety
Table i, Graph 1 I0.4 i4.7"2, " 2 6.8 2.0*
" 3, " 3 3.-* 12.3" 4 ; " 4 12.2 12.0" " 5 8.5 23.1" 1 " 6 6.6 31.8" 7, " 7 3.6*
The tabulation indicates that the device of Graph 2 wasunsafe and those of Graphs 3 and 7 were unreliable; Theother devices may be compared by their values of DR and DSwhich show that the devices for which Graphs I and 4 wereplotted contain the best combination of safety and reliability.
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NAVORD Re ort 2101
CONCLUSIONS
16- Fuze explosive-train reliability or safety estimatesobtained by means of confidence limits cannot establish thehigh assurance necessary for release. Therefore, tests con-ducted with the explosive-safety device in the armed orunarmed position are, for the most part, inadequate for deter-mining statistical estimates of reliability or safety,respectively.
17. The Bruceton or Probit methods of testing at partial-arming positions provide results from which a statisticalestimate of a high assurance of reliability or safety can beestablished. If the degree of reliability (the ratio of the50%-reliable distance to the standard deviation) is greaterthan 5.0, the explosive-train reliability will be at least99.5%. Similarly, if the degree of safety (the ratio of the50%-unsafe distance to the standard deviation) is greaterthan 5.0, the explosive-train safety will be at least 99.5%.
18. The reliability of an explosive train may exceed thesafety to such an extent that the safety requirement isendangered, or conversely. A comparison of the degree ofsafety, DS, with the degree of reliability, DR, will indicatewhether the train is over-designed in favor of either thesafety or reliability requirement.
10
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APPENDIX A
THE BRUCETON METHOD OF SENSITIVITY TESTING
INTRODUCTION
1. In order to il.lustrate the basic statistical applicationof the Bruceton Method, an example of an out-or-line safetydevice arrangement, consisting or a primer, a detonator housedin a slider, and a lead., will be used. Given a sample sizeof 30 units with a distance or one inch between the "armed"and ".safe" positionsj, the following procedure may be used todetermine reliability estimates.
Test Procedure
2. Choose a distance, k., measured from "armed" position, atwhich the slider will be placed for the first sample. Thisdistance should be the position at which roughly 1/2 of thesamples will fire.
3. Ir the first unit fires when tested at a distance k,. thesecond one should be tested at a distance kc + d. where d is apreassigned increment between teat distances. If the firstunit does not fire, the second one should be tested at a dis-tance kc - d. Each succeeding unit will be tested at adistance dependent on the firing behavior of the previous.unit: if the previous unit fired, the next test distancewill be d greater than the previous distance; or, if theprevious unit failed to fire, the next test distance will bed less than the previous distance. Thus, the test resultswill be a sequence of fires (X) and failures (0) centeredabout the 50%-firing distance, XR-
4. For the example described, a distance of 10/32 of an inchwas chosen for the firbt unit and Increments of 1/32 of aninch for d. The test results formed the following sequence:
NAVORD Report 2101
Distance FromArmed Position(in 32nd's of an inch)
k- 2d 8 8
k- d 9 ' X0 XX X9X X
k •1 0 Xa 0 0 0 0 0 X 0 0 X 0-G---
k+ d 1l 0 0 X0
k + 2d =12 0
Calculation of the 50%-Reliable Distance
5. The data are tabulated by recording the number of X's andthe number of O's for each test distance. The test distancesare assigned levels, i (of equal unit increments), with thelevel at which no failures were obtained set equal to zero.(Caution: If no failures were obtained at more than one level,the i = o level is that distance which is only d less than thelevel at which failures were obtained. Data from test distancesless than the i = o level should be disregarded.) The calcu-lations of X' and aR involve only the zeros or only the X's;i.e., the set which contains the fewer number of results, n,is used. The tabulation for the example cited is as follows:
Distance FromArmed Position nI(32ndts of an inch) i 0 1 n, i n
8 o 0 0 0
9 1 10 1 10 10
10 2 3 10 6 12
11 3 1 3 3 9
12 4 0 1 0 0n-15 15 A=19 B=31
2
_____________________
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of = : -,r= - + L S
-r +
.1..6 - 73 -:T d- . I s ofa- n:t
calculation of' the Standard Deviation Lalrl
7. The standard deviation is devermined by the following
steps.
a. Find a value M - (nB - n4.
b. Find the value of s corresponding to the calculated
value M. Table 1 or Graph 1 give s for values of M • .40.or graph 2 gives s for values of M < .40.
c. Determine OR = sd.
For the ... le the calculations are as follows:
M (15 x 31 - (19)2)/(15)2
= (465 - 361)/225 - o.46, then
from table 1,
s .79, and
aR = .79 (1) - 0.8 (in 32nd's of an inch).
3
NAVORD Report 2101
Calculation of Percentage-Reliable Curve
8. Distances at which a given percentage of the devices willfunction may be estimated by adding to or subtracting from
R certain multiples of aR. Percentage-reliable distances maybc est;ýate 2i'0on "R + tOR by choosing the proper value of tsho;:n below for specelfied percent points. The percentagedistances less than 50 are obtained by using the plus signin the above expression, and those greater than 50 (given inthe parentheses below) by using the minus sign.
Percent, p t
50 Q50') 025 75) 0.67510 90o 1.282
5 95) 1.6451 99) 2.3260.1 99.9) 3.0900.01 99.99) 3.719
The percentage-reliable curve for the particular example citedis:
Percentage-Distance (32nd'sof an inch)
Percent, p tOR 3[R ± tOR
99.99 -2.98 6.899.9 -2.47 7.399 -1.86 7.995 -1.32 8.590 -1.03 8.875 -0.54 9.350 0 9.8
25 +0.54 10.310 +!1.03 i0.85 +i1.32 ii1.11 +I1.86 ii1.70.1 +2.47 12o0.01 +2,•98 12:
4
I
NAVORD Report 2101
Sampling Error, ao, for the 50%-Reliable Estimate
9. The estimate of XR is subject to sampling error which maybe 'a!,u'ied from OR by the formula,
OR = 0RG/(n) i/2,
generally referred to as the standard error of 7R, where n isthe number of O's (or X's) used in the calculation of 7 andOR, and G is read from Graph 3 at the value s determined incomputing oR. For the example the error of TR is,
(0o.8)(1.04)/(15)1/2
- 0.22
Sampling Error (ca) of the Standard Deviation
10. The estimate of OR is also subject to sampling error,which is measured by its standard error, ca, given by therelation
0 0 = ORH/(n)1/2
where H is read from Graph 4 using the value of s determinedin computing OR. This measure calculated for the example is
ca = 0.8(1.31)/(15)1/2
Confidence Intervals Determined From oa and a.
11. The standard errors, ax and a., provide measures of pre-cision of the XR estimate and the OR estimate. Since X and
OR are only estimates of their true values, confidence limitsfor these estimates which include the true value will indicatethe precision of the estimates. If y denotes an estimate anday its standard error, confidence limits may be obtained by
y ± tay,
NAVORD Report 2101
where t is a value given in table 2 using N = (n - 1) andP = 1 minus the level of confidence. For the illustratedexample, the 90% confidence limits of YR and aR may be cal.culated as follows:
N = 15 - I = 14,
P 1 1 - .90 .10,
which gives a t = 1.76;
therefore, the limits for XR are
YR ± 1.76c7- = (9.8) ± 1.76(0.22)
= 9.4 to 10.2 (in 32nd's of an inch)
and the limits for aR are
OR ± 1.76c% = (0.8) ± 1.76 (0.27)
= 0.3 to 1.3 (in 32nd's of an inch).
The probability is 10% that a value lies outside of its90% confidence limits. Since this 10% probability is thesum of the 5% probability that a value lies above itsgreater 90% confidence limit and the 5% probability thata value lies below its smaller 90% confidence limit, itcan be seen that the probability is 95% that a value willlie above its smaller 90% confidence limit. In otherwords, a smaller 90% confidence limit is a 95% lowerlimit since the probability is 95% that the true valuelies above it. Similarly, the 'Larger .0 Conf.de.celimit is a 95% upper limit.
Conclusions Concerning the Degree of Reliability
12. When e ..- l " samples are used to estimate thereliability -ýf ,e'•. rve train: the above confidencelimits may be useu to determine the "most pessimistic" DRto be expected. By using the lower limit for 3R and theupper limit of a to calculate DR, the "most pessimistic"value will be obaLined. The example used gave a lower limitfor YR equal to 9.4 with 1.3 as the upper limit for OR. The"most pessimistic" degree of reliability-s, then,
L._ -
NAVORD Report 2101
9.4/1.3 - 7.2,
as compared to its average degree of reliability,
9.8/0.8 = 12.2.
If the "most pessimistic" D is greater than the normaldeviate of 2.58 which corresponds to 99.5%, then the 99.5%reliability is insured.
The Previous Methods Applied to Safety Tests13. The Bruceton Method, as outlined may be readily adapted
to safety tests, by recording "unsafe' and "safe" resultsrather than fires and misfires. If the distances are measuredfrom "armed" position for the safety tests, the method of cal-culation will be identical to the preceding example. But toobtain D X must be converted into units from "safe" posi-tion. A8so,Uthe "most pessimistic" D would be the upperlimit of Yu converted into units fromSsafe position, dividedby the upper limit of aU.
7
NAIJRD R@port 21.01
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NAVORD Report 21-01
TABLE 2
Values of t for N = (n - 1)versus P = 1 - level of confidence
(P)
N .90 .70 .50 .30 .10 .05 .02 .01
1 .16 .51 1.oo 1.96 6.31 12.7 31.8 63.72 .14 44 .82 1.39 2.92 4.30 6.96 9.923 .14 .42 .76 1.25 2.35 3.18 4.54 5.844 .13 .41 .74 1.19 2.13 2.78 3.75 4.605 .13 .41 .73 1.16 2.02 2.57 3.36 4.03
6 .13 .40 .72 1.13 1.94 2.45 3.14 3.717 .13 .40 .71 1.12 1.90 2.36 3.00 3.508 .13 .40 .71 1.11 1.86 2.31 2.90 3.369 .13 .40 .70 1.1o 1.83 2.26 2.82 3.25
10 .13 .40 .70 1.09 1.81 2.23 2.76 3.17
12 .13 .40 .70 1.08 1.78 2.18 2.68 3.0614 .13 -39 .69 1.08 1.76 2.14 2.62 2.9816 .13 .39 .69 1.07 1.75 2.12 2.58 2.9218 .13 .39 .69 1.07 1.73 2.10 2.55 2.8820 .13 .39 .69 1.06 1.72 2.09 2.53 2.84
22 .13 .39 .69 1.06 1.72 2.07 2.51 2.8224 .13 .39 .68 1.06 1.71 2.06 2.49 2.8026 ;13 .39 .68 1.06 1.71 2.06 2.48 2.7828 .13 .39 .68 1.06 1.70 2.05 2.47 2.7630 .13 .39 .68 1.06 1.70 2.04" 2.46 2.75
40 .13 .39 .68 1.05 1.68 2.02 2.42 2.7060 .13 .39 .68 1.05 1.67 2.00 2.39 2.66
120 .13 .39 .68 1.04 1.66 1.98 2.36 2.62
S.126 .385 .674 1.036 1.645 1.960 2.326 2.576
NAVORD REPORT 2101
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NAVORD Report 2101
APPENDIX B
'T'HE PROBIT METHOD OF SENSITIVITY TESTING
INTRODUCTION
1. In order to illustrate the Probit Method, consider asafety test on an explosive train which contains a primer, aslider detonator, and a lead. The slider safety device travelsa distance of 16/32 of an inch between "armed" and "safe"positions. Assume that, when the slider detonator is in anyposition in which it can be initiated by the primer, it willalways initiate the lead. Thus, the safety of the trainis dependent on the slider-detonator to lead transfer, sincethe most unsafe condition exists at this transfer. Therefore,estimates of safety will be obtained by purposely initiatingthe slider detonator at greater distances than the primertransfer allows and recording the effect on the lead as "safe"(0) or "unsafe" (X). In this manner, if spurious initiationof the slider detonator occurs, it may be determined whetherthe safety of the train is adequate or not, as the case may be.
Test Procedure for the Probit Method
2. Determine, roughly, the distance from armed positionwhich produces an unsafe condition about 50% of the timewhen the slider detonator is purposely initiated. The 50%distance may be roughly estimated by testing a small sampleof 5 by means of the up-and-down procedure. For illustra-
ion, the follow-Ing results were obtained from an actual tests
Distance fromArmed Position
5/32 X
6/329
7/32 G
M,1
NAVORD Report 2101
3. Samples of at least 10 each, including the first 5 testedabove, should be tested at distances around the 50% distance,thus obtaining a range of percentage-distances. Such percent-age dlstancF , may be obtained in any order and do not requirean equ.l r±- ,• of samples for each distance tested; nor dothe distances tested need to be equally spaced. The high andlow percentage distances should be estimated by larger samplesthan those near the 50% distance, since they are less accuratewhen estimated from small samples. The results obtained forthe illustrative example are tabulated as follows:
Distance fromArmed Positi•o• Percent(in 32nd's of an Unsafe Safe Unsafeinch) X 0 P
5 XXXXOYXXXXXXXXXXXXX 17 1 94.7
5-3/4 XXXXXOXOXO 7 3 70
6 OXXXXXOXXX 8 2 80
6-1/4 OXOOOXOOOX 3 7 30
7 O00000XO00 1 9 10Total 316--
Description of a "Probit"
4. The Probit Analysis is based on the transformation of thepercentage unsafe into "Probits". To illustrate this stepclearly, some understanding of the theoretical normal curveis necessary. The theoretical normal curve is determined byits two parameters, the 50% unsafe distance, XU, and thestandard deviation, aU. Any specified percentage-unsafedistance may be obtained by integration of the normal curveequation. Tables of this integration are available in moststatistical texts which give the percentage of the area underthe curve for various distances, x, measured from YU in termsof aU, or defined as,
x - XU¶ U
2
NAVORD Report 21.01
Thus, any desired percentage-unsafe distance, x, may beobtained from Xu and au by x = tOU. Now if aU and XUare not known but, say, a 30% distance is obtained, the tableof z, ne- .a curve areas will give a t value that representsthe difference between XU and the 30% distance in terms ofCu. Thus, the "Probit" of a percentage-unsafe distance, x,is the number of standard deviations that x is removed fromthe 50% distance, and, for simplicity in calculation, theprobit of XU is designated as 5.0. For percentages below50% the probits are determined by subtracting the number ofstandard deviations from 5.0, and for percentages above 50%,the number of standard deviations are added to 5.0. A probit,Y, may be defined as,
Y = 5 + 3[u).
Table 1 presents a complete tabulation of the transformationof percentages to probits.
Analysis of Test Data by the Provisional Probit Line
5. Step I of the analysis is the transformation of theobserved percentages to probits, designated as "empiricalprobits". By using Table 1, these values may be read directly.
6. Step 2 involves plotting a graph of the empirical probitsas a function of the distances tested. A provisional straightline is then drawn that best fits the empirical probits,placing the line by eye. For each of the distances tested,the value of the ordinate of the provisional line is read.These are termed tie "provisional probits", Y.
7. If the provisional line is drawn so the difference betweenthe empirical probits and the provisional probits is a minimum,and no other line will fit the empirical probits more accu-rately, the more precise mathematical procedures for deter-mining the probit line may not be needed.
8, The equation for the provisional line may be obtaineddirectly from the graph. The estimate of ,U is the x valuewhich corresponds to the probit 5.0. An estimate of au is
the increase in x for a unit increase of the probit values.
3
NAVORD Report 2101
The slope of the line, b, is given by I'/oa. The equation ofthe line can then be obtained by substituting the values ofXU and b in the equation
Y = 5 + b(x - XU).
9. As an illustration, the test results presented in para-graph 3 may be analyzed by the methods of paragraphs 5through 8 as follows:
Distance fromArmed PosiGion(in 32nd's of Number Percentage Empirical Provisionalan inch) Tested Unsafe Probits Probits
x n p Y
5 18 94.7 6.62 6.6
5-3/4 10 70 5.52 5.5
6 10 80 5.84 5.2
6-1/4 10 30 4.48 4.8
7 10 10 3.72 3.7
The empirical probits were plotted in Graph 1 and the provi-sional line estimated. The provisional probitg were obtainedfrom the provisional line of Graph 1. The X1 is the x valuecorresponding to the probit 5.0, which is 6.Y 2 measured in32nd's of an inch. A unit increase in the probit value ofthe provisional line provides a decrsease in the value of xequal to 0.69, measured in 32nd's of an inch. This value isan estimate of 6U, but since it.represents a decrease in xrather than an increase, it must be considered as a negativevalue in determining b. Thus, the slope, b, of the provi-
sional line is 1/aU or i/-0.69, which is equal to -1,45. Bysubstituting XU = 6.12, and b = -1.45 in the equation
Y = 5 + b(x -XU),
one ,my obtain the equation of the provisional probit linein terms of x, or distance from armed position. Thus,
4
ilVOED ' .... Ort 2101
Y = 5 - 1.4 5 (x - 6.12)
or, Y = 13.87 - 1.45x
10. Thii equation of the provisional line provides a meansfor estimating the distance at which a specified expectedprobit occurs. If the distances related to probits between3.0 and 7.0 are calculated, the percentages correspondingto these probits may be obtained from Table l, and then therelationship of distances versus percenuage-unsafe may beplotted.
11. The foregoing method does not provide a meazls for deter-mining the error of the XU or aU, so it should not be used asthe complete analysis unless D, is considerably larger than5.0. DS is obtained by dividing the XU, measured from safeposition, by aU obtained from the provisional line. For theillustrative example:
safe position is 16/32 of an inch from armed position,and YU is 6.12/32 of an inch from armed position; there-fore, 16 - 6.12
S -0.69 - 9.88/.69 =0.63
which indicates that the safety is more than adequate andh1aL further calculation is not necessary. But for illus-
tration and comparison, this example will be used to demon-strate the calculation of the Probit Line.
The Method of Calculating the Probit Line
12. The method of calculating the Probit Line, based on themaximum likelihood theory of reference (c), has been simpli-fied considerably by use of tables which are appended to thisreport. The explanation of the theoretical basis of themethod has been omitted, so the following example is only anillustration of the calculations and application of the tables.
13. The preceding tabulation in paragraph 9 for estimatingthe provisional line is a necessary part of che calculationsfor the probit line. Thus, steps 1, 2, and 3 are: (1) trans-forming the percentages to empirical probits by means ofTable 1, (2) plotting the empirical probits and estimating
5
NAVORD Report 2101
the provisional probit line by eye, and, (3) reading theprovisional probits, Y, from the provisional line. Step 4incorporates the weighting coefficient, w, which is read
from Table 2 for each provisional probit value. The totalnurbe- of t•'aJ• at each distance is n, which is multipliedby w ani ente_-ed in the tabulation under nw. Step 5 deter-mines a "working probit", y, for each empirical percentageversus its provisional probit. This value is read fromTable 3. For the example cited, the tabulation of the datais as follows:
Distance fromArmed Position Number Percent Empirical Provi- Weighting Working(32nd's of' an Tested Unsafe Probits sional Factor Probitsinch) Probits
x n p Y nw
5 18 94.7 6.62 6.6 4.28 6.62
5.75 10 70 5.52 5.5 5.81 5.52
6 10 80 5.84 5.2 6.27 5.76
6.25 10 30 4.48 4.8 6.27 4.49
7 10 10 3.72 3.7 3 .6 3.72Znw = -57a
14. The products nwx and nwy are tabulated and added; divisionof the totals by Znw gives the means, x and 2y. The values nwxare multiplied by x and summed to give &nwx . Also, they aremultiplied by y and added to give Znwxx. The values nwy aremultiplied by y and added to give Znwy . The sums of squaresand products about the mean are then obtained as follows:
(a) Sxx = Znwx 2 - (Znwx) 2 /2nw,
(b) Sxy = Znwxy - (Znwx)(Zrwy)/Znw, and
(c) Syy = Znwy 2 - (Znwy) 2 /znw.
15. The equation of the Probit Line may be obtained by sub-stituting in the equation Y = y + b(x - i) the calculatedvalues
6
_____________________ _____________________________________________
NAVORD Report 2101y =Znwy
ZnwxXnw'
and b = Sxu/Sxx, which give a linear rela-.:•n>p berween Y, in probits, and x, distance frcm armed
position.
16. The previous example provides the following calculationsas illustration of paragraphs 14 and 15:
nwx nwE nwx 2 nwy2 nLM
21.4000 28.3336 107.000000 187.568432 141.66800033.4075 32.0712 192.093125 177.033024 184.40940037.62o00 36.1152 225.720000 208.023552 216. 6n90039.1875 28.1523 244.921875 126.403827 175.95187523.5200 12.4992 164.640000 46.497024 87.494400
Znwx=155.13-50nwy=1.37-. 7l15Lnwx 2= 3.370Znwy2 =745.525b592nwxy=.T0.214075
.nwx 2 = 934.375000 (Znwx)(Znwy)/•nw = 818.78032
Znwy 2 = 745.525859 (2nwx) 2 /Znw = 926.00493
Znwxy = 806.214875 (Znwy) 2/7nw = 723.97154
thus,
(a) Sxx = 934.375000 - 926.00493 8.37007,
(b) Syy = 745.525859 - 723.97154 = 21.55432, and
(c) Sxy = 806.214875 - 818.78032 12.56544.
The preceding measures allow the calculations of b, x and yfor substitution in the equation for the Probit Line,Y = • + b(x - 3). The calculated values for the example are
7 = 137.1715/25.99 = 5.278
x = 155.1350/25.99 = 5.969
and, b -12.56544/8.37007 = -1.5012.
Therefore, the Probit Line equation for the example is
7
NAVORD Report 2101
Y = 5.278 - 1.50J.(x - 5.969)
or Y = 14.237 - 1.501x
CaiculatIon uo the 50%-Unsafe Distance, YU, and StandardDeviation, oU
17. The estimated YU is obtained by substituting in theProbit Line equation the value 5.0 for the probit value Y andsolving for x. For the example
5.0 = 14.237 - 1.50.1x
or x = 9.237/1.501 = 6.15.
Thus XU equals 6.15, measured in 32nd's of an inch. Theestimated aU is equal to 1/b, or for the example,
1/1.501 which equals 0.67 (in 32nd's of an inch).
The Degree of Safety, DS, and Comparison Between the Provi-
sional Estimates and Calculated Estimates
18. The degree of safety, DS, is obtained by dividing XU,measured from safe position, by aU, obtained from the calcu-lated probit line. From paragraph 17, XU for the example is6.15, measured from armed position in 32nd's of an inch,which gives 9.85 from safe position (16 - 6.15), and oU is0.67. Therefore, .Ds is found to be 9.85/0.67 = 14.7.
19. The method of calculating che probit line is an appro-ximation that results in convergence; therefore, if theprovisional line differs considerably from the calculated one,it is advisable to perform another cycle of corputations andcompare the first cycle with the second to see if convergencehas been approximated. Thus, a comparison between the eyeestimates and the calculated estimates seems in order. Forthe example, the following measures may be compared:
8
NAVORD Report 2101
Provisional Estimates Calculated Estimates
X. = 6.12 Yu = 6.15
O 0 = 0.69 oo = 0.67
Probit Line, Probit Line,
Y = 13.87 - 1.45x Y = 14.24 - 1.50x
DS = 14.3 Ds = 14.7
The difference between the two estimates is so small that nofurther calculation of the Probit Line is needed.- Shouldthere be any doubt concerning the convergence of the provi-sional line and the calculated line, the calculations describedin paragraphs 20 and 21 may be performed to determine whetheranother cycle of computations is necessary.
Sampling Errors of the 50%-Unsafe Distance and Slope, b, ofthe Calculated Prob!t Line
20. The error, a•, to be expected in the calculated estimateof YU may be obtained by
1
a- =b(Znw)1/2
or for the example! 1
OX = 1.50(25.99)172 = 1.50(5.10) = 0.13.
If the limits obtained from 7, + a- include the provisionalXU, a further cycle of computations is not necessary. But ifthe provisional XU is outside these limits another cycle shouldbe performed and comparison made betweer.- cycle 1 and cycle 2estimates. The provisional T.. of 6.12 is well within the limitsof 6.02 and 6.28, indicating Vhat sufficient convergence of the
XU estimates has been obtained.
21. The error, 3b, to be expected in the calculated slope, b,is determined from
9
NAVORD Report 2101a = 1 1/2
ob = ( -- .x
The error for the ex:ample is:
b= (V1 2 = (B7.194)/2 0.35
If the .imit• obtained from b ± !b include the provisionalestimate of b, a further cycle of computations is not necessary.But if these limits do not include the provisional b, anothercycle should be performed. The provisional b of 1.45 for theexample is well within the limits of 1.15 and 1.85, which indi-cates that sufficient convergence has been obtained.
22. The 90% confidence limits of XU may be determined by
XtU- tox,
where t is obtained from Table 2 of Appendix A and is thenormal deviate for P = .10, or t = 1.65. The example gives
6.15 ± 1.65(0.13)
or, 6.15 + 0.21, which gives limits of
5.94 to 6.36.
23. Now, the 90% confidence limits of ab, using t = 1.65,are obtained by
b ± tab,
or, for the example:
1.50 ± 1.65(0.35) = 1.50 ± 0.58,
which gives limits of 0.92 to 2.08. By obtaining the recip-rocals of the confidence limits of b, one may determine thelimits of aU, which are
1/0.92 to 1/2.08,
or, 1.09 to 0.48
24. The "most pessimistic" DS may be calculated from the 90%upper limit of XU, measured from safe pozsition, and the 90%upper limit of aU as follows, where XT represents the totaldistance from armr'A to safe position:
10
NAVORD Report 2101
XT - (Xu + toj
S - 1/(b + tob)
.- the ,'-iple, X + to is 6.36, and l/(b + tob) is 1.09;therefore, Ds is calculated as follows:
16 - 6.36 = 8.81.09
This D is considerably better than 2.58, the normal deviatecorrespondlng to 99.5%, therefore, the device would be con.-sidered safe.
A Test for Goodness of Fit
25. In some cases, it may be doubtful whether the probit lineis actually a measure of the relationship between percentage-unsafe and distance tested. If the empirical probits arescattered in such a fashion that the probit line could bedrawn in several positions by eye estimate, it is advisableto calculate the probit line and obtain a measure of "good-ness of fit" termed the chi-squared test.
26. A chi-squared value,.)X2 , is calculated as follows:/X 2 S 2 /SXX
(k - 2) =Syy - (sxy"S xxwhere (k - 2) refers to the "number of degrees of freedom",or the number of test distances minus 2. By referring to achi-squared table (available in reference (c) or otherstandard statistical references), one may ascertain the pro-bability of obtaining such a value with (k - 2) degrees offreedom, providing the probit line is the true relationshipbetween percentage unsafe and distance tested. For theexample:
- 2)= 2155432 - (-12.56544 ) 2 /8.37
= 21.55432 - 18.86383 = 2.69o49
with (k - 2) degrees of freedom equal to 3. The chi-squaredtable gives a probability of 0.47, which means ýhat practi-cally half the time one would expect a larger/, value than
11
NAVOED Report 2101
2.69 from samples explained by the calculated probit line.In other words, the probit line is a "good fit".
27. A probit 1-.e which gave a chi-squared value corres-pdci:ng pc rcbability less than .05 is considered a poorfit and would not be acceptable as the true relationshipbetween percentage unsafe and distance tested. For suchcases, one might determine some function of x for which acalculated probit line would fit more closely, such as logx, x , or l/x, etc. The log x transformation is used tosome extent; in which case the probit equation takes theform
Y = 7 + b(log x - logx--),
and similarly for other transformations.
12
NAVORD Report 2101
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NAVORD Report 2101
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NAVORD Report 2101
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NAVORD Report 2101
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* 0 r--4H HCi r JC\JC%j CCj0 ~ n--~U '-D -. O0 CY u d..S 0 * * * . . . . . . . . . . . . . . .
H r~. a0
z-Z- CM cr)YDG-zt 000 coC0 --4- H C~j -cO0-10 t t-0-0 m2g0 0 kDCO 0 -uL 0 a~Cf-)tO -t0 tN- (Y 0 O)C -) r-4 C10- * OH HH H rIC\j CUC\jCM('1 ' -4 C -~-4- n '-0\0Nt- 01% CM a)JS0 . . . . .*
rz4 C) m20 ---co~ 00 HC rq : (")0ma o CoC OcOC'j CUjcy)C~U0 t- .114 F-4
e)0 Hc (nt- Nc U, COIC~uO r00 (\I(Y)0 E00CUNM-i-r q4 a 0
N--N-N-N-~~a -N-N -NNN N--) V~
m Cuj Low COt G\ 02\'\" C-i t\O\ N-H m'nrc~C\j H r-cm - ON-0IHzt M - C~ NM t- (n MOO t--j-t ()N0\ C)Hu n 0 4--l
0 u-o C) b* N-0 rV-- cr-i C'-H-. CUj COW0)( e ()-- M Nl- ON FqCdC
N-t-N-- -t--N :NN- -- sC04-) r.F- r-4H 0
co 0 C~j.--- mt-i N-H C\U C'noo HC 0~ H~i -iCO % 0'\(Y--4 -I I rx N-C-UW CU j , 0 ONO (Y) 0 ON (') CUCO CUj 0 CO-HC) 41U) l
n0 C Lf )N IN -4. N- 0 C\4 %) M~ CY)'1W)- H'D -0 H 0 \,a) LrC'. Vr N 020 0OO r- r-lrl -HCUCUCC\UNN rl y- 1nkU\ 1~W)-00CH,4
0 . . .. .HC U.~ . 1WGC~ .OHCCU
t-- COCMCOCC COi()COCOCOCO n, Mr H0 rIcoNCM 0
0~~~~~~~~~~~~~ HY)Z . z .\ot-c ' Z t-CjnNt-00 0c ý
NAVORD REPORT 2101
_C~ ~ -_ H --- ý10 CYn NO~C) 0.J 1 (n -t Y) C o*H o 4 4- H -t 0O\ C)
C)r-I H- 0) (n L- fL, H C)
co H Nu C _-r O c CONo nu -- J- ý'O m\ Lc\ H
00 H C) N-C ON 4-- H-o ON N' '-C 0 t-- N' C
o o n ý1O n H 0 Ca
t-4 OF\ O) ON -Cb~[I- rH '.0 CO 0 nC z- 4- '0
. \10 nC' Ln \1'0 H- t- H Ho C0 t~- fn H- (n C n C0
o \ ' .O n~ cm 0 0
H- co m~ N co (n t.- LC\\10 m '.0ý co U\CO r\ t- Cn
4- H H o C-- t-- m Culo; ) '.0 0 0 trih Y CYI C)
o 0 (Y \1o U-l\ N 0 0
E-4 ~ t nN C- t- ON m 7 0 C- ON jZn o, m ON C7 C7\ ON ry'
0 0 4-. '.0 co CO '.0 4-k 0H C 0 Nu nC n~ N 0 0
CM rL UNý t- n co Cj mN co H
S0 . Cu O C\ - t-- 0 H H 4-tN . o) C) (n ( *LCn C) 0 \'o C)
C! . . . . * . .
E-1- m~ -4- ON ONý ON 4- 4r(Y \.04 t-- UC) 0 CO00
0 * HH C-H \10 nC ~ .H )0 rn) ' H m' tC-C)W 0 0 Nu un '.0 C') 0 0
Lf t-'04 (Y) C-- Cuj
o -0 N r4 n ' \ 0 H CO0
cu m ' -4- 0 -C --r kON 0l
H CO 0 m~ 4-t (n t-- Cuj C*0 <)N 4-. H -4- 4-t 0 r-
C) ) Hl in tý- -.- ) Hl Ho 0 rH ko -.t4 H 0
* -H CO '.0 CO H 4-. 0o 0 H (n (Y) Ce) (Y) (Ir H- 0
C)0 H 4- 0 '.0-r H 0 0
0 C) C)C )C)C )
00>4 * *)S
NAVORD REPORT 2101
TABLE 3
WORKING PROBITS (y)
% ~ pro visional probit, Y% pa 2.0 2.1. 2.2 2.3 1 2.4 2.5 2.6 2.7 2.8 2.9
0 1.o8711877 1.967 2.057 2.146 2.234 2.321 2.408 2.494I 3.951.3.-467 3.141 2.927 2.793 2 I7F 2.661 2,674 2.690 0.7212 6 .2 -7 5 .147 " i ,r• -. P r 3- e: 3 -:) OR o -.7.10 7 a79 1 9 29~~~~~~7~~-l " I.. -I0 , ~*~7 , ,173.027I.7 9JQ3 8.46'3 6.827 5 .6674846 4.265 .85 .574 .380 3.254 3.176
4 8c507 6.931 5.806 5.002 4.428 4.020 .733 .536 .4035 8.19416.765 .738 .998 .46714.086 .818 .631
9.458 7-725 6.474 5.569 4.913 4.440 4.099 .8587 - 8.684 7.210 6.139 5.360 .793 .381 .0858 9.644 .946 .710 .806 5.146 .663 .31329 8.683 7.280 6.253 .4.9 .945 .540
10 9.419 .851 .699 .852 5.227 .767
1l -- 8.421 7.146 6.205 5.509 4.99512 .992 .592 .558 .791 5.222
13 9.562 8.039 .911 6.073 .44914 .486 7.264 .355 .67715 .932 .617 .636 .904
16 9.379 7.970 6 918 6.13217 .825 8.323 7:2001 .35918 .676 .4821 .58619 9.029 .7641 .81420 .382 8.o46 7.041
21 1 9.735 8.328 7.26822 ,I.610 .49623 .892 .72324 9.173 .95025 .455 8.178
26 9.737 8.40527 - .633P8 .86029 9.0873 o I
-I +• ~9.54232
.769
33 .99734_J35 1
"--
NAVORD REPORT 2101
TABLE 3 (CONTINUED)Obs. Spr'ovisional probits Y
1 3. . 3.2 1 3.3 1 3.4 1 3.5 3.6 1 3.7 I; 3.8 3.9
o 2.579 2.662 2.74512.826.2.906,2.984 7.71 2.764 2.815 2.872 2.932 2.996 3.061 3.127 3 193.3.259 3.323,2 .949 -967 .998 3.039 3.0861 .].39 .194 .252 .310 .3693 3.13413.120 3.125 .145 .176 .216 .261 .310 .362 .4154 3191 .272 .252 .251 .267 .293 .328 .369 .413 .461
• 5051 .424 .37 .358 :357 .3701 .395 .427 .465s .503.690 3.577 3.505 3.464 3.447 3.447 3.461 3.485 3.516 3.553
7 .875 .729J .632 .570 .537 .525 .528 .544 .568 .5998 4 060 .8832 .758 .677 .627 .602 .595 .602 .619 .6459 .246 4.034 .885 .783 .717 .679 .662 .66o .671 690
10 .0 4u .186 4.012 .889 .80 .756 .7 .8 719. .722 nL611 4.616 4.339 4.138 3.996 3.898 3.834 3.795 3,777 3.774 3. 78212 .801 .491 .26514.*02 .988 .911 .862 .835 .825 .8281ý .986 .644 .391 .208 4.078 .988 .929 894 2.87401@ 5.172 796 .518 .315 .168 4.065 .996 .952 92 .92015 .357 .948 .645 .421 .25 .142 4,062 4.01 ,98o c)66'lb1 5.542 5.101 4.771 4.927 4.220 4.129 4.069 4.031 4.01217 .727 .253 .898 .634 .439 .297 .196 .127 .083 .05818 .913 .406 5.025 .740 .529 .374 .263 .185 .134 .10k19i 6.098 .5581 .151 .846 .6191 .451 .330 .244 .186 .14920 .283 .710 .278 .9531 .709! .528 .-396 .102 .2ý7 1 9
21 6.68 5.863 5.405 5.059 4.799i4.6o6 4.463 4.361 4.289 4.24122 .653 6.015 .. 531 .165 .889 .683 .530 .419 .340 .28723 o839 .168 .658 .272 ..979 .760 .597 .477 .392 .3332 7.024 .320 .785 .378 5.070 .837 .664 .536 .443 .37925 .209i .4721 .911 .484 .16o .914 .7301 ,594 .49 .42526 7-394 6.625 6.038 5.591 5.250 4.992 4.797 4.652 4.546 4.471
27 .580 .777 .165 .697 .340 5.069 .864 .711 .598 .51728 .765 .930 .291 .803 .430 .146 .931 .769 .649 .56329 .950 7.082 .418 .910 .520 .223 .997 .827 .701 .60o8
01 8.135 -2- .54516.016 .6101 .3001 5.064 .886 .752 .65431I 8.320 7.387 6.671 6;122 5.701 5.378 5.131 4.944'4.804 4.700
32 .506 .539 .798 .9 .7911 .455 .198 5.002 .855c .74633 .691 .692 .925 .335 .881 .532 .2651 0611 .907 .79234 .876 .844 7.051 .441 .971 .609 .331 119 15.958 .8383 51 9.061 .996 .1 7 48 .• 6.0611 .687 .398 .17 .O:1.8361 9.24 8-.149 7.305 6.654 6.151 5.764 5.465 5.236 5.061 4.9301
37 .432 .01 .431 .760 6242 .841 .532 .294 1)38 .617 .#54 .558 .867 .332 .918 .599 .353 :164 5.022139 .802 .606 .685 .973 .422 .995 .6651 4 111 .216 06840l .9871 .758J .81117:0791 .512 6.07ý ,712l .46^1j ý267 Ila41 8 .911 7.938 7.186 6.602 6e150 5.799 5.528 5.319 5.15942 9.063 8.065 .292 .692 .227 .866 .586 .370 .20543 .216 .191 .398 .782 .304 .932 .644 .422 .25144 .368 .318 505 .873 .381 .999 .703 .473 .29745 52_0 .4 b11 .963 .459,6.o66 .76 .529 446 9.673 8.571 7.717 7 ý3 6.536 60.133 5819 5. 576 5.38947 .825 .698 .824 .143 .613 .200 .878 .628 .43548 .978 .825 .930 .233 .690 .266 .936 .679 - 48149 .951 8.036 .323 .767 .333 .994 .71 527
1LSO 9.078 .143 .414 .845 .400 6.053[_.7e 572
NAVORD REPORT 210!
TABLE 3 (CONTINUED)Obs.
%1provisional probits, Y1. 1 .2 4.3 4.4 4.6 4.7 4.9
. 4 .525 577 1.624 3b4 3- .6o8 3.72413.741i 3.387 3.4T6 3.503 3.557 3. 07 3°62361372 .5 .6
2 .427 .487 .538 .589 .637 .680 .719 .751 .775 .7913 .468 .521 .572 .621 .667 .709 .746 .777 .801 .8164 .510 .559 .607 .653 .697 .737 .773 .803 .826 .8415 .551 .596 .641 j685 727 :766 .800 .829 .852 .867
3..592 3.6 37613;7l7 3.757 3.794 3.827 3.856S.634 .671 .710 .749 .787 ,822 .854 .882 .903 .917
. 675 .709 .745 .781 .817 .851 .882 .908 .929 .9429 .716 .747 .779 .813 .847 .879 .909 .934 .954 .9670 .758 .784 .814 .845 .877 .9o8 .936 .96o .80,.991
11 3.799 3.822 3.848 3+877 3490713-936 3.963 3.987 4.0 .012 .84o .859 .883 .909 .937 .964 .990 4.013 .031 .04313 .882 .897 .917 .941 .967 .993 4.017 .039 .057 .06814 .923 .934 .952 .973 .997 4.021 .044 .065 .082 .0931 .964 .972 .986 4.005 4.027 .0501 .072 .092 .18 .11916 4.006 4.010 4.021 4.03814.057.078 4 .099 4.11833 4.14417 .047 .047 .056 .O'., .,n87 .1061 .126 .144 .159 .16918 .088 .085 .090 .102 .117 .135i .153 .170 .184 .19419 .130 .122 .125 .134 .147 163: .180 .196 .210 .21920 .171 .16o lq. q 166 .1771 .192 .2Q .223 .216l .24921 4.212 4.198 4.194 4.198 4.207 4.220!4.235 4.249 4.26114.27022 .253 .235 .228 .230' .237 .248 .262 .275 .2871 .29523 .295 .273 .263 .262 .267 .277 .289 .301 .3121 .32024 .336 .310 .297 .294 .297 .3051 .316 .327 .338 .34525 .377 .3481 .332ý *.261 .327ý .334, .343 .354 .3-63i .370
26 4.4 1 9 14.38514.366 4.358 4.357 4.362 4.37o04.380 43 39ý .39b27 .46o .423i .401 .390 .387 .39 .397 .4o6 .45 .42128 .501 .461 .435 .422 .417 .419 .425 .432 .440 .446291 .5431 .498 .470 .454 .447 .447 .452 .459 .466 .47111 .5841 .536 .504 .486 .477 . .476 9 .485. .491 631 4.25 4.5 4.507 4.506 4 .51 .75 .5232 .667 .611 .573 .550 .537 .533 .533 .537 542 .54733 .708 .649 .608 .582 .567 .561 .560 .563 .568 572
34 .749 .686 .642 .614 .597 .89 588 .590 5 .5975 .1. 64 6 . 6 2 7 .6181 .615 .616 .619 .622
q24.761 4.711, 4. 4.6571 4.646 4.642 4.642 4 64 T.37 .873 .799 .746 .710 .687 .675 .6691 .668 .6701 .673381 .915 .836 .7-80 .742 .717 .703 .696 .695 .6961 .69839 .956 .874 .815 .774 .747 .731 .723 .721 .7211 .72340 .997. .12i .849 .806 .777 .76o .790ý .747 .747174
41 5.039 4.949 4.884 4.838 4.807 4.788 4.778 4.773 4.773 4.774
42 .080 .87 918 .870 .837 .817 .805 .799 .798 .79943 .121 5.024 953 .902 .867 .845 .832 .826 .824 .824
o44 .63 .062 .988 .934 .897 .873 .859 .852 .8491 .849
45 .204 .099 5.0221 .66 .927 .902 .886 88.875 .87414~.0 537 .4900' 4.900
%6 5.245 5.13715-057 4.998 4.957-4.9304.913 490T .40'4.0
47 .287 .175 .091 5.030 .957 .959 .941 .931 .926 .92548 .328 .212 .126 .062 5.017 .987 968 .957 .952 .95049 .369 .250 .160 .094 .047 5.015 .9951 .983 .977 .975
50 .411 .287 .195 .126 .0781 .04415.022 5.009 5.00315.000
NAVORD REPORT 2101
TABLE 3 (CONTINUED)Obs.
orovisional problts, Yp ,.0 , .i _ . . 5.3 5.4 5,5 5.61 5.7 5.8 5.9
,O 3j.7 .74O?.710K680 3.6 20 9 4- 27213.079 8 22 *343.7 72 ,-W - .Z .. . . 30 13 -1 2.871 3.772 3763744 3.7 0 3.06 7 3.56413.452 3J30 4.1 .8
2 .797 .790 .770 .732 .675 .593 .482.336 .148 .909.822 .816 .795 .758 .702 .621 .512 .3G38 .183 .946
4 .847 .841 1.821 .785 .729 .650 .542 .400 .217 .9814.872 .8661 .846 .811 .75' .678 .572 .4 223.021
3.897 3.89113.-72 3.837 3.78333.70636023.4653.287 3.059S.922 .916 .898 .863 .810 .735! .632 .497 .321 .097
.947 a,42ý .923 .890 .838 .763 .662 .529 .356 .134972 967 .949 .916 .865 .792 .692! .561 .390 .17210 -. .892 .820 .722 .593 .425 201111 4.022 4017 4.000 3.9681-3.919 3.848 3.7523 .62 34593[4
12 .047.042 .025 .994 .946 .877 3782 .657 .494 .28413ý 073 o68 .0514.0211 .973 .905 .812 .689 .528 .32214ý o98 :093,077: 047i4.000 "934 .842 "721 "563 "36o5ý 12-3 118 :1021 :073' .028 .962, .872 .753 .597 .39716 4.14814.143 .128 4.099 j.055 3.990 3.902 3.785 3.632 3.435
.173; 168 153 .126 .082 4.019 .932 .817 .666 .47218 .198ý .194 .179 .152 .109 .047 .962 .849 .701 .51019, .223! .219 .204 .178 .136 .076 .992 .881 .735 .548.20 248' .244 ,2%0 .204 .16 .104 4.022 .913. 770 .58521 4.273 4.269 4.25614.230 4.191 4.13214.052 3.945.3.804 3.62322 .298 .294 .281, .257 .218 .161 .082 .9771 .839 .660"3 .323 .320 .3071 .283 .245 .189, .112 4.009 .873 .69824 .348 .345 .3321 .309 .272 .2188 .142 .041 .908 .7352r,37 i.37° .3 8 3 j .299 .1721 .073 .94 .773
2E' 9 .39 954.334.362 .3 "4.2754.202r4.1053.9773.81T
27 .423' .420 .409 .388 .353 .303 .232 .137 4.011 .84844 45.3 414 .381 .331 .2621 .169 .046 .886
29' 474' .471 .460 .44o .4o8 .360 .292 .201 .080 .923IJ0 .499 .496ý .486 .4666 .435 (38 .322 .233 .115 931' 4.524 4.521 4.5114.S3 4.462 441714T52 4.265 4.149 3.99932 .549 546 .537 .519 .489 .445 .382 .297 .184 4.03633 .574 .571 .563 .545 .516 .473 .412 .329 .219 .07434ý .599 .597 .588 .571 .544 .502, .442 .361 .253 .11115 .624 .622 .614 .>598 .571 .,:30 .472±.93 .288 .149•6 4.649 4.647 4.639 4.624 4.598 .4.425 4.322418
7 .674 .66-1 t .650 625 .57 ,532 .457 .357 .224381 .699 .697 .690, .676 .6521 .6151 .562 .489 .391 .26239 .724 .723 .716, .7021 .6791 .644 .592 .521i .4261 .299
4,748 142'7___ .6221 -4a .340 .4 7.7494 .729 ,.706l .6721.2 .553, 40 37
141 4 7 7 4.773 4.7674.7554.734 4.7014.6524.5854.4954.37.8793 .18 .761 .789 .782 .617 .529 .412
4, .825 .823 :.818 .807 .788 4 757 .649 .564 .45044 .85o .849 .844 .133 .815 .786, .742 .682 .598 .487,,ý .875r .874 -.86g .86o, .842 .8141 .7a'2 1.714 !.633ý .525
464.90094.899 4.895 4.886'4.869 4.843 4.,802 8.4 .566 2
47 .925 .924 .921 .912 .897 .871 .832 .778 .702 .6oo48 .950 .949 .946 .938 .924 .899 .862..810 .736 .63749 .975 .975 .972 .965 .951 .928 8921 .842 .771 .67550 5.00015.000 .997 .991 .978 .9566 .922 .874 .805 .713
1---
NAVORD REPORT 2101
TA.LE 3 (CONTINUED)Obs.
%rovisional probits YS6o0 6.2 6.3 6.4 6.5 6.6 I 6.7 6.8 6.9
1 2.523 2.132 " 6 1.030 0.26.1 690 2.564 ! 2.18 1.694 1.088 0.272 .606! .224 .76 146 .394 I3 .647 .270 .797 .205 461014 .688 .316 .849 .263 .5285 .688 .362 .9oo .321 ___5956 2.7.712408 1.952 13b80 .661
7 .UL r .•D •.v U . o teo I
81.854 .500 .055 .496 .?959 .895 .546 .106 .555 .8620 .936 .591 .15o8 .613 .928 0067
11 2.978 -2.637 2.209 1.671 0.995 0.14412 3.019 .683 .261 .730 1.062 .221
*13 .060 .729 .312 .788 .129 .29914 .102 .775 .364 .846 .196 .37611 .141 .821 .415 .905 .262 2453 1163.8 287 2.467 1.963 1.32910-530
"226 .9159 .08 .396 .60718 .267 .959 `•.)0 .080 .463 .685
.8 .308 3.005 .621 .138 .530 .762'If0 .673 .197 .596 .8321 3.391 3.09b 2.724 2.255 1. 663 o.91622 .432 .142 .77-6 .313 .730 .993 0.062
2Z .474 .188 .827 .372 .797 1.071 .15224 .515 .234 .879 .430 .864 .148 .2432c .556 .280 .g30o .488 .930 .225 ,33326 3.598 3.326 2.98212.54-7 1.997 1.302 0.42327 .6391 .372 3.0331 .605 2.064 .379 .51328 .680 .418 .085 .663 .131 .457 .60329 .721 .464 .136 .722 .197 .534 .69330o .63 .509 .188 .780 .264 .611 .78431 3.804 3.555 3.239 2.838 2.331 1.688 0.87432 .845 .601 .291 .897 .398 .766 .96433 .887 .647 .342 .955 .465 .843 1.054 0.05034 .928 .693 .394 3.014 .531 .920 .144 .156H5 .969 .739 - .445 .072 .±98 .997 .234 .262__36 4.011 3.785 3.;,97 3.130 2.665 2.074 1.324 0.36937 .052 .831 .548 .189 .732 .152 .415 .47538 .093 .877 .600 .247 .799 .229 .505 .58139 .135 .923 .651 .305 .865 .306 .595 .688:401 .1761 .9691 -.7031 .364. .932 .383 .685 .794
4i 4.217 4,014 3.754 3.422 2,999 01.7 7 5 0.90042 .259 .060 .806 .480 3.066 .538 .865 1.0074 . 0 .106 .857 .539 .132 .615 .955 .113 0.035
44 341' .152 .909 .597 .199 .692 2.046 .219 .162451 .383 .198 .960 .65 .266 .69 .136 .26 .28946 4.424 4.244 4.012f3.714 3.333 2.8462.226 1.432 0.4154? .465 .290 .063 .772 .400 .924 .316 .538 .54248 .507 .336 .115 .830 .466 3.001 .406 .645 .66949 .548 .382 .i66 .889 .533 .078 .496 .751 .79550 .589. .428 .218 .9471 .600 .155 .586 .857 .922
NAVORD REPORT 2101
TABLE 3 (CONTINUED)Obs.
% provisional probits, Y
p1 3.0 3, 3.2 3.3 3.4 3.5 3.6 3.7 ,.8 j3.9
5 _ - 9.205 8.249 7.504 6.922 6.467 6.111 5.83415.61852 .331 .355 .594 .999 .534 .170 .885 .66453 .458 .462 .684 7.076 .600 .228 .937 .71054 .585 .568 .774 .154 .667 .286 .988 .756
155_ 6__ 1! .4 .864 .2K 4 .34956.o4o .802569.888.7817.9547.386.801 6.403 6.091 5.8-57 .965 .887 8.045 .385 .868 .461 .143 .894
.993 .135 .462 .934 .520 .194 .94059 9.100 .225 .540 7.001 .578 .246 .9866"0'.0 35.1 o8.3 .297 6.031
61 9.312 8.4057.6947.135 6.695 6.-34-9 6.07762 .419 .495 .771 .201 .753 .400 .12363 .525 •585 .848 .268 .811 .452 .16964 .631 .676 .926 .335 .870 .503 .21565 78 .76618.003 .402 .928 .955 - .261
6 9.8•448.8568.080 7.469 6.986 6,.6o6 6.307
67 .950 .946 .157 .535 7.045 .658 .35368 9:036 .234 .602 .103 .709 .39969 .126 .312 .669 .162 761 .445
__0_,216 ,389 36 .220 812 .491
79.307 8.466 7.803 7.278 6.864 6.53672 .397 .543 .869 .337 .915 .58273 .487 .621 .936 .395 .967 .62874 -. 577 .698 8.003 .453 7.018 .674
•_ _ .667 -775 .070 .512 .07 -720
76 9.7578:85278.1367570 7.121 6.76677 .848 .929 .203 .628 .173 .81278 .938 9.007 .270 .687 .224 .85879 :084 .337 .745 .276 .904
180 .161 .404 .803 .327 Q.50T 0 9.238 8.470 7.862 7.379 6.995
8 •315 .537 .920 .430 7.04183 .393 .604 .978 .482 .08784 .470 .671 8.037 .533 .133t r .547 738 .095 85 .179
86 9.624 8.804 8.154 7.36 7.2258 .701 .87'1 .212 .688 .271
8.779 .938 .270 .39 .31780, I I .856 -9.05 3'2^ 3
1 9 1 13 3 .072 .... 445 .4 5
91 9.138 8.;4; * 7. 592 .205 .504 :945 .500
.272 .562 .997 .54693 04 .59294 .339 .62o 8. .9
r_ .4o .605-79 .100 .69b 9.4728Z.737 .151 7.6841
97 539 .795 .203 .730I98 .606 .854 .254 .776
99 .673 .912 .306 .8221100 .739 .9._ 5_ .868.
____________________________________________
4NAVORD REPORT 2101
TABLE 3 (CONTINUED)Obs.
% provisional probits, Y.4.2 4.3 4.4 4.5 '4.6 14.7 4.8 4.9__ _+1. .,: ..... 1I -__ __
51 5.45215.325 5.229 5.158 5.108 5.072 5.049 5.035 5.028 5.02552 4931 .363 .264 .190 .138 .1OI .076 .062 .054 .05153 .535 .400 .298 .222 .168 .129 .103 .088 o079 .07654 .576 .438 .333 .254 .198 .157 .131 .114 .lC5 .10155 .617 .475 .367 .286 .228 .±86 .158 .14 .131 .12656 5.659 5,513 5.402 5.31b 5.258 5.-2-14 -5-Ib5 5 .167 5.156 5.15157 .700 ,550 .436 .351 .288 .243 .212 .193 .182 .177
58 .741 .588 .471 .383 .318 .271 .239 ,219 .207 .20259 .783 .626 .505 .415 .348 .299 .266 .245 .233 .22760 .824 .663 .-40o .447 8M .328 .294 .271 .258 .252
S61 5.865 5.701 5.574 5.479 5.40815.356 5.321 5.298 5.284 5.27762 .907 .738 .609 .511 .438 .385 .348 .324 .310 .3036: .948 .776 .643 .543 .468 .413 .375 .3501 .335 .32864 .989 .814 .678 .575 .498 .441 .402 .376 .361 .35365 6.031 .8-• .712 .607 .928 .470 .429 .402 .380 .378
T.-037-215.889 5.747 5.639 5.5g8 5.498 5.456 5.429 5.412 5.403.67 .113 .926 .781 .671 .5 8 .527 .484 .455 .437 .42968 .155 .964 .816 .703 .618 .555 .511 .481 .463 .45469 .196 6.0Ol .851 .735 .648 .583 .538 .507 .489 .47970 .237° .039 .&85, .767 .678 .6121 .565 .53 .514.5o71 6.279 6.077 5.920 5.799 5.708 5.640 5.592 5.560 5.540 5.529
72 .320 .114 .954 .831 .738 .669 .619 .586 .565 .55573 .361 .152 .989 .8631 .768 .697 .647 .612 .591 .58074 .402 .189 6.023 .895 .798 .725 .674 .638 .617 .60575 .444 .227 .058 .927 .828 .754 .701 .665 .642 .63076 6.48, 6.265 6.092 5.959 5.858 5.782 5.728 5.691 5.668 5.65577 .526 .302 .127 .991 .888 .811 r .717 693 680
78 .568 340 161 6.023 .918 .839 .782 .743 .719 .70679 .609 :377 :1961 .055 .948 .868 .809 .770 .744 .73180 1 .69o .419 230 1087 )978 .8c6 .817 .7Q6 .7701L .l81 6.692 6.452 6.265 6.119 6.008 5.924 5.864 5.822 5.796 5.781182 .733 .490 .299 .151 .038 .953 .891 .848 .821 .80683 .774 .528 .334 .183 .068 .981 .918 .874 .847 .83284 .816 .565 .3681 .215 .098 6.010 .945 .901 .872 .85785 .857 .6o3 .403 .247 .128 ,08 .972 .297 .898 .88286 0.898 6.64o 6.4371 6.279 6.158 6.o66 6.000 5.953 5.923o5.90787] .940 .678 .472 .31L .188 .095 .027 .979 .949 .932
88 .981 .716 .5o6 .343 .218 .123 .054 6.006 .975 .95889 7.022 .753 .541 .375 .248 .152 .o81 .032 6.000 .98390 .o64 .791 .575i .407 .27% .18 o.I8 .q58 .026 6.00891 7.105 6.828 6.610 6.43916.308 6.208 6.135 6.084 6.051 06.03392 .146 .866 .6441 .471 .338 .237 .162 .110 .077 .05893 .188 .903 .679 .503 .368 .265 .190 .137 102 .08494 .229 .941 .7131 .535 .398 .294 .217 .163 :128 .10995 .270 .979 .748 . .428 .22 .244 .189 .154 .1Lt96 7.312 7.016 6.783-6.600 6.458 &.350 6.271 6.215 6.179 6.15997 .3513 .054 .817 .632 .488 .379 .298 .242 .205 .]8498 .394 .091 .852 .664 .518 .407 .325 .268 .230 .210
99 .436 .129 .886 .696 .548 .436 .353 .294 .256 .2356 .921 . 728 .578 .464 .380 .20 .81 .260i00___"__6_ ____2__ . • , .~ .1..
N.',VRD REPORT 2101
TABLE 3 (CONTINUED)Obs.
% ~ provisional pro)bits, Y% 1I 5.0 5.1 5.2 1 5.3 5.4 15.5 . 5.6 5.7 5.81 5.9
51 5.02515 025 o23 5.017 5.005 4.95 4.953 4.906 4.840 4.75052 .0501 .050 *K:O .0 3 2 5.01 3 .933 .938 .874 .788
53 .7 T.5 Q1i . 09 .059 .041 5.013 .970 .909 .825
54 0 :100ioo .o00 .096 .087 .070 .o43 5.002 .943 .863
55 .1251 .126 .125 .122 .114 o098 .073 -34 .978 -9Vo
56S 5.150 .151 5.151 5.148 5.141 5.127 5.103 5.066 5.012 4.938
57 .175 .176 .176 .174 .168 .155 .133 .098 .047 .976
58 .201 .201 .202 .201 .195 .183 .163 .130 .082 5.013
59 .226 .226 .227 .227 .222 .212 .193 .162 .116 .05.60 .51 .252 .5 ? 253 250 .240 .2231 ., .51 .o88
61 5.276 5.277 5.279 5.279 5.277 5.269 5.253 5.226 5.185 5.12662 .301 .302 .304 .305 .3o4 .297 .283 .258 .220 .164
63 .326 .327 .330 .332 .331 .325 .313 .290 .254 .201'64 .351 .352 .355 .358 .358 .354 .343 .322 .289 .23965 .3,76 W87 .381..384 .385 .382 .373 .354 .323 .276166 5.401 5.403 5-.406 5.410 5.412 5.411 5.403 5.386 5.358 5.314
67 426 .428 .432 .437 .440 .439 .433 .418 .392: .351681 .451 .453 .458 ".463 .467 .467 .463' .450 .427 "38969 .476 .478 .483 .489 .434 .496 .493 .482 .461 .427
70 .501 .504 .509 .515 .%1 .524 .521 .514 .496 .464
71 5.526 5.529 5.534 5.541 5.548 5 `53 5.553 5.540 5.530 5.502
72 .551 .554 .560 .568 .575 .591 .583 .578 .565 .539
73 .577 .579 .585 .594 .603 .609 .613 .610 .599 .577
74 '.602 .604 .611 .620 .630 .638 .643 .642 .634 .615
751i .627. .630 63y .646 -657.666 .673 .674 .668 .652'76 5 652 5.655 5.662 5.673 5.684 5.695 5.703 5.706 5703 5:690
77 .677 .680 .688 .699 .711 .723 .733 .738 .737. .727
7P .702 .705 .713 725 .738 .752 .763 .770 .772 .765
79' .727 .730 .739 .751 .765 .780 .793 .802 .8o6 .80280 ,752 5755 .764 .777 .793 .808l .823 ,834 :841 .8409
181 5.777 5.781 5.790 5.804 5.820 5.837 5.853 5.866 5.375 5.878
82 .802 .8o6 .816 .830 .847 .865 .883 .898 .910 .915
83 .827 .831 .841 .856 .874 .894 .913 .930 .944 .953
84 .852 .856 .867 .882 .901 .922 .943 .962 .979 .990
85r .877 .881 .892 q9o8 .9281 Q9 ) .973 .995. 6 .01446 0281
86 5.902 5.907 5.918 5.935 5.956 5.979 6.003 6.027 6.o48 6.066
87 .927 .932 .9431 .961 .983 6.007 .033 .059 .083 .10388 .953 .957 .969 .987 6.010 .036 .063 .091 .117 .14189 .978 .982 .995 6.013 .037 .064 .093 .123 .152 .178
g0 6. 6.00 7 6.020 ,040 .064 092 ,123 .1551 186 .216
91 6.028 6.033 6. o46 6.o66 6.091 6.121 6.153 6.187 6.221 6.253
92 .053 .058 .071 .092, .118 .149 .1831 .219 .255 .291
93 .078, .083 .097 .118 .146 .178 .2131 .251 .290 .32994 .10 .108' .122 .144 .173 .206 .243 .283 ;324 .366
9 5 12 .1331 .48 .171 .200 .234 .271 .315 .359 .404
96 6.153 _.159 6.174 6.19716.227 6.263 .303 6.347 6.39 6.441
97 .178 .184 .199 .223 .254 .291 .333 .379 .42 .479
98 .203 .209 .225 .249 .281 .320 .363 .411 .462 .517
9 .228 .234 .250 .276 .309 .348 .393 .443 .497 .554
0 .253 .2591 .276 .302 .336 .376 .423 .475 .531 .592
00
;,VORD REPORT 2101
TABLE 3 (CONTINUED)Gbs.
% provisional pvhts
1 __ 6.0 6.1 P1 0' _.3 _6.4_ _6.5 6.6 6.7 16.8 6.9
51 4.631 4.473 4.2644006 3o667 3.233 2.677 1.964 1.049
52 ,Of? I59 . c64 .734 .310 .767 2.070 .175 0.022
53 .713 .565 .372 .122 .800 .387 .857 .176 .302 .175
5k -755 .611 .424 181 .867 .464 .947 .283 ..+e9 :)e.796 .657 .47 .239 -934 .5- 3.037 .389 .5f .480
64.37 4.703 4.527 4.297 4.001 3.619 3.127 95 1. 2 0.63;2
57 .879 .749 .578 .356 .068 .696 .218 .602 .809 .78458 .920 .795 .630 .414 .134 .773 .308 .708 .935 .937
59 .961 .841 .681 .472 .201 .850 .398 .814 2.062 1.08960 9.003 .887 .73.3 .531 .268 .92L .488 .921 .189 .24261 5.044 4.932 4.784 4.589 4.335 4.005 3.578 3.027 2.315 1.39462 .085 .978 .836 .647 .4ol .082 .668 .133 .442 .5463 .127 5.024 .887 .706 .468 .159 .758 .240 .569 .699
64 .168 .070 .939 .764 .535 .236 .849 .346 .695 .85165 .209 .116 .990 .23 .602 .-l .939 .452 .822 2.004
6 5.251 5.62 5.042 4 . 669 4.391 .029 3.559 2.949 2.156
67 .292 .208 .093 939 .735 .468 .119 .665 3.075 .308,68 .333 .254 .145 .998 .802 .545 .209 .771 .202 .-,169 .375 .300 .196 5.056 .869 .622 .299 .878 .329 .613
70 .416 .346 .248 .114 .936 .700 :390 .984 .45' .766
71 5.457 5.392 5.299 5.173 5.003 4.777 4.480 4.090 3.582 2.918
72 .499 .437 .351 .231 .069 .854 .570 .197 .709 3.070
73 .540 .483 .402 .289 .136 .931 .660 .303 .835 .223714. .581 .5290 1 C L -.3118.2 5nJ.0U8 .750 .409 .962 .375
7i .62 .75 .505 .406 .270 .086.840 . 4.089. .528S765.664 5.667 5.5! 5 5.336 5 .930 4.622 4.215 3.68o
77 .705 .667 .608 .523 .403 .240 5.021 .728 .342 .832
78 .747 .713 .660 .581 .470 .317 .111 .835 .469 .985
79 .788 .759 .711 .639 .537 .394 .201 .941 .595 4.13780 .82 .805 .6 8 .604 .472 .291 5.047 .7221 29_
7 -5l 70 -i . 5.6710 5 9 5.3 1 5 . 1 54 4.849 4.442-8I5805.851 5.°549.
82 .912 .896 .866 .815 .737 .626 .471 .260 .975 .594
83 .953 .94- .917 .873 .804 .703 .561 .366 5.102 .74784 .994 .988 .969 .931 .871 .780 .r52 .473 .229 .899
85 6.036 06.020 .990.8 9 .858 .742 .579 .355 5.052
6 o.077 6.080 6.072 .048 •-004 5.935 5.32 5.6855.482 5.204
87 .1.8 .126 .1.23 .lO6 .071 6.012 .922 .,92 .609 .356
88 .1600 .172 .175 .165 .138 .-891o6 012 .898 .735 r509
89 .201 .218 .226 .223 .205 .660 .102 6.:004 .862 .66191 .242 .264 .278 .281L .272 ._2;4 .192 .111 .988 .814
91 6.264 6.310 6.329 6.340 6.338 -= 21 6.283 6.217 6.115 5.9
92 .3251 .355 .381 .398 .405 .398 .373 .323 .242 6.11893 3 .401 .'032 . 472 .4751 .463 .430 .368 .271
94 .4o8 .447 .484 .515 .539 .553 .553 .536 .495 .423.449 .493 '35 .5731 .0 .§3o .643 .642 .622 .576
9 6 9 .53935.67 6-b0 567 YT745I6. 797 .532 .585 .638 .690 .739 .784 .824 .855 .875 .880
98 .573 .631 .690 .748 .806 .8611 .914 .961 7.002 7.033
I99 .614~ .677 .741 .807 .87 .93917.00706 .18 as1100 .6561 .72- .793 .865i .939 7.016 .09 4 .174 .255 .338
.L--•-
NAVORD REPORT 2101
TABLE 3 (CONTINUED)Obs.
% provisional probits, YI p 7.0 7.1 1 7.2 7.3 7.4 7.5 1 7.6 7.7 7.8 7.9
51525354
6 o .19862J.3836 .568
5 .753593
61 o.198
67 .309 0.00368 .494 .23169 .680 .45870 .865 .68_71 2,050 0.913
172 .235 1.14073 .420 .36774 .606 .595 0.26375 -791 .822 .54576 2.976 2.0500.77 3-11 71.10878 .347 .504 .39079 .532 .732 .672 0.26579 717 -959 .954 .61881 3.902 3.186 2.236 0.97182 4.087 .414 .518 1.32483 .273 .641 .800 .677 0.17584 .458 .868 3.o82 2.030 .62185 .643 4 o96 364 .,383 1.o6886 4.828 4.323 3.645.2.736 1.51487 5.014 .551 .927 3.089 .961 0.43888 .19 .778 4.209 .4 2 12:1+08 1.00889 .384 5.005 .491 .795 .854 .5799o .569 .233 .773 4.14813.301 2.149 0.581991 5.754 5. 460 5.055 4.501 3.747 2.720 1.317 -92 .940 .687 .337 .854 4.194 3.290 2.054 0.35693 6 .125 .915 .619 5.207 .640 .861 .790 1.31694 .310 6.142 .901 .560 5.087 4.431 3.526 2.275 0.542
"" ý 369 6.182 .. 14 .33 5.002 4 262 3.35_1.8o69 5 , D -3.. . .8 o
66681.976.464 6.267 5.9 5.572 4.998 .1943.069 1.49397 .866 .824 .746 .620 6.426 6.143 5.735 5.154 4.333 3.17398 7.051 7.051 7.028 .973 .873 .713 6.471 6.114 5.596 4.85399 .236 .279 .310 7.326 7.319 7.284 7.20717.073 6.859 6.5331
o00[ .421 .5o6j .5921 .679 .766 .854 .943 18-033 8.123 8.2131
NAVORD REPORT 2101
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iAVORD Report 2101
BIBLIOGRAPHY
I. hAVORD Report 470. "Tables of Confidence Limits forBinomial D1•t,:.ion. 1948 (Unclassified).
2. Naval Ordnance Laboratory Memorandum No. 7062."Confidence Limits for Tests Involving Small Sznples."
1945 (Unclassifieud).
3. Copper -nd Pearson. "The Use of Confidence or FiducialLimits Illustrated in the Case of the Binomial."Biometrika, Vol. 26, pp. 404 - 413 (1934).
4. Naval Ordnance Laboratory Memorandum 9910. "Comparisonof the Probit Method and the Bruceton Up-and-downMethod as Applied to Sensitivity Data." 1948 (Restricted).
5. Columbia University Statistical Research Group. "SelectedTechniques of Statistical Analysis for Scientific andIndustrial Research and Production and ManagementEngineering." McGraw-Hill Book Co., 1947, pp. 339 - 352.
6. NAVORD Report 205-45. "Tpbles to Fecilitate the Anal],y]isof Sensitivity Data." 1945 (Unclassified).
7. Frankford Arsenal Statistical Memorandum No. 4. 'Mechodsof Making Experimental Inferences." 1946 (Unclis:3ified).
8. Fechner, G. T. "Elemente der Psychophysik."r 1,elpz~g:Breitkopf and Hartel, 1860.
9. Goddum, J. H. "Reports on Biological Standards, Ill.Methods of Biological Assay Dependin•g on a QuantalResponse." Medical Research Council, London, Spec. Rep.Ser., No. 183 (1933).
10. Bliss, C. I. "The Method o' Probits." Science, Vol. 79,pp. 38 - 39 (1934).
"The Method of Probits -- A Ctrrection." Science,Vol. 79, pp. 409 - 410 (1934).
"The Calculation of the Doý;age-Mortality Curve."Annals of Applied Biology, Vol. 22, pp. 134 - 167 (1935).
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/ NAvO:'d Report 2]0]
iBIBLIOGRAPHY (Cont'd)
- -- -Th C'rni~r~ ~r!-f Dosage-Mor'tnlity Datta." Annals.oAp€,Ey o. 22, PP. 307 333 (1935).
I1. Hcnmnlngsen, .j,.. M. 'The Accuracy of' Insuilin, Assay onWhite Mice.." Quarterly Journal of Pharmi.-coloy, Vol. 6,Pp. 39 8 0 and 18 2 - 217
12. U.41asser, Ot/o, Editor. "Medical Physics, Volume II." TheYearbook PcIblishers, Inc., l(150; article on "Bioassay",PP. 71 - 8(j).
13. NAVORD RerFort 65-46. "Staircase Method of SensitivityTesting,. " T. W. Anderson, P. J. McCarthy and J. W. Tukey,P1r1nceton Statistical Research Group, 1946.
14. D. F. Votlaw, Jr. "The Effect of Non-normality on 'Stair.-case' Me]/hod2 of Sensitivity Testing." PrincetonSat:1t1Qst/a1 Research Group, 1948.
I1_' NAMTC Tc/chnical Report No. 75. "A Study of Methods forAchlevi.,]Lg Reliab:!lity of Guided Missiles." 10 July 1950
6. ,NAMTC ,J.'echnIca] Report No. 94. "Gener'al Specifications
for ]I'(,, Safety Margins Required for' Guided MissileCorpoe intr,1 t. 10 July 1951 (Restricted).
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NAVORD Report 2101
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