
COMMON 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 a single sample of normally distributed data, the command
can be used to determine a confidence interval (this command supports six different methods for obtaining the confidence interval). The command documented here is for the case where there are two or more groups of data and we want to find a confidence interval for the common coefficient of variation for all groups. This command assumes that the data for each group is approximately normally distributed and that each group has the same coefficient of variation. For that reason, it is desirable to plot the coefficient of variation for each group (see the program example below) to check if this is a reasonable assumption. The biased common coefficient of variation is
where
A bias correccted estimate of the common coefficient of variation is
The confidence limits are computed as
Upper Limit = \( \sqrt{ \frac{\sum_{i}^{k}{(N_{i}  1) u_{i}}} {\chi_{\alpha/2}^{2}  \sum_{i}^{k}{(N_{i}  1) u_{i}}} } \) where
This confidence interval was proposed by Forkman (see References below).
LIMITS <y> <x> <SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; <x> is the groupid 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 does not support matrix arguments for the response and groupid variables.
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 can be used if the groups are in separate variables. You can 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. This syntax supports matrix arguments for the response variables.
COMMON COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y1 X1 ... SUBSET X1 > 2 MULTIPLE COMMON COEFFICIENT OF VARIATION CONFIDENCE LIMITS ... Y1 TO Y5
LET A = COMMON COEFFICIENT OF VARIATION Y X The LET ALPHA command specifies the significance level. In addition to the above LET commands, builtin statistics are supported for 20+ different commands (enter HELP STATISTICS for details).
McKay (1932), "Distributions of the Coefficient of Variation and the Extended 't' Distribution", Journal of the Royal Statistical Society, Vol. 95, pp. 695698.
. Step 1: Create the data . skip 25 read gear.dat y x skip 0 set write decimals 6 . . Step 2: Define plot control settings and plot the data . title case asis title offset 2 label case asis . y1label Coefficient of Variation x1label Group title Coefficient of Variation for GEAR.DAT let ngroup = unique x xlimits 1 ngroup major x1tic mark number ngroup minor x1tic mark number 0 tic mark offset units data x1tic mark offset 0.5 0.5 y1tic mark label decimals 3 . character X line blank . set statistic plot reference line average . coefficient of variation plot y x . . Step 3: Define plot control settings and plot the data . common coefficient of variation confidence limits y x TwoSided Confidence Limits for the Common Coefficient of Variation for Normally Distributed Data Method: Forkman Response Variable: Y GroupID Variable: X Number of Distinct Groups: 10 Number of Groups Included in Analysis: 10 Common Coefficient of Variation: 0.005953 Bias Corrected Estimate: 0.005969  Confidence Coefficient Lower Upper Value (%) of Variation Limit Limit  50.0 0.005953 0.005686 0.006289 80.0 0.005953 0.005445 0.006596 90.0 0.005953 0.005309 0.006792 95.0 0.005953 0.005196 0.006970 99.0 0.005953 0.004985 0.007340 99.9 0.005953 0.004759 0.007810  
Date created: 06/23/2017 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 