
TOLERANCE LIMITSName:
There are two numbers for the tolerance interval:
Tolerance limits are given by
with \( \bar{X} \) and s denoting the sample mean and the sample standard deviation, respectively, and where k is determined so that one can state with (1\( \alpha \))% confidence that at least \( \phi \)% 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". Dataplot computes the tolerance interval for three confidence levels (90%, 95%, and 99%) and five coverage percentages (50.0, 75.0, 90.0, 95.0, 99.9). In addition, Dataplot computes nonparametric tolerance intervals. These may be preferred if the data are not adequately approximated by a normal distribution. In this case, the tables have been developed based on the smallest and largest data values in the sample.
where <y> is the response variable, and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax generates both the normal and the nonparametric tolerance limits.
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable, and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax generates only the normal tolerance limits. If the keyword LOGNORMAL is present, the log of the data will be taken, then the normal tolerance limits will be computed, and then the computed normal lower and upper limits will be exponentiated to obtain the lognormal tolerance limits. Similarly, if the keyword BOXCOX is present, a BoxCox transformation to normality will be applied to the data before computing the normal tolerance limits. The computed lower and upper limits will then be transformed back to the original scale.
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable, and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax generates only the nonparametric tolerance limits.
TOLERANCE LIMITS Y1 SUBSET TAG > 2 NORMAL TOLERANCE LIMITS Y1 SUBSET TAG > 2 NONPARAMETRIC TOLERANCE LIMITS Y1 SUBSET TAG > 2
In reliability and lifetime applications, onesided tolerance limits are more common. In these cases, we typically want coverage intervals that are greater than a given value (lower tolerance intervals) or smaller than a given value (upper tolerance intervals). These tolerance intervals are equivalent to onesided confidence limits for percentiles of the specified distribution. Dataplot can compute onesided (or twosided) confidence limits for percentiles for a number of distributions commonly used in reliability applications. For example, to compute lower onesided tolerance limits for the 2parameter Weibull distribution, you can do the following
set distributional percentile lower weibull maximum likelihood y
set maximum likelihood percentiles default
LET A = NORMAL TOLERANCE ONE SIDED K FACTOR Y
LET A = NORMAL TOLERANCE LOWER LIMIT Y The above commands are for the raw data case (i.e., a a single response variable).
LET A = SUMMARY NORMAL TOLERANCE K FACTOR MEAN SD N
LET A = SUMMARY NORMAL TOLERANCE LOWER LIMIT MEAN SD N The above commands are for the summary data case. The three arguments can be either parameters or variables. If a variable rather than a parameter is given, the first element of the variable is extracted. The three values denote the mean, standard deviation, and sample size of the original data. To specify the coverage and confidence, enter the commands
LET GAMMA = <value> where ALPHA specifies the confidence level and GAMMA specifies the coverage level. The defaults values are 0.95 for both the confidence and the coverage. In addition to the above LET command, builtin statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).
Hahn and Meeker (1991), "Statistical Intervals: A Guide for Practitioners", Wiley.
2006/3: Allow only the normal or only the nonparametric limits to be generated 2014/06: Support for LOGNORMAL and BOXCOX tolerance limits SKIP 25 READ ZARR13.DAT Y SET WRITE DECIMALS 4 TOLERANCE LIMITS YThe following output is generated: TwoSided Normal Tolerance Limits: (XBAR +/ K*S) Response Variable: Y Summary Statistics: Number of Observations: 195 Sample Mean: 9.2615 Sample Standard Deviation: 0.0228 Confidence = 90%  Coverage k Lower Upper Value (%) Factor Limit Limit  50.0 0.7240 9.2450 9.2780 75.0 1.2349 9.2333 9.2896 90.0 1.7657 9.2212 9.3017 95.0 2.1040 9.2135 9.3094 99.0 2.7650 9.1985 9.3245 99.9 3.5321 9.1810 9.3420 Confidence = 95%  Coverage k Lower Upper Value (%) Factor Limit Limit  50.0 0.7382 9.2446 9.2783 75.0 1.2591 9.2328 9.2902 90.0 1.8003 9.2204 9.3025 95.0 2.1452 9.2126 9.3103 99.0 2.8192 9.1972 9.3257 99.9 3.6014 9.1794 9.3435 Confidence = 99%  Coverage k Lower Upper Value (%) Factor Limit Limit  50.0 0.7661 9.2440 9.2789 75.0 1.3067 9.2317 9.2912 90.0 1.8684 9.2189 9.3040 95.0 2.2263 9.2107 9.3122 99.0 2.9258 9.1948 9.3281 99.9 3.7375 9.1763 9.3466 TwoSided DistributionFree Tolerance Limits Response Variable: Y Summary Statistics: Number of Observations: 195 Sample Mean: 9.2615 Sample Standard Deviation: 0.0228 Involving X(3) = 9.207325 Involving X(N2) = 9.310506  Confidence Coverage Value (%) Value (%)  100.00 50.00 100.00 75.00 99.99 90.00 92.80 95.00 36.18 97.50 1.43 99.00 0.05 99.50 0.00 99.90 0.00 99.95 0.00 99.99 Involving X(2) = 9.206343 Involving X(N1) = 9.320067  Confidence Coverage Value (%) Value (%)  100.00 50.00 100.00 75.00 100.00 90.00 98.91 95.00 72.05 97.50 13.30 99.00 1.72 99.50 0.01 99.90 0.00 99.95 0.00 99.99 Involving X(1) = 9.196848 Involving X(N) = 9.327973  Confidence Coverage Value (%) Value (%)  100.00 50.00 100.00 75.00 100.00 90.00 99.95 95.00 95.69 97.50 58.16 99.00 25.50 99.50 1.66 99.90 0.44 99.95 0.02 99.99  
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Date created: 06/05/2001 