 Dataplot Vol 1 Vol 2

# COEFFICIENT OF VARIATION CONFIDENCE LIMITS

Name:
COEFFICIENT OF VARIATION CONFIDENCE LIMITS
Type:
Analysis Command
Purpose:
Generates a confidence interval for the coefficient of variation for normally distributed data.
Description:
The coefficient of variation is defined as the ratio of the standard deviation to the mean

$$\mbox{cv} = \frac{\sigma} {\mu}$$

where $$\sigma$$ and $$\mu$$ denote the population standard deviation and population mean, respectively. The sample coefficient of variation is defined as

$$\hat{\mbox{cv}} = \frac{s} {\bar{x}}$$

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 non-negative 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 chi-square percent point function and $$\alpha$$ denotes the significance level.

The supported methods are

1. The "naive" method is based on dividing the standard confidence limit for the standard deviation by the sample mean.

$\mbox{lcl} = \frac{s}{\bar{x}} \sqrt{\frac{n-1}{\chi^{2}_{(1-\alpha/2;n-1)}}}$

$\mbox{ucl} = \frac{s}{\bar{x}} \sqrt{\frac{n-1}{\chi^{2}_{(\alpha/2;n-1)}}}$

This method is generally not as accurate as the McKay and modified McKay methods. However, it is sometimes used in practice.

This method can be specified with the command

SET COEFFICIENT OF VARIATION CONFIDENCE LIMIT METHOD ...             NAIVE

2. The McKay confidence limit is

$\mbox{lcl} = \frac{K} {\sqrt{\left( \frac{u_1}{n} - 1 \right) K^2 + \frac{u_1}{n-1}}}$

$\mbox{ucl} = \frac{K} {\sqrt{\left( \frac{u_2}{n} - 1 \right) K^2 + \frac{u_2}{n-1}}}$

where

$u_1 = \chi^2_{1 - \alpha/2,n-1}$

$u_2 = \chi^2_{\alpha/2,n-1}$

McKay's approximation is asymptotically exact as n goes to infinity. McKay recommends this approximation only if the coefficient of variation is less than 0.33. Note that if the coefficient of variation is greater than 0.33, either the normality of the data is suspect or the probability of negative values in the data is non-neglible. In this case, McKay's approximation may not be valid. Also, it is generally recommended that the sample size should be at least 10 before using McKay's approximation.

This method can be specified with the command

SET COEFFICIENT OF VARIATION CONFIDENCE LIMIT METHOD ...             MCKAY

3. Vangel proposed the following modification to McKay's method.

$\mbox{lcl} = \frac{K} {\sqrt{\left( \frac{u_1 + 2}{n} - 1 \right) K^2 + \frac{u_1}{n-1}}}$

$\mbox{ucl} = \frac{K} {\sqrt{\left( \frac{u_2 + 2}{n} - 1 \right) K^2 + \frac{u_2}{n-1}}}$

where

$u_1 = \chi^2_{1 - \alpha/2,n-1}$

$u_2 = \chi^2_{\alpha/2,n-1}$

Vangel's modified McKay method is more accurate than the McKay in most cases, particilarly for small samples.. According to Vangel, the unmodified McKay is only more accurate when both the coefficient of variation and alpha are large. However, if the coefficient of variation is large, then this implies either that the data contains negative values or the data does not follow a normal distribution. In this case, neither the McKay or the modified McKay should be used.

In general, the Vangel's modified McKay method is recommended over the McKay method. It generally provides good approximations as long as the data is approximately normal and the coefficient of variation is less than 0.33.

This is the default method.

This method can be specified with the command

SET COEFFICIENT OF VARIATION CONFIDENCE LIMIT METHOD ...             VANGEL

4. Panichkitkitkosolkul proposed the maximum likelihood based method. This uses Vangel's formula. However, it replaces the sample coefficient of variation with the maximum likelihood estimate of the coefficient of variation. Specifically,

$K = \frac{\sqrt{\sum_{i=1}^{n}{(X_i - \bar{x})^2}}} {\sqrt{n} \bar{x}}$

That is, we use the n rathar than (n-1) in the denominator when computing the standard deviation.

As with the McKay and modified McKay methods, it is recommended that this method only be used when the coefficient of variation is less than 0.33 and the data are normally distributed.

This method can be specified with the command

SET COEFFICIENT OF VARIATION CONFIDENCE LIMIT METHOD ...
MAXIMUM LIKELIHOOD

5. McKay also provided an exact method. The exact method involves solving a non-linear equation with the non-central t distribution. The details are given in the McKay and Verrill papers. Verrill also provides a Fortran code for implementing the exact method. Dataplot's implementation is based on this code.

Verrill gives justification for preferring the exact method to the McKay and modified McKay approximations. If the coefficient of variation is greater than 0.33 the exact method is preferred to the McKay and modified McKay approximations. If the sample size is greater than 3,000, the exact method reverts to the Vangel modified McKay method.

This method can be specified with the command

SET COEFFICIENT OF VARIATION CONFIDENCE LIMIT METHOD ...             EXACT

6. Liu proposed a generalized confidence interval approach. The details for this method are given in her paper.

This method can be specified with the command

SET COEFFICIENT OF VARIATION CONFIDENCE LIMIT METHOD ...             GPQ

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

$\mbox{lcl} = \sqrt{\exp(a_L) - 1}$

$\mbox{ucl} = \sqrt{\exp(a_U) - 1}$

where

 $$a_L$$ = $$\frac{(n-1)S_{n}^{2}} {\chi_{n-1,1-\alpha/2}}$$ $$a_U$$ = $$\frac{(n-1)S_{n}^{2}} {\chi_{n-1,\alpha/2}}$$ $$S_{n}^2$$ = the variance of the log of the data

The derivation of this is given in the Koopmans, Owen and Rosenblatt and the Verrill papers.

Syntax 1:
<LOWER/UPPER> <LOGNORMAL> COEFFICIENT OF VARIATION
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 one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

If LOGNORMAL is specified, the formula for the log normal-based confidence limits are used. If LOGNORMAL is omitted, the formulas for the normal-based confidence limits are used.

This syntax supports matrix arguments for the response variable.

Syntax 2:
MULTIPLE <LOWER/UPPER> <LOGNORMAL> COEFFICIENT OF
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,

MULTIPLE COEFFICIENT OF VARIATION CONFIDENCE LIMITS ...
Y1 Y2 Y3

is equivalent to

COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y1 Y2 Y3

You can also use the TO syntax as in

COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y1 TO Y10

If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

If LOGNORMAL is specified, the formula for the log normal-based confidence limits are used. If LOGNORMAL is omitted, the formulas for the normal-based confidence limits are used.

This syntax supports matrix arguments for the response variables.

Syntax 3:
REPLICATED <LOWER/UPPER> <LOGNORMAL> COEFFICIENT OF
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 group-id variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax performs a cross-tabulation of the <x1> ... <xk> and generates a confidence interval for each unique combination of the cross-tabulated values. For example, if X1 has three levels and X2 has two levels, six confidence intervals will be generated.

If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

If LOGNORMAL is specified, the formula for the log normal-based confidence limits are used. If LOGNORMAL is omitted, the formulas for the normal-based confidence limits are used.

This syntax does not support matrix arguments.

Examples:
COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y1
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
Note:
A table of confidence limits is printed for alpha levels of 50.0, 80.0, 90.0, 95.0, 99.0, and 99.9. If the exact method is specified, the 99.9 level is omitted.
Note:
In addition to the COEFFICIENT OF VARIATION CONFIDENCE LIMIT command, the following commands can also be used:

LET ALPHA = 0.05

LET A = LOWER COEFFICIENT OF VARIATION CONFIDENCE LIMIT Y
LET A = UPPPER COEFFICIENT OF VARIATION CONFIDENCE LIMIT Y
LET A = LOWER ONESIDED COEFFICIENT OF VARIATION ...
CONFIDENCE LIMIT Y
LET A = UPPER ONESIDED COEFFICIENT OF VARIATION ...
CONFIDENCE LIMIT Y

LET A = SUMMARY LOWER COEFFICIENT OF VARIATION ...
CONFIDENCE LIMIT YMEAN YSD N
LET A = SUMMARY UPPPER COEFFICIENT OF VARIATION ...
CONFIDENCE LIMIT YMEAN YSD N

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, built-in statistics are supported for 20+ different commands (enter HELP STATISTICS for details).

Default:
None
Synonyms:
CONFIDENCE INTERVAL is a synonym for CONFIDENCE LIMITS
Related Commands:
 COEFFICIENT OF VARIATION = Compute the coefficient of variation. CONFIDENCE LIMITS = Generate a confidence limit for the mean. SD CONFIDENCE LIMITS = Generate a confidence limit for the standard deviation. PREDICTION LIMITS = Generate prediction limits for the mean. TOLERANCE LIMITS = Generate a tolerance limit.
References:
McKay (1932), "Distributions of the Coefficient of Variation and the Extended 't' Distribution", Journal of the Royal Statistical Society, Vol. 95, pp. 695-698.

Koopmans, Owen, and Rosenblatt (1964), "Confidence Intervals for the Coefficient of Variation for the Normal and Log Normal Distributions", Biometrika, 51, 1, pp. 25-31.

Mark Vangel (1996), "Confidence Intervals for a Normal Coefficient of Variation", American Statistician, Vol. 15, No. 1, pp. 21-26.

Panichkitkitkosolkul (2009), "Improved Confidence Intervals for a Coefficient of Variation of a Normal Distribution", Thailand Statistician, 7(2), pp. 193-199.

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 2187-2206.

Liu (2012), "Confidence Interval Estimation for Coefficient of Variation", Thesis, Georgia State University, http://scholarworks.gsu.edu/math_theses/124.

Applications:
Confirmatory Data Analysis
Implementation Date:
2017/01
Program 1:

SKIP 25
SET WRITE DECIMALS 5
.
COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y
LOWER COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y
UPPER COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y

The following output is generated
            Two-Sided 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

One-Sided 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

One-Sided 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        0.00291

Program 2:

. Step 1:   Create the data
.
let y = data 326 302 307 299 329
set write decimals 4
.
. Step 2:   Compute the built-in 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: One-Sided"
print cvlow9 cvupp9
print " "
.
set coefficient of variation confidence limit method vangel
coefficient of variation confidence limits y

The 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: One-Sided

PARAMETERS AND CONSTANTS--

CVLOW9  --         0.0289
CVUPP9  --         0.1061

Two-Sided 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         0.3664

Program 3:

SKIP 25
LET NNEW = 3
.
REPLICATED COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y X

The following output is generated
            Two-Sided 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

Two-Sided 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

Two-Sided 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

Two-Sided 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

Two-Sided 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

Two-Sided 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

Two-Sided 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

Two-Sided 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

Two-Sided 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

Two-Sided 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        0.01630

Program 4:

skip 25
.
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 y

The following output is generated
 PARAMETERS AND CONSTANTS--

LCV     --  0.1923724E+02
LCVL    --  0.5442615E+01
UCVL    --  0.5503915E+03

Two-Sided 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):                0.5866234E-01
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


NIST is an agency of the U.S. Commerce Department.

Date created: 02/01/2017
Last updated: 02/01/2017