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Probabilities of Misclassification¶

This page defines the five probabilities the misclassification block reports, and the mean shift sweep the matching charts draw. They answer what the measurement system does to accept and reject decisions: how often a good part is rejected, and how often a bad part is accepted.

Requires a specification limit, and is not produced by the XbarR method

At least one specification limit must be supplied. The block is also refused outright under the XbarR method, which produces none of it.

Notation¶

Term	Description
\(X\)	the true value of a part
\(Y\)	the measured value of that part
\(\bar{x}\)	the study mean, the mean the probabilities are evaluated at
\(\sigma_{\text{part}}\)	the part to part standard deviation used
\(\sigma_{\text{gage}}\)	the total gage standard deviation used
\(\sigma_{\text{hist}}\)	a supplied historical process standard deviation
\(s\)	a mean shift, in part to part standard deviations
in spec	lying between whichever specification limits were supplied

The two distributions¶

The misclassification block asks what the measurement system does to accept and reject decisions. A part has a true value and a measured value, and they differ by measurement error, so a good part can be rejected and a bad part accepted.

Both quantities are treated as normal. The true value has the study mean and the part to part standard deviation; the measured value differs from the true value by measurement error with the total gage standard deviation:

\[ X \sim N\!\left(\bar{x},\; \sigma^2_{\text{part}}\right), \qquad Y \mid X \sim N\!\left(X,\; \sigma^2_{\text{gage}}\right) \]

\(X\) is the true value and \(Y\) the measured value. "In specification" means lying between whichever specification limits were supplied.

Every probability below is a probability over that pair, so it is a statement about the population of parts and the gage together, not about any part that was actually measured.

The joint probabilities¶

The two joint probabilities each cover one way of getting the decision wrong, as a fraction of all parts:

\[ P(\text{good part rejected}) = P\big(X \in \text{spec} \;\text{ and }\; Y \notin \text{spec}\big) \]

\[ P(\text{bad part accepted}) = P\big(X \notin \text{spec} \;\text{ and }\; Y \in \text{spec}\big) \]

Because they are fractions of all parts, they can be added to each other and compared directly against each other.

The conditional probabilities¶

The two conditional probabilities ask the same two questions of a restricted population: of the parts that really are good, what fraction get rejected, and of the parts that really are bad, what fraction get accepted.

\[ P(\text{reject} \mid \text{good}) = P\big(Y \notin \text{spec} \;\big|\; X \in \text{spec}\big) \]

\[ P(\text{accept} \mid \text{bad}) = P\big(Y \in \text{spec} \;\big|\; X \notin \text{spec}\big) \]

Each is its joint probability divided by the probability of the condition, so the two families carry the same information on different denominators. The conditional numbers are typically much larger than the joint ones, and the difference is entirely the denominator: a conditional probability discards every part that was not in the group it conditions on.

The probability a part is good¶

The fifth reported probability is the denominator the conditionals use:

\[ P(\text{part is good}) = P\big(X \in \text{spec}\big) \]

It describes the parts alone and says nothing about the gage. It is reported because without it the conditional probabilities cannot be turned back into fractions of all parts.

The historical standard deviation¶

When a historical standard deviation is supplied, it replaces the estimated part to part standard deviation in this block only, and it does so by subtraction rather than substitution:

\[ \sigma_{\text{part}} \;\longleftarrow\; \sqrt{\sigma^2_{\text{hist}} - \sigma^2_{\text{gage}}} \]

The historical value is a process standard deviation, meaning it already contains the measurement error, so the gage variance is removed from it to leave the part to part variation. This is why the dialog requires the historical value to be larger than the total gage standard deviation: otherwise the quantity under the square root is not positive.

Three consequences, each of which a reader can see on the report:

The report records which standard deviations were actually used, so the substitution is visible.
If the historical value fails the requirement, nothing is substituted and nothing is clamped. The block is refused with a reason naming that failure, and the estimated part to part standard deviation is left alone.
A historical value that passes the requirement will produce the block even when the estimated part to part variance is zero, which would otherwise have refused it, because the historical path never reads the estimated part variance at all.

The historical value does not enter the number of distinct categories, and it does not change the percent of tolerance columns. It affects this block and the sweep charts.

When a conditional probability is withheld¶

A conditional probability needs a population to condition on. When the specification limits sit far enough from the study mean, one of those two populations effectively does not exist, and the corresponding conditional probability is withheld rather than reported as a number computed from almost nothing.

The rule: once the distance from the study mean to the nearest specification limit exceeds 8 part to part standard deviations, the conditional probability for the empty population is not reported, and the report names which case it is:

Situation	Withheld	Still reported
every part is good	the probability a bad part is accepted	the other conditional, both joint probabilities, and the probability the part is good
every part is bad	the probability a good part is rejected	the other conditional, both joint probabilities, and the probability the part is good

The rule is symmetric between the two, and 8 standard deviations is a long way out: the fraction of parts beyond it is about \(6.2 \times 10^{-16}\), roughly one part in \(1.6 \times 10^{15}\).

When the whole block is refused¶

The whole block is refused, with a reason, in five situations. They are checked in this order, so the reported reason is the first one that applies and not necessarily the only one that would have:

Order	Reason reported	Condition
1	the estimation method	the XbarR method, which produces no misclassification at all
2	no specification limit	neither limit was supplied
3	the part to part variance is zero	and no supplied historical value passes its requirement
4	the historical standard deviation	one was supplied but it does not exceed the gage standard deviation
5	could not be computed	the calculation itself was refused or failed

The XbarR check comes first, which is why an XbarR run with specification limits supplied still reports the method as the reason rather than reporting the limits as missing.

A refusal for reason 3, 4 or 5 also prints a note, because in each of those the user supplied something that could not be used. Reason 2 prints no note: an input that was never supplied is an absent section, not a problem to report.

The mean shift sweep¶

The sweep charts ask what the five probabilities would be if the process mean were somewhere other than where it is. The mean is shifted across a range measured in part to part standard deviations, and each probability is recomputed at every step:

\[ \bar{x} \;\longrightarrow\; \bar{x} + s\,\sigma_{\text{part}}, \qquad s = -1.5,\; -1.4,\; \ldots,\; +1.5 \]

The shift runs from \(-1.5\) to \(+1.5\) part to part standard deviations in steps of \(0.1\), which is 31 points on each curve. Everything else about the study is held fixed: the same components, the same specification limits, the same gage standard deviation.

Each sweep chart is titled with the matching row label from the Probabilities of Misclassification table, so a chart can be read against the single number the table reports for the study's own mean. That number is the point on the curve at \(s = 0\).