
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).
For twosided intervals, the WaldWolfowitz method provides the basic approach. However, this method is computationally expensive. Weisberg and Beatty (1960) published tables based on this method. Gardiner and Hull (1966) proposed an approximation that replaced an integration with algebraic formulas. Howe (1969) proposed a simpler approximation for the tolerance limits that is considered to be more accurate than the Weisberg and Beatty method. Guenther (1977) proposed a correction term for Howe's method. Howe's approximation is
where
The degrees of freedom parameter is N  1 by default. However, if the standard deviation is based on historical data rather than the current data set, then an independent value for the degrees of freedom may be given. In Dataplot, you can specify the degrees of freedom by entering the command
If this command is not given, N  1 will be used. The Guenther correction is
where
The details for the Gardinar method can be found in the Gardiner paper. Dataplot supports both the Gardiner method and the Howe method. The default for the 2018/05 version is the Howe method. Prior versions use the Gardiner method. To specify the method in Dataplot, enter the command
To specify whether the Guenther correction will be applied to Howe's method, enter the command
The default is OFF. Dataplot supports two methods for onesided intervals. The first method uses the formula
where t is the noncentral t distribution with noncentrality parameter
The noncentral t distribution can lose accuracy as N gets large. The second method only uses the percent point function for the normal distribution and has the formula
where
\( b = z_{\gamma}^2  \frac{ z_{\alpha}^2}{N} \) To specify the onesided method, enter
<NONCENTRAL T/NORMAL/DEFAULT> The default is to use the noncentral t based approximation for N ≤ 100 and to use the normal based approximation for N > 100.
Weisberg and Beatty (1960), "Tables of ToleranceLimit Factors for Normal Distributions", Technometrics, Vol. 2, pp. 483500. Gardiner and Hull (1966), "An Approximation to TwoSided Tolerance Limits for Normal Populations", Technometrics, Vol. 8, No. 1, pp. 115122. Howe (1969), "TwoSided Tolerance Limits for Normal Populations  Some Improvements", Journal of the American Statistical Association, Vol. 64, pp. 610620. Guenther (1977), "Sampling Inspection in Statistical Quality Control", Griffin's Statistical Monographs, Number 37, London. Natrella (1966), "Experimental Statistics: NBS Handbook 91", National Institute of Standards and Technology (formerly National Bureau of Standards), pp. 213  215. 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 2018/05: Support for Howe method and Guenther correction for twosided limits 2018/05: Support for normal based approximation for onesided limits 2018/05: Some tweaks to the output format SKIP 25 READ ZARR13.DAT Y SET WRITE DECIMALS 4 TOLERANCE LIMITS YThe following output is generated: TwoSided Normal Tolerance Limits: (XBAR +/ K*S) Howe Method Response Variable: Y Summary Statistics: Number of Observations: 195 Degrees of Freedom: 194 Sample Mean: 9.2615 Sample Standard Deviation: 0.0228 Coverage = 90%  Confidence k Lower Upper Value (%) Factor Limit Limit  50.0 1.6519 9.2238 9.2991 75.0 1.7102 9.2225 9.3004 90.0 1.7657 9.2212 9.3017 95.0 1.8003 9.2204 9.3025 99.0 1.8683 9.2189 9.3040 99.9 1.9498 9.2170 9.3059 Coverage = 95%  Confidence k Lower Upper Value (%) Factor Limit Limit  50.0 1.9684 9.2166 9.3063 75.0 2.0378 9.2150 9.3079 90.0 2.1039 9.2135 9.3094 95.0 2.1452 9.2126 9.3103 99.0 2.2263 9.2107 9.3122 99.9 2.3233 9.2085 9.3144 Coverage = 99%  Confidence k Lower Upper Value (%) Factor Limit Limit  50.0 2.5869 9.2025 9.3204 75.0 2.6782 9.2004 9.3225 90.0 2.7650 9.1984 9.3245 95.0 2.8192 9.1972 9.3257 99.0 2.9258 9.1948 9.3281 99.9 3.0533 9.1919 9.3310 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  
Date created: 06/05/2001 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 