
SD CONFIDENCE LIMITSName:
\( \mbox{upper confidence limit} = s \sqrt{\frac{n1}{\chi^{2}_{(\alpha/2;n1)}}} \) with \( \chi^{2} \) denoting the percent point function of the chisquare distribution. In these formulas, \( \alpha \) is less than 0.5 (i.e., for a 95% confidence interval, we are using \( \alpha \) = 0.05). The onesided lower confidence limit is
The onesided upper confidence limit is
This confidence interval is based on the assumption that the underlying data is approximately normally distributed. The confidence interval for the standard deviation is highly sensitive to nonnormality in the data. It is recommended that the original data be tested for normality before using these normal based intervals. If the data is not approximately normal, an alternative is to use the command
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional. If LOWER is specified, a onesided lower confidence limit is returned. If UPPER is specified, a onesided upper confidence limit is returned. If neither is specified, a twosided limit is returned. This syntax supports matrix arguments for the response variable.
<SUBSET/EXCEPT/FOR qualification> where <y1> .... <yk> is a list of 1 to 30 response variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax will generate a confidence interval for each of the response variables. The word MULTIPLOT is optional. That is,
is equivalent to
If LOWER is specified, a onesided lower confidence limit is returned. If UPPER is specified, a onesided upper confidence limit is returned. If neither is specified, a twosided limit is returned. This syntax supports matrix arguments for the response variables.
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; <x1> .... <xk> is a list of 1 to 6 groupid variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax performs a crosstabulation of the <x1> ... <xk> and generates a confidence interval for each unique combination of the crosstabulated values. For example, if X1 has 3 levels and X2 has 2 levels, six confidence intervals will be generated. If LOWER is specified, a onesided lower confidence limit is returned. If UPPER is specified, a onesided upper confidence limit is returned. If neither is specified, a twosided limit is returned. This syntax does not support matrix arguments.
SD CONFIDENCE LIMITS Y1 SUBSET TAG > 2 SD CONFIDENCE LIMITS Y1 TO Y5 REPLICATED SD CONFIDENCE LIMITS Y X
where
The use of the trimmed mean reduces the bias of the kurtosis estimate for heavy tailed and skewed data. The justification and derivation of this interval is given in Bonett's paper. To request that the Bonett interval be generated, enter
Based on simulation studies by Bonett, this interval results in greatly improved coverage properties for moderately nonnormal data. For more extreme nonnormality, large sample sizes may be required to obtain good coverage properties. Often a transformation to reduce skewness (which may reduce the heavy tailedness as well), such as the LOG or square root, can significantly reduce the sample size required to obtain good coverage properties. Bonett also suggests that improved estimates for the kurtosis can significantly improve the coverage properties. Bonett's example is quality control applications where much historical data is frequently available. If a prior estimate of kurtosis is available, then the above formula pools this prior estimate with the kurtosis estimate from the data using
with \( \tilde{\gamma}_{4} \) and \( n_{0} \) denoting the prior estimate of kurtosis and the associated sample size, respectively. Bonett gives guidance on pooling multiple estimates of kurtosis based on several small samples (it is often the case in quality control applications that a large number of small samples are available). In Dataplot, you can specify a prior estimate of kurtosis by entering the commands
LET N0 = <value> Niwitpong and Kirdwichai (2008) suggested two modifications to Bonett's interval to improve the coverage for data that are skewed and heavy tailed. Specifically, the following modifications are made to Bonett's interval
This interval will be more conservative than the original Bonett data. Based on simulation studies in their paper, these adjustments can improve the nominal coverage for data that are skewed. However, it may be overly conservative for data that are only moderately nonnormal. To use the Niwitpong and Kirdwichai adjusted interval, enter
ADJUSTED ON
LET A = LOWER STANDARD DEVIATION CONFIDENCE LIMIT Y
LET A = SUMMARY LOWER STANDARD DEVIATION CONFIDENCE The first command specifies the significance level. The next six commands are used when you have raw data. The last four commands are used when only summary data ( standard deviation, sample size) is available. In addition to the above LET command, builtin statistics are supported for about 20 different commands (enter HELP STATISTICS for details).
SD CONFIDENCE LIMIT is a synonym for STANDARD DEVIATION CONFIDENCE LIMIT
Bonett (2006), "Approximate Confidence Interval for Standard Deviation of Nonnormal Distributions", Computational Statistics and Data Analysis, Vol. 50, pp. 775  782. Niwitpong and Kirdwichai (2008), "Adjusted Bonett Confidence Interval for Standard Deviation of NonNormal Distributions", Thailand Statistician, Vol. 6, No. 1, pp. 16.
2017/12: Support for Bonett intervals SKIP 25 READ ZARR13.DAT Y SET WRITE DECIMALS 5 . SD CONFIDENCE LIMITS Y LOWER SD CONFIDENCE LIMITS Y UPPER SD CONFIDENCE LIMITS YThe following output is generated TwoSided Confidence Limits for the SD Response Variable: Y Summary Statistics: Number of Observations: 195 Sample Mean: 9.26146 Sample Standard Deviation: 0.02278 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.02206 0.02363 80.0 0.02141 0.02440 90.0 0.02104 0.02487 95.0 0.02072 0.02530 99.0 0.02013 0.02617 99.9 0.01948 0.02725 OneSided Lower Confidence Limits for the SD Response Variable: Y Summary Statistics: Number of Observations: 195 Sample Mean: 9.26146 Sample Standard Deviation: 0.02278 OneSided Lower Confidence Limits for the SD  Confidence Lower Value (%) Limit  50.0 0.02282 80.0 0.02188 90.0 0.02141 95.0 0.02104 99.0 0.02037 99.9 0.01966 OneSided Upper Confidence Limits for the SD Response Variable: Y Summary Statistics: Number of Observations: 195 Sample Mean: 9.26146 Sample Standard Deviation: 0.02278 OneSided Upper Confidence Limits for the SD  Confidence Upper Value (%) Limit  50.0 0.02282 80.0 0.02384 90.0 0.02440 95.0 0.02487 99.0 0.02581 99.9 0.02694Program 2: SKIP 25 READ GEAR.DAT Y X . SET WRITE DECIMALS 5 REPLICATED SD CONFIDENCE LIMITS Y XThe following output is generated TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 1.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99800 Sample Standard Deviation: 0.00435 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00386 0.00537 80.0 0.00340 0.00639 90.0 0.00317 0.00715 95.0 0.00299 0.00793 99.0 0.00268 0.00990 99.9 0.00239 0.01323 TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 2.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99910 Sample Standard Deviation: 0.00522 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00464 0.00644 80.0 0.00408 0.00767 90.0 0.00380 0.00858 95.0 0.00359 0.00952 99.0 0.00322 0.01188 99.9 0.00287 0.01588 TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 3.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99540 Sample Standard Deviation: 0.00398 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00354 0.00491 80.0 0.00311 0.00584 90.0 0.00290 0.00654 95.0 0.00274 0.00726 99.0 0.00246 0.00906 99.9 0.00219 0.01211 TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 4.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99820 Sample Standard Deviation: 0.00385 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00343 0.00476 80.0 0.00302 0.00566 90.0 0.00281 0.00634 95.0 0.00265 0.00703 99.0 0.00238 0.00878 99.9 0.00212 0.01173 TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 5.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99190 Sample Standard Deviation: 0.00758 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00674 0.00936 80.0 0.00593 0.01114 90.0 0.00553 0.01247 95.0 0.00521 0.01384 99.0 0.00468 0.01726 99.9 0.00417 0.02306 TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 6.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99880 Sample Standard Deviation: 0.00989 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00879 0.01221 80.0 0.00774 0.01453 90.0 0.00721 0.01626 95.0 0.00680 0.01805 99.0 0.00611 0.02252 99.9 0.00545 0.03009 TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 7.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 1.00150 Sample Standard Deviation: 0.00788 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00700 0.00973 80.0 0.00617 0.01158 90.0 0.00575 0.01296 95.0 0.00542 0.01438 99.0 0.00487 0.01794 99.9 0.00434 0.02397 TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 8.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 1.00040 Sample Standard Deviation: 0.00363 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00322 0.00448 80.0 0.00284 0.00533 90.0 0.00265 0.00597 95.0 0.00249 0.00662 99.0 0.00224 0.00826 99.9 0.00200 0.01104 TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 9.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99830 Sample Standard Deviation: 0.00414 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00368 0.00511 80.0 0.00324 0.00608 90.0 0.00302 0.00681 95.0 0.00285 0.00755 99.0 0.00256 0.00942 99.9 0.00228 0.01259 TwoSided Confidence Limits for the SD Response Variable: Y Factor Variable 1: X 10.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99480 Sample Standard Deviation: 0.00533 TwoSided Confidence Limits for the SD  Confidence Lower Upper Value (%) Limit Limit  50.0 0.00474 0.00658 80.0 0.00417 0.00783 90.0 0.00389 0.00877 95.0 0.00367 0.00973 99.0 0.00329 0.01214 99.9 0.00294 0.01622Program 3: . Following example from Hahn and Meeker's book. . let ymean = 50.10 let ysd = 1.31 let n1 = 5 let alpha = 0.05 . set write decimals 5 let slow1 = summary lower sd confidence limits ysd n1 let supp1 = summary upper sd confidence limits ysd n1 let slow2 = summary one sided lower sd confidence limits ysd n1 let supp2 = summary one sided upper sd confidence limits ysd n1 print slow1 supp1 slow2 supp2The following output is generated. PARAMETERS AND CONSTANTS SLOW1  0.78486 SUPP1  3.76436 SLOW2  0.85059 SUPP2  3.10779Program 3: . Step 1: Read the data (example from Bonett paper) . let y = data 15.83 16.01 16.24 16.42 15.33 15.44 16.88 16.31 . . Step 2: Compute the statistics . set write decimals 4 let ysd = standard deviation y let lcl = lower bonett standard deviation confidence limit y let ucl = upper bonett standard deviation confidence limit y print ysd lcl ucl . set bonett standard deviation confidence limit on standard deviation confidence limits yThe following output is generated PARAMETERS AND CONSTANTS YSD  0.5168 LCL  0.3263 UCL  1.0841 TwoSided Confidence Limits for the SD Response Variable: Y Summary Statistics: Number of Observations: 8 Sample Mean: 16.0575 Sample Standard Deviation: 0.5168  Confidence Standard Lower Upper Value (%) Deviation Limit Limit  50.0 0.5168 0.4548 0.6629 80.0 0.5168 0.3944 0.8123 90.0 0.5168 0.3646 0.9288 95.0 0.5168 0.3417 1.0518 99.0 0.5168 0.3036 1.3747 99.9 0.5168 0.2681 1.9636 TwoSided Confidence Limits for the SD Bonett Interval for NonNormality Response Variable: Y Summary Statistics: Number of Observations: 8 Sample Mean: 16.0575 Sample Standard Deviation: 0.5168  Confidence Standard Lower Upper Value (%) Deviation Limit Limit  50.0 0.5168 0.4555 0.6404 80.0 0.5168 0.3963 0.8026 90.0 0.5168 0.3592 0.9359 95.0 0.5168 0.3263 1.0841 99.0 0.5168 0.2607 1.5109 99.9 0.5168 0.1849 2.4533  
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Date created: 04/15/2013 