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BASIS TOLERANCE LIMITS

Name:
    <A/B> BASIS <DIST> TOLERANCE LIMITS
Type:
    Analysis Command
Purpose:
    Generates A-basis and B-basis tolerance intervals for the Weibull, normal, and lognormal distributions.
Description:
    Standard tolerance intervals calculate a confidence interval that contains a fixed percentage (or proportion) of the data. This is related to, but distinct from, the confidence interval for the mean.

    There are two numbers for the tolerance interval:

    1. The coverage probability is the fixed percentage of the data to be covered.
    2. The confidence level.

    Standard tolerance limits are given by

      \( \bar{X} \pm k*s \)

    where \( \bar{X} \) is the sample mean, s is the sample standard deviation, and k is determined so that one can state with (1- α)% confidence that at least Φ% of the data fall within the given limits. The values for k, assuming a normal distribution, have been numerically tabulated.

    This is commonly stated as something like "a 95% confidence interval for 90% coverage".

    A and B basis values are a special case of this. Specifically, the B basis value is a 95% lower confidence bound on the tenth percentile of a specified population of measurements and the A basis value is a 95% lower confidence bound of the first percentile. Alternatively, this can be stated as the B basis value is a 95% lower tolerance bound for the upper 90% of a specified population and the A basis value is a 95% lower tolerance bound for the upper 99% of a specified population.

    Note that the A and B basis values are one sided intervals (the standard tolerance limits are two sided). Also, the standard tolerance limits are typically based on a normality assumption while the A and B basis values can be computed for Weibull, normal, or lognormal distributions or they can be computed non-parametrically if none of these distributions provide an adequate fit.

    A and B basis values were added to support the MIL-17 Handbook standard (see the Reference section below). The mathematics of computing these basis values are given in the MIL-17 Handbook and are not given here.

    A and B basis values are used for the case where the data can be considered unstructured. That is, the data are either univariate to start with or the Anderson-Darling k-sample test has determined that the data can be treated as coming from a common sample. Also, the appropriate distribution should be determined first. The MIL-17 Handbook recommends using the Anderson-Darling goodness of fit test. It also recommends trying the Weibull, then the lognormal, then the normal. If all of these fail, then the non-parametric case can be used.

Syntax 1:
    BBASIS <dist> TOLERANCE LIMITS <y> <SUBSET/EXCEPT/FOR qualification>
    where <dist> is WEIBULL, NORMAL, LOGNORMAL, or NONPARAMETRIC;
              <y> is the response variable,
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax computes B basis values.

Syntax 2:
    ABASIS <dist> TOLERANCE LIMITS <y> <SUBSET/EXCEPT/FOR qualification>
    where <dist> is WEIBULL, NORMAL, LOGNORMAL, or NONPARAMETRIC;
              <y> is the response variable,
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax computes A basis values.

Examples:
    BBASIS WEIBULL TOLERANCE LIMITS Y1
    BBASIS LOGNORMAL TOLERANCE LIMITS Y1
    BBASIS NORMAL TOLERANCE LIMITS Y1
    BBASIS NONPARAMETRIC TOLERANCE LIMITS Y1
    ABASIS WEIBULL TOLERANCE LIMITS Y1
    ABASIS WEIBULL TOLERANCE LIMITS Y1 SUBSET BATCH > 1
Note:
    The following statistics are also supported:

      LET A = NORMAL A BASIS Y
      LET A = LOGNORMAL A BASIS Y
      LET A = WEIUBULL A BASIS Y
      LET A = NONPARAMETRIC A BASIS Y
      LET A = NORMAL B BASIS Y
      LET A = LOGNORMAL B BASIS Y
      LET A = WEIUBULL B BASIS Y
      LET A = NONPARAMETRIC B BASIS Y

    Enter HELP STATISTICS to see what commands can use these statistics.

Default:
    If no distribution is specified, the normal distribution is assumed.
Synonyms:
    The following are synonyms for ABASIS <dist> TOLERANCE LIMIT:

      ABASIS <dist>
      ABASIS <dist> TOLERANCE
      A BASIS <dist>
      A BASIS <dist> TOLERANCE

    The following are synonyms for BBASIS TOLERANCE LIMIT:

      BBASIS <dist>
      BBASIS <dist> TOLERANCE
      B BASIS <dist>
      B BASIS <dist> TOLERANCE
Related Commands: Reference:
    "MIL-HDBK-17 Volume 1: Guidelines for Characterization of Structural Materials", Depeartment of Defense, chapter 8. The URL for MIL-HDBK-17 is http://mil-17.udel.edu/.
Applications:
    Reliability of Materials
Implementation Date:
    1998/04
    2016/12: Corrected tolerance limit factor for Weibull A-Basis
Program:
     
    SKIP 25
    READ VANGEL31.DAT Y
    SET WRITE DECIMALS 4
    BBASIS WEIBULL TOLERANCE LIMITS Y
    ABASIS WEIBULL TOLERANCE LIMITS Y
        
    The following output is generated:
                Weibull B Basis Tolerance Limits
     
    Summary Statistics:
    Number of Observations:                  38
    Sample Mean:                             185.7895
    Sample Standard Deviation:               18.5955
    Sample Minimum:                          147.0000
    Sample Maximum:                          231.0000
     
    Tolerance Values:
    Confidence Value:                        0.9500
    Coverage Value:                          0.9000
    Shape Parameter:                         10.5732
    Scale Parameter:                         194.2046
    Tolerance Limit Factor:                  4.8513
    B Basis Value:                           145.7135
     
     
                Weibull A Basis Tolerance Limits
     
    Summary Statistics:
    Number of Observations:                  38
    Sample Mean:                             185.7895
    Sample Standard Deviation:               18.5955
    Sample Minimum:                          147.0000
    Sample Maximum:                          231.0000
     
    Tolerance Values:
    Confidence Value:                        0.9500
    Coverage Value:                          0.9900
    Shape Parameter:                         10.5732
    Scale Parameter:                         194.2046
    Tolerance Limit Factor:                  8.7913
    A Basis Value:                           109.8321
        

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Date created: 06/05/2001
Last updated: 12/08/2016

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