1、Designation: G16 95 (Reapproved 2010)G16 13Standard Guide forApplying Statistics to Analysis of Corrosion Data1This standard is issued under the fixed designation G16; the number immediately following the designation indicates the year of originaladoption or, in the case of revision, the year of las
2、t revision.Anumber in parentheses indicates the year of last reapproval.Asuperscriptepsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This guide covers and presents briefly some generally accepted methods of statistical analyses which are useful in theinterpr
3、etation of corrosion test results.1.2 This guide does not cover detailed calculations and methods, but rather covers a range of approaches which have foundapplication in corrosion testing.1.3 Only those statistical methods that have found wide acceptance in corrosion testing have been considered in
4、this guide.1.4 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.2. Referenced Documents2.1 ASTM Standards:2E178 Practice for Dealing With Outlying ObservationsE691 Practice for Conducting an Interlaboratory Study to Determine t
5、he Precision of a Test MethodG46 Guide for Examination and Evaluation of Pitting CorrosionIEEE/ASTM SI 10 American National Standard for Use of the International System of Units (SI): The Modern Metric System3. Significance and Use3.1 Corrosion test results often show more scatter than many other ty
6、pes of tests because of a variety of factors, including thefact that minor impurities often play a decisive role in controlling corrosion rates. Statistical analysis can be very helpful inallowing investigators to interpret such results, especially in determining when test results differ from one an
7、other significantly.This can be a difficult task when a variety of materials are under test, but statistical methods provide a rational approach to thisproblem.3.2 Modern data reduction programs in combination with computers have allowed sophisticated statistical analyses on data setswith relative e
8、ase. This capability permits investigators to determine if associations exist between many variables and, if so, todevelop quantitative expressions relating the variables.3.3 Statistical evaluation is a necessary step in the analysis of results from any procedure which provides quantitativeinformati
9、on. This analysis allows confidence intervals to be estimated from the measured results.4. Errors4.1 DistributionsIn the measurement of values associated with the corrosion of metals, a variety of factors act to producemeasured values that deviate from expected values for the conditions that are pre
10、sent. Usually the factors which contribute to theerror of measured values act in a more or less random way so that the average of several values approximates the expected valuebetter than a single measurement. The pattern in which data are scattered is called its distribution, and a variety of distr
11、ibutionsare seen in corrosion work.4.2 HistogramsA bar graph called a histogram may be used to display the scatter of the data. A histogram is constructed bydividing the range of data values into equal intervals on the abscissa axis and then placing a bar over each interval of a height equalto the n
12、umber of data points within that interval. The number of intervals should be few enough so that almost all intervals contain1 This guide is under the jurisdiction of ASTM Committee G01 on Corrosion of Metals and is the direct responsibility of Subcommittee G01.05 on Laboratory CorrosionTests.Current
13、 edition approved Feb. 1, 2010Dec. 1, 2013. Published March 2010December 2013. Originally approved in 1971. Last previous edition approved in 20042010 asG1695(2004).G1695 (2010). DOI: 10.1520/G0016-95R10.10.1520/G0016-13.2 For referencedASTM standards, visit theASTM website, www.astm.org, or contact
14、ASTM Customer Service at serviceastm.org. For Annual Book of ASTM Standardsvolume information, refer to the standards Document Summary page on the ASTM website.This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been m
15、ade to the previous version. Becauseit may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current versionof the standard as published by ASTM is to be considered the official document.Co
16、pyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1at least three points, howeverpoints; however, there should be a sufficient number of intervals to facilitate visualization of the shapeand symmetry of the bar heights. Twenty intervals ar
17、e usually recommended for a histogram. Because so many points are requiredto construct a histogram, it is unusual to find data sets in corrosion work that lend themselves to this type of analysis.4.3 Normal DistributionMany statistical techniques are based on the normal distribution. This distributi
18、on is bell-shaped andsymmetrical. Use of analysis techniques developed for the normal distribution on data distributed in another manner can lead togrossly erroneous conclusions. Thus, before attempting data analysis, the data should either be verified as being scattered like anormal distribution, o
19、r a transformation should be used to obtain a data set which is approximately normally distributed.Transformed data may be analyzed statistically and the results transformed back to give the desired results, although the processof transforming the data back can create problems in terms of not having
20、 symmetrical confidence intervals.4.4 Normal Probability PaperIf the histogram is not confirmatory in terms of the shape of the distribution, the data may beexamined further to see if it is normally distributed by constructing a normal probability plot as described as follows (1).34.4.1 It is easies
21、t to construct a normal probability plot if normal probability paper is available. This paper has one linear axis,and one axis which is arranged to reflect the shape of the cumulative area under the normal distribution. In practice, the“probability” axis has 0.5 or 50 % at the center, a number appro
22、aching 0 percent at one end, and a number approaching 1.0 or100 % at the other end. The marks are spaced far apart in the center and close together at the ends. A normal probability plot maybe constructed as follows with normal probability paper.NOTE 1Data that plot approximately on a straight line
23、on the probability plot may be considered to be normally distributed. Deviations from a normaldistribution may be recognized by the presence of deviations from a straight line, usually most noticeable at the extreme ends of the data.4.4.1.1 Number the data points starting at the largest negative val
24、ue and proceeding to the largest positive value. The numbersof the data points thus obtained are called the ranks of the points.4.4.1.2 Plot each point on the normal probability paper such that when the data are arranged in order: y (1), y (2), y (3), ., thesevalues are called the order statistics;
25、the linear axis reflects the value of the data, while the probability axis location is calculatedby subtracting 0.5 from the number (rank) of that point and dividing by the total number of points in the data set.NOTE 2Occasionally two or more identical values are obtained in a set of results. In thi
26、s case, each point may be plotted, or a composite point maybe located at the average of the plotting positions for all the identical values.4.4.2 If normal probability paper is not available, the location of each point on the probability plot may be determined asfollows:4.4.2.1 Mark the probability
27、axis using linear graduations from 0.0 to 1.0.4.4.2.2 For each point, subtract 0.5 from the rank and divide the result by the total number of points in the data set. This is thearea to the left of that value under the standardized normal distribution. The cumulative distribution function is the numb
28、er, alwaysbetween 0 and 1, that is plotted on the probability axis.4.4.2.3 The value of the data point defines its location on the other axis of the graph.4.5 Other Probability PaperIf the histogram is not symmetrical and bell-shaped, or if the probability plot shows nonlinearity,a transformation ma
29、y be used to obtain a new, transformed data set that may be normally distributed. Although it is sometimespossible to guess at the type of distribution by looking at the histogram, and thus determine the exact transformation to be used,it is usually just as easy to use a computer to calculate a numb
30、er of different transformations and to check each for the normalityof the transformed data. Some transformations based on known non-normal distributions, or that have been found to work in somesituations, are listed as follows:y = log x y = exp xy5x y = x2y = 1/x y5sin21 x/nwhere:y = transformed dat
31、um,x = original datum, andn = number of data points.Time to failure in stress corrosion cracking usually is best fitted with a log x transformation (2, 3).Once a set of transformed data is found that yields an approximately straight line on a probability plot, the statistical proceduresof interest c
32、an be carried out on the transformed data. Results, such as predicted data values or confidence intervals, must betransformed back using the reverse transformation.4.6 Unknown DistributionIf there are insufficient data points, or if for any other reason, the distribution type of the data cannotbe de
33、termined, then two possibilities exist for analysis:3 The boldface numbers in parentheses refer to a list of references at the end of this standard.G16 1324.6.1 A distribution type may be hypothesized based on the behavior of similar types of data. If this distribution is not normal,a transformation
34、 may be sought which will normalize that particular distribution. See 4.5 above for suggestions.Analysis may thenbe conducted on the transformed data.4.6.2 Statistical analysis procedures that do not require any specific data distribution type, known as non-parametric methods,may be used to analyze
35、the data. Non-parametric tests do not use the data as efficiently.4.7 Extreme Value AnalysisIn the case of determining the probability of perforation by a pitting or cracking mechanism, theusual descriptive statistics for the normal distribution are not the most useful. In this case, Guide G46 shoul
36、d be consulted for theprocedure (4).4.8 Significant DigitsIEEE/ASTM SI 10 should be followed to determine the proper number of significant digits whenreporting numerical results.4.9 Propagation of VarianceIf a calculated value is a function of several independent variables and those variables have e
37、rrorsassociated with them, the error of the calculated value can be estimated by a propagation of variance technique. See Refs (5) and(6) for details.4.10 MistakesMistakes either in carrying out an experiment or in calculations are not a characteristic of the population andcan preclude statistical t
38、reatment of data or lead to erroneous conclusions if included in the analysis. Sometimes mistakes can beidentified by statistical methods by recognizing that the probability of obtaining a particular result is very low.4.11 Outlying ObservationsSee Practice E178 for procedures for dealing with outly
39、ing observations.5. Central Measures5.1 It is accepted practice to employ several independent (replicate) measurements of any experimental quantity to improve theestimate of precision and to reduce the variance of the average value. If it is assumed that the processes operating to create errorin the
40、 measurement are random in nature and are as likely to overestimate the true unknown value as to underestimate it, then theaverage value is the best estimate of the unknown value in question. The average value is usually indicated by placing a bar overthe symbol representing the measured variable.NO
41、TE 3In this standard, the term “mean” is reserved to describe a central measure of a population, while average refers to a sample.5.2 If processes operate to exaggerate the magnitude of the error either in overestimating or underestimating the correctmeasurement, then the median value is usually a b
42、etter estimate.5.3 If the processes operating to create error affect both the probability and magnitude of the error, then other approaches mustbe employed to find the best estimation procedure. A qualified statistician should be consulted in this case.5.4 In corrosion testing, it is generally obser
43、ved that average values are useful in characterizing corrosion rates. In cases ofpenetration from pitting and cracking, failure is often defined as the first through penetration and in these cases, average penetrationrates or times are of little value. Extreme value analysis has been used in these c
44、ases, see Guide G46.5.5 When the average value is calculated and reported as the only result in experiments when several replicate runs were made,information on the scatter of data is lost.6. Variability Measures6.1 Several measures of distribution variability are available which can be useful in es
45、timating confidence intervals and makingpredictions from the observed data. In the case of normal distribution, a number of procedures are available and can be handledwith computer programs. These measures include the following: variance, standard deviation, and coefficient of variation. Therange is
46、 a useful non-parametric estimate of variability and can be used with both normal and other distributions.6.2 VarianceVariance, 2, may be estimated for an experimental data set of n observations by computing the sample estimatedvariance, S2, assuming all observations are subject to the same errors:S
47、25(d2n 21 (1)where:d = the difference between the average and the measured value,n 1 = the degrees of freedom available.Variance is a useful measure because it is additive in systems that can be described by a normal distribution,distribution;however, the dimensions of variance are square of units.A
48、procedure known as analysis of variance (ANOVA) has been developedfor data sets involving several factors at different levels in order to estimate the effects of these factors. (See Section 9.)6.3 Standard DeviationStandard deviation, , is defined as the square root of the variance. It has the prope
49、rty of having thesame dimensions as the average value and the original measurements from which it was calculated and is generally used to describethe scatter of the observations.G16 1336.3.1 Standard Deviation of the AverageThe standard deviation of an average, Sx, is different from the standard deviation ofa single measured value, but the two standard deviations are related as in (Eq 2):Sx 5 S=n(2)where:n = the total number of measurements which were used to calculate the average value.When reporting standard deviation calculations, it is important to n
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