
COEFFICIENT OF VARIATION CONFIDENCE LIMITSName:
where \( \sigma \) and \( \mu \) denote the population standard deviation and population mean, respectively. The sample coefficient of variation is defined as
where s and \( \bar{x} \) denote the sample standard deviation and sample mean respectively. The coefficient of variation should typically only be used for ratio data. That is, the data should be continuous and have a meaningful zero. Although the coefficient of variation statistic can be computed for data that is not on a ratio scale, the interpretation of the coeffcient of variation may not be meaningful. Currently, this command is only supported for nonnegative data. If the response variable contains one or more negative numbers, an error message will be returned. For normally distributed data, a number of methods for determining confidence intervals for the coefficient of variation have been proposed. Dataplot currently supports six different methods. In the following, \( \bar{x} \), s, n, and K denote the sample mean, sample standard deviation, sample size, and sample coefficient of variation, respectively. CHSPPF denotes the chisquare percent point function and \( \alpha \) denotes the significance level. The supported methods are
The default method is the Vangel method. For most applications, this choice should be reasonable as long as the data are approximately normally distributed. If the data follow a lognormal distribution, then a confidence interval for the coefficient of variation is
\[ \mbox{ucl} = \sqrt{\exp(a_U)  1} \] where
The derivation of this is given in the Koopmans, Owen and Rosenblatt and the Verrill papers.
CONFIDENCE LIMITS <y> <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. If LOGNORMAL is specified, the formula for the log normalbased confidence limits are used. If LOGNORMAL is omitted, the formulas for the normalbased confidence limits are used. This syntax supports matrix arguments for the response variable.
VARIATION CONFIDENCE LIMITS <y1> ... <yk> <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,
Y1 Y2 Y3 is equivalent to
You can also use the TO syntax as in
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. If LOGNORMAL is specified, the formula for the log normalbased confidence limits are used. If LOGNORMAL is omitted, the formulas for the normalbased confidence limits are used. This syntax supports matrix arguments for the response variables.
VARIATION CONFIDENCE LIMITS <y> <x1> ... <xk> <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 three levels and X2 has two 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. If LOGNORMAL is specified, the formula for the log normalbased confidence limits are used. If LOGNORMAL is omitted, the formulas for the normalbased confidence limits are used. This syntax does not support matrix arguments.
COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y1 SUBSET TAG > 2 MULTIPLE COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y1 TO Y5 REPLICATED COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y X
LET A = LOWER COEFFICIENT OF VARIATION CONFIDENCE LIMIT Y
LET A = SUMMARY LOWER COEFFICIENT OF VARIATION ... The first command specifies the significance level. The next four commands are used when you have raw data. The last two commands are used when only summary data (mean, standard deviation, sample size) is available. In addition to the above LET commands, builtin statistics are supported for 20+ different commands (enter HELP STATISTICS for details).
Koopmans, Owen, and Rosenblatt (1964), "Confidence Intervals for the Coefficient of Variation for the Normal and Log Normal Distributions", Biometrika, 51, 1, pp. 2531. Mark Vangel (1996), "Confidence Intervals for a Normal Coefficient of Variation", American Statistician, Vol. 15, No. 1, pp. 2126. Panichkitkitkosolkul (2009), "Improved Confidence Intervals for a Coefficient of Variation of a Normal Distribution", Thailand Statistician, 7(2), pp. 193199. Steve Verrill (2003), "Confidence Bounds for Normal and Lognormal Distribution Coefficients of Variation", Research Paper 609, USDA Forest Products Laboratory, Madison, Wisconsin. Verrill, S. and Johnson, R.A. (2007), "Confidence Bounds and Hypothesis Tests for Normal Distribution Coefficients of Variation", Communications in Statistics Theory and Methods, Volume 36, No. 12, pp 21872206. Liu (2012), "Confidence Interval Estimation for Coefficient of Variation", Thesis, Georgia State University, http://scholarworks.gsu.edu/math_theses/124.
SKIP 25 READ ZARR13.DAT Y SET WRITE DECIMALS 5 . COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y LOWER COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y UPPER COEFFICIENT OF VARIATION CONFIDENCE LIMITS YThe following output is generated TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Summary Statistics: Number of Observations: 195 Sample Mean: 9.26146 Sample Standard Deviation: 0.02279 Sample Coefficient of Variation: 0.00246  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00246 0.00238 0.00255 80.0 0.00246 0.00231 0.00263 90.0 0.00246 0.00227 0.00269 95.0 0.00246 0.00224 0.00273 99.0 0.00246 0.00217 0.00283 99.9 0.00246 0.00210 0.00294 OneSided Lower Confidence Limits for the Coefficient of Variation (for Normally Distributed Data) Method: Vangel (Modified McKay) Response Variable: Y Summary Statistics: Number of Observations: 195 Sample Mean: 9.26146 Sample Standard Deviation: 0.02279 Sample Coefficient of Variation: 0.00246  Confidence Coefficient Lower Value (%) of Variation Limit  50.0 0.00246 0.00246 80.0 0.00246 0.00236 90.0 0.00246 0.00231 95.0 0.00246 0.00227 99.0 0.00246 0.00220 99.9 0.00246 0.00212 OneSided Upper Confidence Limits for the Coefficient of Variation (for Normally Distributed Data) Method: Vangel (Modified McKay) Response Variable: Y Summary Statistics: Number of Observations: 195 Sample Mean: 9.26146 Sample Standard Deviation: 0.02279 Sample Coefficient of Variation: 0.00246  Confidence Coefficient Upper Value (%) of Variation Limit  50.0 0.00246 0.00246 80.0 0.00246 0.00257 90.0 0.00246 0.00263 95.0 0.00246 0.00269 99.0 0.00246 0.00279 99.9 0.00246 0.00291Program 2: . Step 1: Create the data . let y = data 326 302 307 299 329 set write decimals 4 . . Step 2: Compute the builtin intervals . let cvlow = lower coefficient of variation confidence limit y let cvupp = upper coefficient of variation confidence limit y set coefficient of variation confidence limit method mckay let cvlow2 = lower coefficient of variation confidence limit y let cvupp2 = upper coefficient of variation confidence limit y set coefficient of variation confidence limit method maximum likelihood let cvlow3 = lower coefficient of variation confidence limit y let cvupp3 = upper coefficient of variation confidence limit y set coefficient of variation confidence limit method naive let cvlow6 = lower coefficient of variation confidence limit y let cvupp6 = upper coefficient of variation confidence limit y set coefficient of variation confidence limit method gpq let cvlow7 = lower coefficient of variation confidence limit y let cvupp7 = upper coefficient of variation confidence limit y set coefficient of variation confidence limit method exact let cvlow8 = lower coefficient of variation confidence limit y let cvupp8 = upper coefficient of variation confidence limit y let cvlow9 = lower onesided coefficient of variation confidence limit y let cvupp9 = upper onesided coefficient of variation confidence limit y . print "Vangel Method" print cvlow cvupp print " " print "McKay Method" print cvlow2 cvupp2 print " " print "Maximum Likelihood Method" print cvlow3 cvupp3 print " " print "Naive Method" print cvlow6 cvupp6 print " " print "GPQ Method" print cvlow7 cvupp7 print " " print "Exact Method" print cvlow8 cvupp8 print " " . print "Exact Method: OneSided" print cvlow9 cvupp9 print " " . set coefficient of variation confidence limit method vangel coefficient of variation confidence limits yThe following output is generated Vangel Method PARAMETERS AND CONSTANTS CVLOW  0.0267 CVUPP  0.1287 McKay Method PARAMETERS AND CONSTANTS CVLOW2  0.0267 CVUPP2  0.1291 Maximum Likelihood Method PARAMETERS AND CONSTANTS CVLOW3  0.0239 CVUPP3  0.1150 Naive Method PARAMETERS AND CONSTANTS CVLOW6  0.0267 CVUPP6  0.1281 GPQ Method PARAMETERS AND CONSTANTS CVLOW7  0.0268 CVUPP7  0.1306 Exact Method PARAMETERS AND CONSTANTS CVLOW8  0.0267 CVUPP8  0.1287 Exact Method: OneSided PARAMETERS AND CONSTANTS CVLOW9  0.0289 CVUPP9  0.1061 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Summary Statistics: Number of Observations: 5 Sample Mean: 312.6000 Sample Standard Deviation: 13.9392 Sample Coefficient of Variation: 0.0446  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.0446 0.0384 0.0643 80.0 0.0446 0.0320 0.0866 90.0 0.0446 0.0289 0.1061 95.0 0.0446 0.0267 0.1287 99.0 0.0446 0.0231 0.1982 99.9 0.0446 0.0199 0.3664Program 3: SKIP 25 READ GEAR.DAT Y X LET NNEW = 3 . REPLICATED COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y XThe following output is generated TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 1.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99800 Sample Standard Deviation: 0.00435 Sample Coefficient of Variation: 0.00435  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00435 0.00387 0.00000 80.0 0.00435 0.00341 0.00640 90.0 0.00435 0.00318 0.00716 95.0 0.00435 0.00300 0.00795 99.0 0.00435 0.00269 0.00992 99.9 0.00435 0.00240 0.01325 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 2.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99910 Sample Standard Deviation: 0.00522 Sample Coefficient of Variation: 0.00522  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00522 0.00464 0.00000 80.0 0.00522 0.00409 0.00767 90.0 0.00522 0.00381 0.00859 95.0 0.00522 0.00359 0.00953 99.0 0.00522 0.00322 0.01189 99.9 0.00522 0.00288 0.01589 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 3.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99540 Sample Standard Deviation: 0.00398 Sample Coefficient of Variation: 0.00400  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00400 0.00355 0.00000 80.0 0.00400 0.00313 0.00587 90.0 0.00400 0.00291 0.00657 95.0 0.00400 0.00275 0.00730 99.0 0.00400 0.00247 0.00910 99.9 0.00400 0.00220 0.01216 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 4.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99820 Sample Standard Deviation: 0.00385 Sample Coefficient of Variation: 0.00386  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00386 0.00343 0.00000 80.0 0.00386 0.00302 0.00567 90.0 0.00386 0.00282 0.00635 95.0 0.00386 0.00265 0.00705 99.0 0.00386 0.00238 0.00879 99.9 0.00386 0.00213 0.01175 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 5.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99190 Sample Standard Deviation: 0.00758 Sample Coefficient of Variation: 0.00764  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00764 0.00679 0.00000 80.0 0.00764 0.00598 0.01123 90.0 0.00764 0.00557 0.01257 95.0 0.00764 0.00526 0.01395 99.0 0.00764 0.00472 0.01740 99.9 0.00764 0.00421 0.02326 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 6.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99880 Sample Standard Deviation: 0.00989 Sample Coefficient of Variation: 0.00990  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00990 0.00880 0.00000 80.0 0.00990 0.00775 0.01454 90.0 0.00990 0.00722 0.01629 95.0 0.00990 0.00681 0.01807 99.0 0.00990 0.00611 0.02255 99.9 0.00990 0.00545 0.03013 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 7.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 1.00150 Sample Standard Deviation: 0.00788 Sample Coefficient of Variation: 0.00787  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00787 0.00699 0.00000 80.0 0.00787 0.00616 0.01156 90.0 0.00787 0.00574 0.01294 95.0 0.00787 0.00541 0.01436 99.0 0.00787 0.00486 0.01792 99.9 0.00787 0.00433 0.02394 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 8.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 1.00040 Sample Standard Deviation: 0.00363 Sample Coefficient of Variation: 0.00363  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00363 0.00322 0.00000 80.0 0.00363 0.00284 0.00533 90.0 0.00363 0.00264 0.00596 95.0 0.00363 0.00249 0.00662 99.0 0.00363 0.00224 0.00826 99.9 0.00363 0.00200 0.01103 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 9.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99830 Sample Standard Deviation: 0.00414 Sample Coefficient of Variation: 0.00414  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00414 0.00368 0.00000 80.0 0.00414 0.00325 0.00609 90.0 0.00414 0.00302 0.00682 95.0 0.00414 0.00285 0.00757 99.0 0.00414 0.00256 0.00944 99.9 0.00414 0.00228 0.01262 TwoSided Confidence Limits for the Coefficient of Variation for Normally Distributed Data Method: Vangel (Modified McKay) Response Variable: Y Factor Variable 1: X 10.00000 Summary Statistics: Number of Observations: 10 Sample Mean: 0.99480 Sample Standard Deviation: 0.00533 Sample Coefficient of Variation: 0.00536  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.00536 0.00476 0.00000 80.0 0.00536 0.00419 0.00787 90.0 0.00536 0.00391 0.00881 95.0 0.00536 0.00368 0.00978 99.0 0.00536 0.00331 0.01220 99.9 0.00536 0.00295 0.01630Program 4: skip 25 read LGN.DAT . let lcv = lognormal coef of vari y let lcvl = lower lognormal coef of vari conf limit y let ucvl = upper lognormal coef of vari conf limit y . set write decimals 3 print lcv lcvl ucvl . lognormal coefficient of variation confidence limits yThe following output is generated PARAMETERS AND CONSTANTS LCV  0.1923724E+02 LCVL  0.5442615E+01 UCVL  0.5503915E+03 TwoSided Confidence Limits for the Coefficient of Variation for Lognormally Distributed Data Method: Koopmans, Owen, and Rosenblatt Response Variable: Y Summary Statistics: Number of Observations: 20 Sample Mean (Log of Data): 0.5866234E01 Sample Standard Deviation (Log of Data): 0.2432364E+01 Sample Coefficient of Variation: 0.1923724E+02  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.1923724E+02 0.1182862E+02 0.4744320E+02 80.0 0.1923724E+02 0.7830511E+01 0.1244767E+03 90.0 0.1923724E+02 0.6375431E+01 0.2586717E+03 95.0 0.1923724E+02 0.5442615E+01 0.5503915E+03 99.0 0.1923724E+02 0.4173985E+01 0.3686548E+04 99.9 0.1923724E+02 0.3245359E+01 0.9312940E+05  
Date created: 02/01/2017 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 