
RATIO OF STANDARD DEVIATIONS CONFIDENCE LIMITS
Name:

\( \gamma_{4} \)  =  \( (n_1 + n_2) \frac{ \sum_{i=1}^{2}{\sum_{j=1}^{n_i}{(y_{ij}  m_{i})^{4}}}} {(\sum_{i=1}^{2}{\sum_{j=1}^{n_i}{(y_{ij}  \mu_{i})^{2}}} )^{2}} \) 
\( m_{i} \)  =  sample trimmed mean of the ith variable with trimming proportion \( \frac{1}{\sqrt{2 (n{i}  4)}} \) 
\( \bar{y_{i}} \)  =  sample mean of the ith variable 
se  =  \( \sqrt{ \frac{\gamma_{4}  k_{1}} {n_{1}  1} + \frac{\gamma_{4}  k_{2}} {n_{2}  1} } \) 
\( k_1 \)  =  \( \frac{n_{1}3}{n_{1}} \) 
\( k_2 \)  =  \( \frac{n_{2}3}{n_{2}} \) 
c  =  \( \frac{\frac{n_{1}} {n_{1}  \Phi^{1}(\alpha/2)}} {\frac{n_{2}} {n_{2}  \Phi^{1}(\alpha/2)}} \) 
\( \Phi^{1} \)  =  the percent point function of the normal distribution 
The parameter c is a small sample adjustment to help equalize the tail probabilities. It equals 1 when n1 and n2 equal and approaches 1 as n1 and n2 get large.
The above formula is for the variance. For the standard deviation, take the square root.
This interval is based on a standard normal distribution rather than an F distribution.
The justification and derivation of this test is given in the Bonett paper. Based on his simulations, he claims this method improves somewhat on the Shoemaker modification for heavytailed (particularly skewed) distributions.
To specify Shoemaker's modification with fractional degrees of freedom, enter
To specify Shoemaker's modification with integral degrees of freedom as suggested by Bonett, enter
To specify Bonett's method, enter
\( \sigma^{2} = \frac{(n_{1}  1) s_{1}^{2} + (n_{2}  1) s_{2}^{2} + ... + (n_{k}  1) s_{k}} {n_{1} + n_{2} + ... + n_{k}} \)
To use these values, enter the commands
To reset the default of Dataplot computing these from the two groups being tested, enter
CUTLOW90    lower 90% confidence limit 
CUTUPP90    upper 90% confidence limit 
CUTLOW95    lower 95% confidence limit 
CUTUPP95    upper 95% confidence limit 
CUTLOW99    lower 99% confidence limit 
CUTUPP99    upper 99% confidence limit 
CUTLW999    lower 99.5% confidence limit 
CUTUP999    upper 99.5% confidence limit 
In addition to the above LET commands, builtin statistics are supported for 30+ different commands (enter HELP STATISTICS for details).
F TEST  =  Perform a normalbased test that two standard deviations are equal. 
SIEGEL TUKEY TEST  =  Perform a ranks based test that two standard deviations are equal. 
BARTLETT TEST  =  Perform a ksample Bartlett test for homogeneous variances. 
LEVENE TEST  =  Perform a ksample Levene test for homogeneous variances. 
SQUARED RANKS TEST  =  Perform a ksample squared ranks test for homogeneous variances. 
KLOTZ TEST  =  Perform a ksample Klotz test for homogeneous variances. 
SD CONFIDENCE LIMITS  =  Compute the confidence limits for the standard deviation. 
CHISQUARE TEST  =  Performs a one sample chisquare test that the standard deviation is equal to a given value. 
T TEST  =  Performs a twosample t test for equal means. 
CONFIDENCE LIMITS  =  Compute the confidence limits for the mean of a sample. 
STANDARD DEVIATION  =  Computes the standard deviation of a variable. 
Shoemaker (2003), "Fixing the FTest for Equal Variances," The American Statistician, Vol. 57, pp. 105114.
Bonett (2006), "Robust Confidence Intervals for a Ratio of Standard Deviations," Applied Pyschological Measurement, Vol. 30, No. 5, pp. 432439.
Program 2:SKIP 25 READ AUTO83B.DAT Y1 Y2 DELETE Y2 SUBSET Y2 < 0 . RATIO OF SD CONFIDENCE LIMITS Y1 Y2The following output is generatedConfidence Limits for the Ratio of Standard Deviations Numerator Variable: Y1 Denominator Variable: Y2 Summary Statistics for Numerator Variable: Number of Observations: 249 Sample Mean: 20.14458 Sample Standard Deviation: 6.41470 Summary Statistics for Variable 2: Number of Observations: 79 Sample Mean: 30.48101 Sample Standard Deviation: 6.10771 Degrees of Freedom (Numerator): 248 Degrees of Freedom (Denomerator): 78 Ratio of Standard Deviations: 1.05026  Confidence Lower Upper Value (%) Ratio Limit Limit  50.000 1.050 0.99058 1.12161 80.000 1.050 0.93827 1.18830 90.000 1.050 0.90869 1.23086 95.000 1.050 0.88401 1.26950 99.000 1.050 0.83820 1.35006 99.900 1.050 0.78873 1.45285
. Step 1: Read the data . skip 25 read shoemake.dat bottom mid surface let y x = stack bottom mid surface let temp1 = cross tabulate mean y x let temp2 = (y  temp1)**4 let num = sum temp2 let den = size y let mu4 = num/den let y1var = variance bottom let y2var = variance mid let y3var = variance surface let n1 = size bottom let n2 = size mid let n3 = size surface let num = (n1  1)*y1var + (n2 1)*y2var + (n3  1)*y3var let poolvar = num/den . . Step 2: Generate a box plot . character box plot line box plot xlimits 1 3 major xtic mark number 3 minor xtic mark number 0 xtic mark offset 0.5 0.5 x1tic mark label format alpha x1tic mark label content Bottom Middepth Surface tic mark label case asis set box plot fences on box plot y x
. . Step 3: Default confidence intercal . set f test degrees of freedom default print "Standard F interval" ratio of sd confidence limits bottom mid let lcl = ratio of sd lower confidence limit bottom mid let ucl = ratio of sd upper confidence limit bottom mid print lcl uclThe following output is generated
Standard F interval Confidence Limits for the Ratio of Standard Deviations Numerator Variable: BOTTOM Denominator Variable: MID Summary Statistics for Numerator Variable: Number of Observations: 10 Sample Mean: 6.02100 Sample Standard Deviation: 1.58184 Summary Statistics for Variable 2: Number of Observations: 10 Sample Mean: 5.01900 Sample Standard Deviation: 1.10440 Degrees of Freedom (Numerator): 9 Degrees of Freedom (Denomerator): 9 Ratio of Standard Deviations: 1.43231  Confidence Lower Upper Value (%) Ratio Limit Limit  50.000 1.432 1.13558 1.80659 80.000 1.432 0.91688 2.23750 90.000 1.432 0.80334 2.55373 95.000 1.432 0.71384 2.87392 99.000 1.432 0.56003 3.66322 99.900 1.432 0.41217 4.97734 THE COMPUTED VALUE OF THE CONSTANT LCL = 0.7138401 THE COMPUTED VALUE OF THE CONSTANT UCL = 2.873916 PARAMETERS AND CONSTANTS LCL  0.71384 UCL  2.87392
. . Step 4: Shoemaker Ftest . set f test degrees of freedom shoemaker print "Shoemaker degrees of freedom interval" ratio of sd confidence limits bottom mid print "Use pooled location and variance for Shoemaker" set shoemaker f test pooled variance poolvar set shoemaker f test pooled mu mu4 ratio of sd confidence limits bottom mid print "Shoemaker rounded degrees of freedom F Test" set f test degrees of freedom shoemaker rounded ratio of sd confidence limits bottom mid set shoemaker f test pooled variance 0 set shoemaker f test pooled mu The following output is generatedShoemaker degrees of freedom interval Confidence Limits for the Ratio of Standard Deviations (Use Shoemaker Modifications to Degrees of Freedom) Numerator Variable: BOTTOM Denominator Variable: MID Summary Statistics for Numerator Variable: Number of Observations: 10 Sample Mean: 6.02100 Sample Standard Deviation: 1.58184 Summary Statistics for Variable 2: Number of Observations: 10 Sample Mean: 5.01900 Sample Standard Deviation: 1.10440 Degrees of Freedom (Numerator): 12.42315 Degrees of Freedom (Denomerator): 12.42315 Ratio of Standard Deviations: 1.43231  Confidence Lower Upper Value (%) Ratio Limit Limit  50.000 1.432 1.17755 1.74220 80.000 1.432 0.98426 2.08433 90.000 1.432 0.88188 2.32631 95.000 1.432 0.80006 2.56420 99.000 1.432 0.65671 3.12394 99.900 1.432 0.51437 3.98837 Use pooled location and variance for Shoemaker THE FORTRAN COMMON SCALAR SHOEF HAS JUST BEEN SET TO 0.1251562E+01 THE FORTRAN COMMON SCALAR SHOEF HAS JUST BEEN SET TO 0.4585221E+01 Confidence Limits for the Ratio of Standard Deviations (Use Shoemaker Modifications to Degrees of Freedom) Numerator Variable: BOTTOM Denominator Variable: MID Summary Statistics for Numerator Variable: Number of Observations: 10 Sample Mean: 6.02100 Sample Standard Deviation: 1.58184 Summary Statistics for Variable 2: Number of Observations: 10 Sample Mean: 5.01900 Sample Standard Deviation: 1.10440 Degrees of Freedom (Numerator): 9.30474 Degrees of Freedom (Denomerator): 9.30474 Ratio of Standard Deviations: 1.43231  Confidence Lower Upper Value (%) Ratio Limit Limit  50.000 1.432 1.14021 1.79924 80.000 1.432 0.92423 2.21969 90.000 1.432 0.81186 2.52695 95.000 1.432 0.72313 2.83698 99.000 1.432 0.57032 3.59714 99.900 1.432 0.42285 4.85169 Shoemaker rounded degrees of freedom F Test THE FORTRAN COMMON CHARACTER VARIABLE F TEST HAS JUST BEEN SET TO SHO2 Confidence Limits for the Ratio of Standard Deviations (Shoemaker Modifications with Rounded Degrees of Freedom) Numerator Variable: BOTTOM Denominator Variable: MID Summary Statistics for Numerator Variable: Number of Observations: 10 Sample Mean: 6.02100 Sample Standard Deviation: 1.58184 Summary Statistics for Variable 2: Number of Observations: 10 Sample Mean: 5.01900 Sample Standard Deviation: 1.10440 Degrees of Freedom (Numerator): 10 Degrees of Freedom (Denomerator): 9 Ratio of Standard Deviations: 1.43231  Confidence Lower Upper Value (%) Ratio Limit Limit  50.000 1.432 1.31822 3.25440 80.000 1.432 0.87399 4.95711 90.000 1.432 0.67922 6.43618 95.000 1.432 0.54288 8.13193 99.000 1.432 0.34378 13.16491 99.900 1.432 0.19419 24.23791 THE FORTRAN COMMON SCALAR SHOEF HAS JUST BEEN SET TO 0.0000000E+00 THE FORTRAN COMMON SCALAR SHOEF HAS JUST BEEN SET TO 0.1000000E+01. . Step 5: Bonett . set f test degrees of freedom bonett print "Bonett method" ratio of sd confidence limits bottom mid The following output is generatedBonett method Confidence Limits for the Ratio of Standard Deviations (Bonett Method for Robustness) Numerator Variable: BOTTOM Denominator Variable: MID Summary Statistics for Numerator Variable: Number of Observations: 10 Sample Mean: 6.02100 Sample Standard Deviation: 1.58184 Summary Statistics for Variable 2: Number of Observations: 10 Sample Mean: 5.01900 Sample Standard Deviation: 1.10440 Ratio of Standard Deviations: 1.43231  Confidence Lower Upper Value (%) Ratio Limit Limit  50.000 1.432 1.28002 1.56990 80.000 1.432 1.12547 1.68414 90.000 1.432 1.02186 1.74895 95.000 1.432 0.92262 1.80328 99.000 1.432 0.68853 1.90498 99.900 1.432 0.19075 2.01659