Dataplot Vol 1 Vol 2

# TOLERANCE LIMITS

Name:
TOLERANCE LIMITS
Type:
Analysis Command
Purpose:
Generates normal and non-parameteric tolerance intervals.
Description:
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.

Tolerance limits are given by

$$\bar{X} \pm ks$$

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 non-parametric 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.

Syntax 1:
TOLERANCE LIMITS <y>             <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable,
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax generates both the normal and the non-parametric tolerance limits.

Syntax 2:
<NORMAL/LOGNORMAL/BOXCOX> TOLERANCE LIMITS <y>
<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 Box-Cox 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.

Syntax 3:
NONPARAMETRIC TOLERANCE LIMITS <y>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable,
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax generates only the non-parametric tolerance limits.

Examples:
TOLERANCE LIMITS Y1
TOLERANCE LIMITS Y1 SUBSET TAG > 2
NORMAL TOLERANCE LIMITS Y1 SUBSET TAG > 2
NONPARAMETRIC TOLERANCE LIMITS Y1 SUBSET TAG > 2
Note:
Two-sided tolerance limits are used when symmetric coverage intervals from the mean are desired.

In reliability and lifetime applications, one-sided 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 one-sided confidence limits for percentiles of the specified distribution.

Dataplot can compute one-sided (or two-sided) confidence limits for percentiles for a number of distributions commonly used in reliability applications. For example, to compute lower one-sided tolerance limits for the 2-parameter Weibull distribution, you can do the following

set maximum likelihood percentiles default
set distributional percentile lower
weibull maximum likelihood y

set maximum likelihood percentiles default
set bootstrap distributional percentile lower
bootstrap weibull maximum likelihood plot y

Note:
The following statistics are also supported:

LET A = NORMAL TOLERANCE K FACTOR Y
LET A = NORMAL TOLERANCE ONE SIDED K FACTOR Y

LET A = NORMAL TOLERANCE LOWER LIMIT Y
LET A = NORMAL TOLERANCE UPPER LIMIT Y
LET A = NORMAL TOLERANCE ONE SIDED LOWER LIMIT Y
LET A = NORMAL TOLERANCE ONE SIDED UPPER 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 ONE SIDED K FACTOR MEAN SD N

LET A = SUMMARY NORMAL TOLERANCE LOWER LIMIT MEAN SD N
LET A = SUMMARY NORMAL TOLERANCE UPPER LIMIT MEAN SD N
LET A = SUMMARY NORMAL TOLERANCE ONE SIDED LOWER LIMIT ...
MEAN SD N
LET A = SUMMARY NORMAL TOLERANCE ONE SIDED UPPER 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 ALPHA = <value>
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, built-in statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).

Default:
None
Synonyms:
None
Related Commands:
 CONFIDENCE LIMITS = Generate the confidence limits for the mean. PREDICTION LIMITS = Generate prediction limits for the mean. MAXIMUM LIKELIHOOD = Generate maximum likelihood estimates for a distributional fit. T-TEST = Perform a t-test.
Reference:
"Experimental Statistics: NBS Handbook 91", Natrella, National Institute of Standards and Technology (formerly National Bureau of Standards), October, 1966, pp. 2-13 - 2-15.

Hahn and Meeker (1991), "Statistical Intervals: A Guide for Practitioners", Wiley.

Applications:
Confirmatory Data Analysis
Implementation Date:
1998/12
2006/3: Allow only the normal or only the non-parametric limits to be generated
2014/06: Support for LOGNORMAL and BOXCOX tolerance limits
Program:

SKIP 25
SET WRITE DECIMALS 4
TOLERANCE LIMITS Y

The following output is generated:
             Two-Sided 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

Two-Sided Distribution-Free 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(N-2) =    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(N-1) =    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
Last updated: 02/10/2015