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XbarR Method¶

This page defines the XbarR method, which estimates the variance components from ranges rather than by fitting a model. It is the method the older published gage study procedures describe, so it is the one to use when a result has to match a worked example computed that way by hand.

Crossed studies only

The XbarR method is offered only for a Crossed study, and only when that study also meets the conditions below. It is refused for a Nested study and for an Extended study.

Notation¶

Term	Description
\(p\)	the number of parts
\(o\)	the number of operators
\(r\)	the number of replicates in each part and operator cell
\(y_{ijm}\)	replicate \(m\) of part \(i\) measured by operator \(j\)
\(R_{ij}\)	the range of the replicates in the part \(i\), operator \(j\) cell
\(\overline{\overline{R}}\)	the mean of all \(p \times o\) cell ranges
\(\bar{y}_i, \bar{y}_j\)	the mean of part \(i\), and the mean of operator \(j\)
\(R_{\text{oper}}\)	the range of the \(o\) operator means
\(d_2^{*}(n_{\text{sub}}, n)\)	the two-index range constant, defined below
\(\sigma_{\text{repeat}}, \sigma_{\text{reprod}}, \sigma_{\text{part}}\)	the estimated component standard deviations

Scope and requirements¶

The XbarR method estimates the variance components from ranges, without fitting a model and without an analysis of variance. It is available only for a crossed study, and the engine refuses it with a named reason for each condition it requires: a crossed study, exactly one part factor and one operator factor, both random rather than fixed, no additional factor, no nesting, balanced data, and at least two replicates in every cell.

Throughout this section \(p\) is the number of parts, \(o\) the number of operators, and \(r\) the number of replicates in each part and operator cell. Because the method requires balanced data, \(r\) is the same for every cell.

Repeatability¶

Each part and operator cell holds \(r\) replicate measurements. The cell range is the largest minus the smallest of them:

\[ R_{ij} = \max_{m} y_{ijm} - \min_{m} y_{ijm} \]

\(\overline{\overline{R}}\) is the mean of all \(p \times o\) cell ranges, and repeatability is that mean divided by a range constant:

\[ \sigma_{\text{repeat}} = \frac{\overline{\overline{R}}}{d_2^{*}(p\,o,\; r)} \]

The constant is indexed by both the number of ranges being averaged and the size of each range, which is why it is written \(d_2^{*}\) with two arguments rather than the single-argument \(d_2\) used by the control charts.

Reproducibility¶

Each operator has a mean taken over all \(p \times r\) of that operator's measurements. The range of those \(o\) operator means is

\[ R_{\text{oper}} = \max_{j} \bar{y}_{j} - \min_{j} \bar{y}_{j} \]

Reproducibility is then computed as a variance, by scaling that range and subtracting the part of it that repeatability alone would have produced:

\[ \sigma^2_{\text{reprod}} = \left(\frac{R_{\text{oper}}}{d_2^{*}(1,\; o)}\right)^{2} - \frac{\sigma^2_{\text{repeat}}}{p\,r} \]

The subtracted term is there because an operator mean is itself an average of \(p\,r\) measurements, so even operators who behave identically would show some spread purely from measurement error. The subtraction removes that expected spread.

The quantity under the square root can come out negative, and then reproducibility is reported as zero and the row is flagged as truncated at zero. The unfloored quantity is still reported beside it, so the size of the negative result is visible rather than hidden.

With one operator¶

With exactly one operator there is no operator to operator difference to measure, and the reproducibility step is skipped entirely rather than evaluated and found to be zero. No operator range constant is looked up, and the unfloored quantity is not reported at all. The study reports two components, part and repeatability, rather than three.

Part to part variation¶

Each part has a mean taken over all \(o \times r\) of that part's measurements. Part to part variation is the range of those \(p\) part means, scaled by the same one-argument form of the constant:

\[ \sigma_{\text{part}} = \frac{\max_{i} \bar{y}_{i} - \min_{i} \bar{y}_{i}}{d_2^{*}(1,\; p)} \]

Unlike reproducibility, no measurement-error term is subtracted here.

The range constant¶

The constant \(d_2^{*}\) takes two arguments, the number of ranges being averaged and the size of each range. It is not the same function as the single-argument \(d_2\) used by the control charts, and the two must not be substituted for one another.

Its values come from the AIAG Measurement Systems Analysis manual, fourth edition, Appendix C, tabulated for 1 to 20 ranges of size 2 to 20. Beyond that table the value is computed rather than clamped to the table's last entry:

for a range size from 2 to 100, the full-precision single-argument \(d_2\) of that size
above 100, the exact expected range of that many independent standard normal values

\[ d_2(n) = \int_{-\infty}^{\infty} \left[\,1 - \Phi(x)^n - \left(1 - \Phi(x)\right)^n\,\right] dx \]

Below the floor, meaning fewer than one range or a range of size below two, no value is defined and nothing is reported.

What this method does not produce¶

Three things that appear on every other method's report are absent under the XbarR method, and in each case because the method does not produce the quantity they are built from rather than because they were switched off:

Absent	Why
the analysis of variance table	the method computes no sums of squares and no mean squares
every confidence interval	with no mean squares there is no sampling variance for an interval
the whole misclassification block	it is refused before any other condition is even checked

The misclassification refusal is worth being precise about: it is checked first, ahead of whether specification limits were supplied. So an XbarR run reports the misclassification block as not computed for the method, not as missing limits, even when limits were supplied.