ROBUST POOLED STANDARD DEVIATION

ROBUST POOLED STANDARD DEVIATION (LET)

Let Subcommand

Compute the robust pooled standard deviation of a variable.

The definition for the robust pooled standard deviation used here is from the ISO 13528 standard. It is computed as follows.

Let w₁, w₂, ... , w_p be the standard deviations from the p labs. The degrees of freedom for w_i = n_i - 1 where n_i is the number of replications for lab i.

Compute the limit factor, \( \eta \),

\( \eta = \sqrt{\frac{\chi_{\nu}^{2}(0.9)} {\nu}} \)

with \( \chi_{\nu}^{2} \) denoting the percent point function of the chi-square distribution

and the adjustment factor, \( \xi \),

\( \xi = \frac{1} {\sqrt{\chi_{\nu+2}^{2}(\nu \eta^2) + 0.1 \eta^2}} \)

with \( \chi_{\nu}^{2} \) denoting the cumulative distribution function of the chi-square distribution.

The limit factor and adjustment factor assume that each lab has the same number of replications. If the number of replications are not equal, Dataplot will use the average number of replications.

The initual value of w* is set to the median of the w_i's.

To update the value of w* compute

\( \psi = \eta w^{*} \)

For each w_i,

\( \begin{array}{lcll} w_{i}^{*} & = & \psi \hspace{10pt} & \mbox{if } w_{i} > \psi \\ & = & w_{i} \hspace{10pt} & \mbox{if } w_{i} \le \psi \end{array} \)

The updated value of w* is

\( w^{*} = \xi \sqrt{\sum_{i=1}^{p}{w_{i}^2}/p} \)

The value of w* is iterated until the difference between two successive values of w* is sufficiently small.

The ISO 13528 standard also allows this computation to be performed on the lab ranges. In this case, the robust pooled range uses the same computation as above with the exception that the degrees of freedom, \( \nu \), is set to 1.

The input for this command is the sample standard deviations. To specify the sample size, enter

LET NREPL = <value>

LET <par> = ROBUST POOLED STANDARD DEVIATION <ysd> <x>
                        <SUBSET/EXCEPT/FOR qualification>
where <ysd> is the variable containing the sample standard deviations;
            <x> is the lab-id variable;
            <par> is a parameter where the computed robust pooled sd is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = ROBUST POOLED RANGE <yrange> <x>
                        <SUBSET/EXCEPT/FOR qualification>
where <yrange> is the variable containing the sample ranges;
            <x> is the lab-id variable;
            <par> is a parameter where the computed robust pooled range is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET NREPL = 10
LET SD = ROBUST POOLED STANDARD DEVIATION YSD
LET SD = ROBUST POOLED STANDARD DEVIATION YSD ...
                  SUBSET TAG > 2
LET SD = ROBUST POOLED RANGE YRANGE

Dataplot statistics can be used in a number of commands. For details, enter
HELP STATISTICS

None

ROBUST POOLED SD is a synonym for ROBUST POOLED STANDARD DEVIATION

MEAN	= Compute the mean of a variable.
MEDIAN	= Compute the median of a variable.
RANGE	= Compute the range of a variable.
STANDARD DEVIATION	= Compute the standard deviation of a variable.
H15 SCALE	= Compute the H15 scale estimate of a variable.
WEIGHTED STANDARD DEVIATION	= Compute the weighted standard deviation of a variable.
HOMOSCEDASTICITY PLOT	= Compute the standard deviation of a variable.

Data Analysis, Proficiency Analysis

ISO 13528 (2005), "Statistical Methods for use in proficiency testing by interlaboratory comparisons," Section C.2 Algorithm S.

2010/12

 
SKIP 25
READ GEAR.DAT Y X
.
SET LET CROSS TABULATE COLLAPSE
LET YSD = CROSS TABULATE SD Y X
LET NSIZE = CROSS TABULATE SIZE X
LET NREPL = MEAN NSIZE
.
LET A = ROBUST POOLED STANDARD DEVIATION YSD
    

The returned value is 0.5335508E-02.