Reliability.has.two.meanings—a.qualitative.and.a.quantitative.meaning..
The.former.denotes.the.likelihood.of.item.success;.the.latter.is.the.prob-ability. expressing. the. degree. of. success.. ReliThe.former.denotes.the.likelihood.of.item.success;.the.latter.is.the.prob-ability. in. the. quantitative.
sense.is.defined.as.the.ability.of.an.item.to.perform.a.required.function.
under. stated. environmental. and. operational. conditions. for. a. specified.
period.of.time..Thus,.reliability.corresponds.to.the.probability.that.failure.
will.not.occur.until.a.given.time,.and.unreliability.to.the.probability.of.
failure.occurring,.which.is.equal.to.the.cumulative.failure.rate;.the.sum.
of.the.two.probabilities.equals.1.†
*. Perhaps.the.MTTF.should.be.considered,.instead,.as.the.time.for.replacement.or.preven-tive.maintenance..For.example,.assume.that.100.machines.are.deployed.and.operating..At.
the.point.when.60.machines.are.in.failure,.the.failed.items.should.be.replaced.with.new.
ones.. As. the. number. of. failures. becomes. too. great,. the. standard. time. for. replacement.
should.be.adjusted.with.respect.to.the.failure.mode.and.its.effects..Setting.the.target.of.X.
in.the.BX.life.is.described.in.Chapter 3.
†.Reliability.(R(t)).and.cumulative.failure.rate.(F(t)).have.the.following.relation:.R(t).+.F(t).=.1.
Table 2.2 Misinterpretation.of.Reliability.Using.the.MTTF.Index Index Estimation Conversion Appraisal Remark
MTTF 40,000.h 20.years Accept Cumulative.failure.rate:.63%
B10.Life . 4,000.h . 2.years Reject Failure.rate:.10%/2.years (Annual.failure.rate:.5%)
The.confidence.level.denotes.the.adequacy.of.the.estimated.conclu-sions.*.Let’s.say.that.the.reliability.of.a.certain.item.is.99%.for.1.year,.with.
a.confidence.level.of.60%..The.phrases.“reliability.of.99%.for.1.year”.or.
“cumulative.failure.rate.of.1%.for.1.year”.are.estimates..There.is.a.confi-dence.level.of.60%.that.this.conclusion.will.prove.valid,.or.in.other.words,.
the.probability.of.this.conclusion.being.correct.is.60%..The.meaning.is.the.
same.whether.we.use.the.term.assurance level.or.adequacy level.
Let’s.consider.the.confidence.level.in.detail..Assume.that.there.is.a.
population.consisting.of.tens.of.thousands.of.products,.including.defec-tive.ones,.in.which.the.defect.rate.is.exactly.1%..We.pick.100.items.from.
the. population. at. random. and. inspect. them. to. find. defects,. then. carry.
out.this.action.repeatedly..Since.the.defect.rate.of.the.population.is.1%,.
common.sense.leads.us.to.expect.to.find.one.defect.among.100.samples..
But.when.we.continue.to.consecutively.sample.groups.of.100,.one.group.
after.another,.there.are.actually.various.findings:.In.one.group.there.are.
no.defects,.in.another.there.is.one.defect,.in.yet.another.two.defects.occur.
(perhaps. rarely),. and. so. on.. In. this. case,. the. probability. of. finding. one.
defect. or. more. is. calculated. as. 63%,. using. the. Poisson. distribution.†. In.
other.words,.if.we.repeat.these.samplings.of.a.hundred.100.times,.we.are.
likely.to.find.one.defect.or.more.63.times.and.to.find.no.defects.37.times..
Since.we.do.not.really.repeat.the.sampling.100.times.and.actually.do.it.
only. once. or. twice,. if.we. find. no.failure,. we. can.regard.the.product. as.
good.quality.
From.the.point.of.view.of.the.consumer,.the.probability.of.correctly.
judging.the.quality.of.the.population—that.is,.63%—is.called.the.confi-dence level.‡ .In.the.above.population.with.a.1%.defect.rate,.the.probabil-ity. of. finding. two. defects. or. more. in. a. sample. of. 200. units. is. slightly.
*.In.Korea.and.Japan,.reliability.has.been.translated.into.Japanese.as.shinrai-sei,.and.the.
confidence. level. in. Japanese. is. shinrai-suijun.. Using. exactly. the. same. word—shinrai—.
confuses.both.Japanese.and.Korean.people..But.the.two.concepts.are.totally.different.
†.The.probability.(P).of.finding.c.or.more.defects.is.
. P r c n p
. where.r.is.the.number.of.defects,.n.is.the.number.of.samples,.p.is.the.defect.rate,.and.c.is.
the.integer.nearest.to.(n.·.p)..When.n.=.100,.p.=.0.01,.and.c.=.1,.then.P.=.0.63,.or.63%.
‡.The.lot.acceptance.rate.(L).is.given.as.follows:.
. L p n pr e CL
. where.p.is.the.LTPD.(lot.tolerance.percent.defective),.c.is.the.acceptance.number,.CL.is.the.
confidence.level,.and.(1.–.CL).is.the.consumer’s.risk.rate.
decreased.to.59%,.and.the.probability.of.finding.three.defects.or.more.
in. a. sample. of. 300. is. also. decreased. to. 58%.. The. term. “defect. rate. of.
1%”.means.1.defect.occurs.among.100.sample.items,.2.among.200,.and.
3.among.300,.which.is.simple.common.sense..Therefore,.I.call.this.con-fidence. level. the. “commonsense. level. of. confidence,”. which. can. reach.
rate.is.0%?.The.results.are.perplexing..The.variation.between.the.two.pro-cedures. requires. different. conclusions.. In. order. to. create. a. meaningful.
mathematical. model. that. accounts. for. this. difference,. let’s. estimate. the.
defect.rate.by.adding.1.to.the.defect.number,.then.dividing.by.the.number.
of.samples..That.is,.if.no.defect.is.found.among.100.units,.the.defect.rate.
is.calculated.as.1%.of.the.population,.and.when.no.defects.are.found.in.a.
population.of.1,000,.the.defect.rate.is.0.1%..These.reasonable.results.have.
Table 2.3 Anticipated.Numbers.of.Defects.and.Confidence.Level
Defect rate
(p)
Sample number
(n)
Anticipated number of defects
(n · p)
Probability of finding that
number of defects or more
Commonsense level of
a.commonsense.level.of.confidence.*.If.there.is.one.defect.among.100.units,.
the.estimate.of.the.defect.rate.having.a.commonsense.level.of.confidence.
would.be.2%,.adding.1.to.the.one.defect.found.and.dividing.it.by.100.
Let’s.think.about.this.concept.from.another.angle..Assume.that.one.
defect. is. found. after. inspecting. 100. samples. of. a. population. for. which.
the.defect.rate.is.not.known..Then.the.defect.rate.can.be.predicted.in.three.
cases:.1%.and.below,.2%.and.below,.3%.and.below..All.the.answers.are.
right,.but.their.confidence.levels.are.different..What.are.the.correspond-ing.confidence.levels?.They.would.be.26,.59,.and.80%,.respectively..Since.
the. estimation. interval. of. the. last. case,. 3%. and. below,. is. very. wide,. its.
confidence.level.is.very.high.(80%),.which.equals.the.addition.of.2.to.the.
one.defect..Adding.1.to.the.one.defect,.or.2%.and.below,.has.a.confidence.
level.of.59%..In.the.first.case,.without.any.addition,.the.number.of.defects.
found.divided.by.100—or.1%—results.in.a.very.low.confidence.level.of.
26%,.which.makes.no.sense.
Now.let’s.review.the.confidence.level.when.we.add.1.to.the.number.of.
defects.found.†.Assume.that.there.are.three.populations.for.which.defect.
rates.are.not.known.and.that.we.pick.up.and.inspect.100.samples.respec-tively.from.three.populations..And.assume.that.the.numbers.of.defects.
found.in.the.three.groups.are,.respectively,.none,.one,.and.two.per.100..
The.confidence.levels.using.the.“add.1”.estimation.method—that.is,.1%.
and.below,.2%.and.below,.and.3%.and.below—are.then.63,.59,.and.58%..
These.are.the.same.as.the.confidence.levels.calculated.according.to.the.
anticipated.defect.number.in.Table 2.3..Thus,.the.result.of.using.the.“add.1.
to. the. number. of. defects. found”. method. becomes. the. estimated. value.
with.a.commonsense.level.of.confidence..A.summary.is.in.Table 2.4.
*.The.upper.limit.of.the.defect.rate.(p).with.the.confidence.level.(CL).is.estimated.as.follows:
. p r
n
= CL( + )
− ⋅
( )
χ21 2 2 2
1.
. If.we.use.in.turn.CL.=.0.63,.r.=.0.and.CL.=.0.59,.r.=.1.and.CL.=.0.58,.r.=.2,.the.first.chi-square.
term,.or.
. χ2(1−CL)2(2r+2)
,
. becomes.(r.+.1),.or.1,.2,.3,.respectively..Therefore,.the.defect.rate.with.a.commonsense.level.
is.given.as.follows:.p.≅.(r.+.1).·.1/n.
†.In.the.equation.for.the.lot.acceptance.rate.in.footnote.(double.cross.on.page.17),.the.term.
(1.–.L(p)).becomes.the.commonsense.level.of.confidence.if.substituting.the.number.of.fail-ure.found.for.c,.and.if.p.is.estimated.with.the.equation.as.follows:.(c.+.1).=.n.·.p..This.is.the.
statistical.basis.for.adding.one,.or.(r.+.1).
The. commonsense. level. of. confidence. is. significant. because,. using.
this.concept,.everyone.in.the.organization.can.review.and.validate.qual-ity.test.methods.with.a.simple.calculation.
Former.U.S..president.Franklin.Roosevelt.once.confessed.that.it.was.
his.highest.aspiration.that.75%.of.his.ideas.would.turn.out.to.be.successful..
And.motivational.speaker.Dale.Carnegie.once.said.that.being.confident.of.
55%.of.his.ideas.would.make.him.rich.beyond.any.worries.*.In.light.of.this.
view,.a.confidence.level.of.around.60%—that.is,.the.commonsense.level.of.
confidence—is.reasonably.high..However,.since.any.conclusion.using.the.
commonsense.level.of.confidence.would.be.right.by.60%.and.conversely.
wrong.by.40%,.it.should.not.be.used.with.too.much.assurance..In.order.to.
find.the.exact.confidence.level.of.100%.and.narrow.down.the.estimation.
interval,.validation.should.be.performed.with.multilateral.and.in-depth.
scientific.procedures.
Let’s.say.that.a.certain.company.applies.a.confidence.level.of.50%.to.
the.establishment.of.its.test.methods,.which.means,.basically,.that.it.is.
half.confident.and.half.doubtful..Thus,.a.confidence.level.that.exceeds.
50%.would.be.good,.so.it.is.useful.to.apply.the.commonsense.level.of.
confidence.. In. insurance,. a. confidence. level. of. 95%. would. usually. be.
applied,. but. in. the. case. of. product. reliability,. achieving. such. a. level.
would.cost.more.than.twice.as.much.as.aiming.for.the.commonsense.
level.of.confidence..This.would.be.unacceptable..It.is.sufficient.to.start.
with. the. commonsense. level. of. confidence. to. confirm. new-product.
quality.problems.at.an.early.stage..Since.making.mockups.and.manag-ing.test.procedures.create.expenses,.after.identifying.problems.with.a.
small.sample.fit.to.the.commonsense.level.of.confidence,.it.is.good.to.
increase.the.sample.size.and.the.confidence.level..Note.that.if.the.test.
is. repeated. twice. with. exactly. the. same. conditions. and. produces. the.
*.Dale. Carnegie,. How to Win Friends and Influence People (translated. into. Korean),. Seoul,.
Korea:.Dale.Carnegie.Training,.1995,.p. 180.
Table 2.4 Confidence Level for the Estimated Value (Defect Rate
Sample number
Defect number
found (r (r + 1 Estimated value Confidence level
Commonsense level of confidence?
100 0 — 0.5%.and.below 39% No
100 0 1 1%.and.below 63% Yes
100 1 — 1%.and.below 26% No
100 1 2 2%.and.below 59% Yes
100 1 — 3%.and.below 80% No
100 2 3 3%.and.below 58% Yes
same.results,.the.confidence.level.for.that.conclusion.would.increase.to.
around.90%.