OPR
E 63
641
Stat
istica
l Pr
oces
s Co
ntro
l
OPR
E 63
642
Sta
tist
ical
QA A
ppro
aches
•St
atis
tical
pro
cess
con
trol (
SPC
)–
Mon
itors
pro
duct
ion
proc
ess
to p
reve
nt p
oor
qual
ity
•Ac
cept
ance
sam
plin
g–I
nspe
cts
rand
om s
ampl
e of
pro
duct
to
dete
rmin
e if
a lo
t is
acce
ptab
le
•D
esig
n of
Exp
erim
ents
OPR
E 63
643
Sta
tist
ical
Qual
ity
Ass
ura
nce
●Pu
rpos
e: A
ssur
e th
at p
roce
sses
are
per
form
ing
in a
n ac
cept
able
man
ner
●M
etho
dolo
gy:
Mon
itorp
roce
ss o
utpu
t usi
ng s
tatis
tical
te
chni
ques
–If
resu
lts a
re a
ccep
tabl
e, n
o fu
rther
act
ion
is re
quire
d –
Una
ccep
tabl
e re
sults
cal
l for
cor
rect
ive
actio
nA
ccep
tanc
e Sa
mpl
ing:
Q
ualit
y as
sura
nce
that
relie
s pr
imar
ily o
n in
spec
tion
befo
rean
d af
ter p
rodu
ctio
nSt
atis
tical
Pro
cess
Con
trol
(SPC
):Q
ualit
y co
ntro
l effo
rts th
at o
ccur
dur
ing
prod
uctio
n
OPR
E 63
644
What
is
SPC
?
●A
sim
ple,
yet
pow
erfu
l, co
llect
ion
of to
ols
for g
raph
ical
ly
anal
yzin
g pr
oces
s da
ta
●H
as o
ne p
rimar
y pu
rpos
e: to
tell
you
whe
n yo
u ha
ve a
pr
oble
m.
●In
vent
ed b
y W
alte
r She
wha
rtat
AT&
T to
min
imiz
e pr
oces
s ta
mpe
ring
●Im
porta
nt b
ecau
se u
nnec
essa
ry p
roce
ss c
hang
es
incr
ease
inst
abilit
y an
d in
crea
se th
e er
ror r
ate
●SP
C w
ill id
entif
y w
hen
a pr
oble
m (o
r spe
cial
cau
se
varia
tion)
occ
urs
OPR
E 63
645
To
cont
rol,
you
have
to m
easu
re!
OPR
E 63
646
Product
ion D
ata always
has
some
Var
iabili
ty
Plot
of R
aw P
roce
ss M
easu
rem
ents
0246810
14
710
1316
1922
2528
3134
3740
4346
49Ti
me
X
OPR
E 63
647
Chan
ce a
nd A
ssig
nab
le C
ause
s of
Qual
ity
Var
iation
OPR
E 63
648
Acc
ura
cy a
nd
Pre
cisi
on
Exam
ples
of q
ualit
y ch
arac
teris
tics:
Pain
ted
surfa
ce, t
hick
ness
, har
dnes
s, a
nd re
sist
ance
to
fadi
ng o
r chi
ppin
g, v
isco
sity
, sw
eetn
ess,
ele
ctric
al
resi
stan
ce, f
requ
ency
, …W
e ca
n co
ntro
l onl
y th
ose
char
acte
ristic
s th
at c
an
be c
ount
ed, e
valu
ated
or m
easu
red
Engi
neer
ing
char
acte
ristic
s m
ay s
how
pro
blem
s w
ith
accu
racy
or w
ith p
reci
sion
AP
AP
AP
AP
OPR
E 63
649
No
rmal
Dis
trib
uti
on
Sh
aft
Dia
mete
r(W
hat
is t
his
plo
t o
f d
ata
tell
ing
us?
)
µ+3
+2+1
-1-2
-3
6.00
cm
Targ
etU
pp
er
Sp
ec
Lim
it
Lo
wer
Sp
ec
Lim
it
Off
spec
Off
spec
OPR
E 63
6410
Cau
ses
of V
aria
tion
•C
omm
on C
ause
s–V
aria
tion
inhe
rent
in a
pro
cess
–Can
be
elim
inat
ed o
nly
thro
ugh
impr
ovem
ents
in
the
syst
em
•As
sign
able
Cau
ses
–Var
iatio
n du
e to
iden
tifia
ble
fact
ors
–Can
be
mod
ified
thro
ugh
oper
ator
or
man
agem
ent a
ctio
n
OPR
E 63
6411
Assi
gnab
le C
ause
s ar
e co
ntro
lled
by S
PC
•Ta
ke p
erio
dic
sam
ples
from
pro
cess
•Pl
ot s
ampl
e po
ints
on
a co
ntro
l cha
rt•
Det
erm
ine
if pr
oces
s is
with
in li
mits
•Pr
even
t qua
lity
prob
lem
s
OPR
E 63
6412
Plot
of Sam
ple
Ave
rages
0246810
13
57
911
1315
17Sa
mpl
e #
Xbar
OPR
E 63
6413
Plot
of Sam
ple
Sta
ndar
d D
evia
tion
00.
20.
40.
60.
811.
2
13
57
911
1315
17
Sam
ple
#
Std Deviation
OPR
E 63
6414
Con
trol C
harts
•A
key
tool
in S
PC•
Gra
ph e
stab
lishi
ng p
roce
ss c
ontro
l lim
its•
Cha
rts fo
r var
iabl
es–M
ean
(X-b
ar),
Ran
ge (R
), EW
MA,
CU
SUM
•C
harts
for a
ttrib
utes
–p, n
pan
d c
OPR
E 63
6415
The
Shew
har
tContr
ol Char
t
•A
time-
orde
red
plot
of s
ampl
e st
atis
tics
•W
hen
char
t is
with
in c
ontro
l lim
its–
Onl
y ra
ndom
or c
omm
on c
ause
s pr
esen
t–
We
leav
e th
e pr
oces
s al
one
•Pl
ot o
f eac
h po
inti
s th
e te
st o
f hyp
othe
sis:
H0:
Pro
cess
is “i
n co
ntro
l”vs
.H
1: P
roce
ss is
out
of c
ontro
l and
requ
ires
inve
stig
atio
n
OPR
E 63
6416
Rel
atio
nsh
ip b
etw
een t
he
pro
cess
an
d t
he
contr
ol ch
art
OPR
E 63
6417
How
Does
the
Char
t W
ork
?
Ou
t o
f co
ntr
ol
poin
ts c
ause
d b
y ass
ign
ab
le
cau
ses
Dis
trib
uti
on
of
pro
cess
sta
tist
ic
Up
per
con
tro
l li
mit
Natu
ral
vari
ati
on
µ±
3σ
Lo
wer
con
tro
l li
mit
Tim
e
OPR
E 63
6418
A Pr
oces
s Is
“In
Con
trol”
If
•N
o sa
mpl
e po
ints
out
side
lim
its•
Mos
t poi
nts
near
pro
cess
ave
rage
•Ab
out e
qual
num
ber o
f poi
nts
abov
e &
belo
w c
ente
rline
•Po
ints
app
ear r
ando
mly
dis
tribu
ted
•A
proc
ess
“in c
ontro
l” is
sup
pose
d to
be
unde
r the
influ
ence
of r
ando
m c
ause
s on
ly
OPR
E 63
6419
The
Sign
al fr
om a
Con
trol C
hart
12
34
56
78
910
Sam
ple
num
ber
Upp
erco
ntro
llim
it
Proc
ess
aver
age
Low
erco
ntro
llim
it
Rej
ect H
0be
caus
e
Proc
ess
is li
kely
ou
t of c
ontr
ol
OPR
E 63
6420
Pote
ntial
Rea
sons
for
Var
iation
•Th
e O
pera
tor:
Tra
inin
g,
Super
visi
on,
Tec
hniq
ue.
•Th
e M
eth
od
:Pr
oce
dure
s, S
et-u
p,
Tem
per
ature
, Cutt
ing S
pee
ds.
•Th
e M
ate
rial:
Mois
ture
conte
nt,
Ble
ndin
g,
Conta
min
atio
n.
•Th
e M
ach
ine:
Set
-up,
Mac
hin
e co
nditio
n,
Inher
ent
Prec
isio
n
•M
an
ag
em
en
t:
Po
or
pro
cess
man
ag
em
en
t; p
oo
r sy
stem
s
Equ
ipm
ent
Mat
eria
l
Proc
edur
ePe
rson
nel
Qu
ality
Vari
ati
on
Shee
t Met
alV
endo
rFa
ulty
Spec
s
Lac
k of
T
rain
ing
Lac
k of
Mai
nten
ance
Ince
ntiv
esD
ocum
enta
tion
OPR
E 63
6421
Cha
rts m
ay s
igna
l inc
orre
ctly
!
Ch
art
s re
peate
dly
ap
ply
hyp
oth
esi
s te
stin
g!
Typ
e I
err
or
wit
h c
hart
s:Concl
udin
g t
hat
a p
roce
ss is
not
in c
ontr
ol
when
it
actu
ally
is
Typ
e I
I err
or
wit
h c
hart
s:Concl
udin
g t
hat
a p
roce
ss is
in c
ontr
ol
when
it
is n
ot
OPR
E 63
6422
Tw
o T
ypes
of
Proce
ss D
ata
•Num
ber o
r per
cent
of d
efec
tive
item
s in
a lo
t.• N
umbe
r of d
efec
ts p
er it
em.
• Typ
es o
f def
ects
.• V
alue
ass
igne
d to
def
ects
(min
or =
1, m
ajor
= 5
, crit
ical
= 1
0)
•Len
gth
• Wei
ght
• Tim
e
•Dia
met
er• T
ensi
le S
tren
gth
• Str
engt
h of
Sol
utio
n
•Blo
od p
ress
ure
• Vol
ume
• Tem
pera
ture
“Thi
ngs
we
coun
t”At
tribu
tes
Varia
bles
“Thi
ngs
we
mea
sure
”
OPR
E 63
6423
Type
s of
Con
trol C
harts
•Ba
sic
Type
s–
Mos
t typ
ical
thre
e•
X-Ba
r and
R•
p ch
art
•c
char
t–
Dep
end
Upo
n D
ata
Type
•Va
riabl
es•
Attri
bute
•Ad
vanc
es T
ypes
: C
USU
M, E
WM
A, M
ultiv
aria
te•
Rec
all t
hat p
lotti
ng p
oint
s on
a c
ontro
l cha
rt is
the
repe
ated
app
licat
ion
of H
ypot
hesi
s Te
stin
g
OPR
E 63
6424
Typ
es o
f Shew
har
tContr
ol Char
ts
p ch
arts
: pro
port
ion
of u
nits
non
conf
orm
ing.
npch
arts
: num
ber o
f uni
ts n
onco
nfor
min
g.c
char
ts: n
umbe
r of n
onco
nfor
miti
es.
u ch
arts
: num
ber o
f non
conf
orm
ities
per
uni
t.
Co
ntr
ol
Ch
art
s fo
r V
ari
ab
les
Data
X an
d R
cha
rts:
for s
ampl
e av
erag
es a
nd ra
nges
.
Md
and
R c
hart
s: fo
r sam
ple
med
ians
and
rang
es.
X an
d s
char
ts: f
or s
ampl
e m
eans
and
sta
ndar
d de
viat
ions
.
X ch
arts
: for
indi
vidu
al m
easu
res;
use
s m
ovin
g ra
nges
.
Co
ntr
ol
Ch
art
s fo
r A
ttri
bu
tes
Data
OPR
E 63
6425
Con
trol C
harts
For
Var
iabl
es
•M
ean
char
t (X-
Bar C
hart)
fo
r acc
urac
yU
ses
aver
age
of a
sam
ple:
X-Ba
r = (x
1+x 2
+x3+
x 4+x
5)/5
•R
ange
cha
rt (R
-Cha
rt)
for p
reci
sion
Use
s am
ount
of d
ispe
rsio
n in
a s
ampl
eR
= m
ax (x
i) –
min
(xi)
OPR
E 63
6426
Xb
ar
Ch
art
help
s co
ntr
olA
ccu
racy
•Av
erag
e Xb
ar=
82.5
kg
•St
anda
rd D
evia
tion
of X
bar
= σ
xbar
= 1.
6 kg
•C
ontr
ol L
imits
= Av
erag
e Xb
ar+
3 σ x
bar
= 82
.5 ±
3 ×
1.6
= [7
7.7,
87.
3]H
ere,
the
proc
ess
is “i
n co
ntro
l” (i.
e., t
he m
ean
is s
tabl
e)
767880828486
135
791113151719
XbarU
CL
LCL
Sam
ple
#
OPR
E 63
6427
Cen
tral
Lim
it T
heor
em
99.7
% o
f all
sam
ple
mea
ns
Popu
latio
n,In
divi
dual
item
s
Sam
ple
mea
ns
µ-3σ
xµ+
3σx
µ
(Bas
is f
or
speci
fica
tio
nlim
its)
(Bas
is f
or
con
tro
llim
its)
OPR
E 63
6428
Dis
trib
ution o
f Xbar
--a
Proce
ss S
tatist
ic
Dis
trib
uti
on
of
Xb
ar:
No
rmal(µ,
σ2/
n)
Dis
trib
uti
on
of
X:
No
rmal(µ,
σ2)
Mea
n
OPR
E 63
6429
SPC
Contr
ol Li
mits
Popu
latio
n of
pr
oces
s ou
tput
x
µ-3σ
x
µ+3σ
xU
CL
Dis
tribu
tion
ofpr
oces
s st
atis
tic x
bar
µ
LCL
OPR
E 63
6430
Proc
ess C
ontr
ol b
y C
ontr
ol L
imits
LCL
UC
L•
•
•••
•
••
••
•In c
ontr
ol
Proc
ess
is s
tabl
e
Proc
ess
cent
erha
s sh
ifted
Out
of c
ontr
ol
µ-3σ
x
µ+3σ
x µ
OPR
E 63
6431
Routine
use
of th
ePr
oce
ss C
ontr
ol Char
t
•D
ata/
Info
rmat
ion:
Mon
itor p
roce
ss v
aria
bilit
y ov
er ti
me
•C
ontr
ol L
imits
:
Avera
ge +
z ×
No
rmal V
ari
ab
ilit
y
•D
ecis
ion
Rul
e:
●
Igno
re v
aria
bilit
y w
hen
poin
ts a
re w
ithin
lim
its
●In
vest
igat
e va
riatio
n w
hen
outs
ide
as “a
bnor
mal
”
•
Erro
rs:
Type
I-F
alse
ala
rm (u
nnec
essa
ry in
vest
igat
ion)
Type
II-M
isse
d si
gnal
(to
iden
tify
and
corre
ct)
Sam
ple
s
Process Measure
Up
per
con
tro
l lim
it
Lo
wer
con
tro
l lim
it
Out
of co
ntr
ol sa
mple
s
OPR
E 63
6432
SPC
App
lied
To S
ervi
ces
•N
atur
e of
def
ect i
s di
ffere
nt in
ser
vice
s
•Se
rvic
e de
fect
is a
failu
re to
mee
t cu
stom
er re
quire
men
ts
•M
onito
r tim
es, c
usto
mer
sat
isfa
ctio
n
OPR
E 63
6433
Serv
ice
SPC
Exa
mpl
es•
Hos
pita
ls
–Ti
mel
ines
s, re
spon
sive
ness
, ac
cura
cy•
Gro
cery
Sto
res
–C
heck
-out
tim
e, s
tock
ing,
cle
anlin
ess
•Ai
rline
s–
Lugg
age
hand
ling,
wai
ting
times
, co
urte
sy•
Fast
food
rest
aura
nts
–W
aitin
g tim
es, f
ood
qual
ity,
clea
nlin
ess
•Ba
nks
–D
aily
bal
ance
erro
rs, #
of c
usto
mer
s se
rved
, tra
nsac
tions
com
plet
ed,
cour
tesy
OPR
E 63
6434
Con
trol C
harts
•Ba
sic
Type
s–
Mos
t typ
ical
thre
e•
X-Ba
r and
R•
p ch
art
•c
char
t–
Dep
end
Upo
n D
ata
Type
•Va
riabl
es•
Attri
bute
•Al
l are
App
licat
ions
of H
ypot
hesi
s Te
stin
g
OPR
E 63
6435
Vari
ati
on
s an
d C
on
tro
l
Ran
do
m o
r C
om
mo
n V
ari
ati
on
:N
atura
l or
inher
ent
variat
ions
in t
he
outp
ut
of pro
cess
are
cr
eate
d b
y co
un
tless
min
or
fact
ors
, to
o m
an
y t
o
invest
igate
eco
no
mic
all
y
Ass
ign
ab
le o
r S
peci
al
Vari
ati
on
:
A v
aria
tion w
hose
cau
se c
an
be i
den
tifi
ed
⇒Ass
ignab
le v
aria
tions
push
the
char
ts b
eyond c
ontr
ol lim
its
⇒Thei
r ca
use
s m
ust
be i
nvest
igate
d,
dete
cted
an
d r
em
oved
Ass
ign
ab
le c
au
se e
xam
ple
s: T
ool w
ear,
equip
men
t th
at
nee
ds
adju
stm
ent,
def
ective
mat
eria
ls,
hum
an fac
tors
(c
arel
essn
ess,
fat
igue,
nois
e an
d o
ther
dis
trac
tions,
fai
lure
to
follo
w c
orr
ect
pro
cedure
s),
failu
re o
f pum
ps,
hea
ters
, et
c.
OPR
E 63
6436
Spec
ial C
ause
s of
Var
iatio
n
●Al
so c
alle
d as
sign
able
cau
se o
f var
iatio
n●
Whe
n an
ass
igna
ble
caus
e is
act
ive,
the
char
t goe
s be
yond
con
trol l
imits
●In
SPC
, whe
n so
me
unus
ual o
r ext
erna
l cau
se o
ccur
s,
the
caus
e is
iden
tifie
d an
d da
ta p
oint
rem
oved
to
calc
ulat
e tru
e co
ntro
l lim
its●
Atte
mpt
ing
to im
prov
e a
proc
ess
(con
tain
ing
spec
ial
caus
e va
riatio
n) w
ithou
t rem
ovin
g th
e sp
ecia
l cau
se o
nly
incr
ease
s th
e in
stab
ility
and
varia
tion
of th
e pr
oces
s
OPR
E 63
6437
Com
mon
Cau
ses
of V
aria
tion
●Al
so c
alle
d ra
ndom
cau
ses
of v
aria
tion
●W
hen
only
com
mon
cau
ses
are
activ
e, th
e ch
art r
emai
ns
stab
le a
nd w
ithin
con
trol l
imits
●In
SPC
, whe
n on
ly ra
ndom
cau
ses
are
activ
e, n
o si
ngle
ca
use
is a
t fau
lt. A
ny p
roce
ss im
prov
emen
t effo
rt no
w
mus
t con
side
r all
sour
ces
of v
aria
tion,
gen
eral
ly th
e fa
ctor
s in
here
nt in
the
tech
nolo
gy o
f the
pro
cess
●A
proc
ess
with
onl
y co
mm
on c
ause
of v
aria
tion
is s
tabl
e an
d pr
edic
tabl
e an
d it
form
s th
e ba
sis
for m
easu
ring
proc
ess
capa
bilit
y
OPR
E 63
6438
We
can u
se R
ange
in p
lace
of
Std
Dev
iation t
o c
ontr
ol Pr
ecis
ion
00.
20.
40.
60.
811.
2
13
57
911
1315
17
Sam
ple
#
Std Deviation
00.
511.
522.
5
13
57
911
1315
17
Sam
ple
#
Range (R)
Sca
tter P
lot o
f Sig
ma
and
R
0
0.51
1.52
2.5
00.
51
1.5
Sigm
a
Range Cor
rela
tion(
s, R
) = 0
.993
4
OPR
E 63
6439
Con
trol C
harts
for V
aria
bles
•M
ean
char
t (X-
Bar C
hart)
–Use
s av
erag
e of
a s
ampl
e
•R
ange
cha
rt (R
-Cha
rt)–U
ses
amou
nt o
f dis
pers
ion
in a
sam
ple
OPR
E 63
6440
Con
stru
ctio
n of
Con
trol C
hart
●C
ontro
l lim
its m
ust b
e ba
sed
only
on
hist
oric
pr
oces
s da
ta th
at a
re “i
n-co
ntro
l”●
We
draw
tent
ativ
e lim
it lin
es a
nd c
heck
if a
ny
poin
ts fa
ll ou
tsid
e th
e lim
its●
If so
me
poin
ts fa
ll ou
tsid
e, n
on-ra
ndom
ca
uses
are
pre
sent
; dis
card
thos
e da
ta
poin
ts a
nd re
-cal
cula
te c
ontro
l lim
its●
Rep
eat c
alcu
latio
n of
lim
its if
nec
essa
ry
OPR
E 63
6441
Th
ree S
igm
a C
on
tro
l Lim
its
•Th
e us
e of
3-s
igm
a lim
its g
ener
ally
giv
es
good
resu
lts in
pra
ctic
e (A
RL
= 1/
(α/2
).
•If
the
dist
ribut
ion
of th
e qu
ality
cha
ract
eris
tic
is re
ason
ably
wel
l app
roxi
mat
ed b
y th
e no
rmal
dis
tribu
tion,
then
the
use
of 3
-sig
ma
limits
is a
pplic
able
.•
Thes
e lim
its a
re o
ften
refe
rred
to a
s ac
tion
limits
.
OPR
E 63
6442
War
nin
g L
imits
on C
ontr
ol Char
ts
•W
arni
ng li
mits
(if u
sed)
are
typi
cally
set
at 2
st
anda
rd d
evia
tions
from
the
mea
n.•
If on
e or
mor
e po
ints
fall
betw
een
the
war
ning
lim
its
and
the
cont
rol l
imits
, or c
lose
to th
e w
arni
ng li
mits
th
e pr
oces
s m
ay n
ot b
e op
erat
ing
prop
erly
.•
Goo
d th
ing:
War
ning
lim
its o
ften
incr
ease
the
sens
itivi
tyof
the
cont
rol c
hart.
•Ba
d th
ing:
War
ning
lim
its c
ould
resu
lt in
an
incr
ease
d ris
k of
fals
e al
arm
s.
OPR
E 63
6443
Calc
ula
tio
n o
f X
bar
Ch
art
Co
ntr
ol
Lim
its
tabl
e.a
from
foun
dis
an
d
rang
es
sam
ple
of
Ave
rage
w
here
LCL
lim
it,
cont
rol
Low
er
UC
Llim
it,
cont
rol
Upp
er
ity.
var
iabi
lpr
oces
s
of m
easu
re a
as
rang
e
sam
ple
av
erag
e
us
e
tois lim
its
cont
rol
fin
ding
for
m
etho
dqu
ick
A
Ran
ge
5/)(x
X
bar
:D
ef
2
22
54
32
1
AR
RA
xz
x
RA
xz
x
R
xM
inx
Max
R
xx
xx
x
xbar
xbar
ii
=
−=
−=
+=
+=−
=
++
++
==
σσ
OPR
E 63
6444
Proc
ess C
ontr
ol C
hart
Fac
tors
LCL
Fact
orfo
r Ran
ges
(Ran
geC
hart
s)(D
3)
UC
L Fa
ctor
for R
ange
s(R
ange
Cha
rts)
(D4)
Con
trol
Lim
itFa
ctor
for
Ave
rage
s(M
ean
Cha
rts)
(A2)
Fact
or fo
rEs
timat
ing
Sigm
a(
= R
/d2)
(d2)
Sam
ple
(Sub
grou
p)Si
ze (n) 2
1.88
03.
267
01.
128
31.
023
2.57
50
1.69
32.
282
40.
729
02.
059
50.
577
2.11
50
2.32
66
0.48
32.
004
02.
534
70.
419
1.92
40.
076
2.70
48
0.37
31.
864
0.13
62.
847
90.
337
1.81
60.
184
2.97
010
0.30
81.
777
0.22
33.
078
OPR
E 63
6445
Sele
ct 2
5 sm
all s
ampl
es(in
this
cas
e, n
= 4
)
Find
X a
nd R
of e
ach
sam
ple.
The
X ch
art i
s us
ed to
cont
rol t
he p
roce
ss m
ean.
The
R c
hart
is u
sed
toco
ntro
l pro
cess
var
iatio
n.
1
2
3
4
2
5
4
7
6
7
6
3
9
6
5
8
8
6
5
6
9
5
20 2
4 3
2 2
4
2
8To
tal
5
6
8
6
7
15
0
2
5
3
2
3
7
5
Sam
ple
Num
ber
X Values
Sum X R
Proc
ess D
ata
Exa
mpl
e:
5-31
OPR
E 63
6446
Sum X R
1
2
3
4
2
54
7
6
7
6
3
9
6
5
8
8
6
5
6
9
5
20
24
32
24
28
Tota
l5
6
8
6
7
150
2
5
3
2
3
7
5
Sam
ple
Num
ber
Values
2 3 4
0 0 0
1.88
01.
023
0.72
9
3.26
72.
575
2.28
2
1.12
81.
693
2.05
9
X a
nd R
Cha
rts F
acto
rsn
A2
D3
D4
d 2
5-32a
OPR
E 63
6447
Sum X R
1
2
3
4
2
54
7
6
7
6
3
9
6
5
8
8
6
5
6
9
5
20
24
32
24
28
Tota
l5
6
8
6
7
150
2
5
3
2
3
7
5
Sam
ple
Num
ber
Values
2 3 4
0 0 0
1.88
01.
023
0.72
9
3.26
72.
575
2.28
2
1.12
81.
693
2.05
9
– – X =
150
/ 25
= 6
R =
75
/ 25
= 3
A2R
= 0
.729
(3) =
2.2
UC
L X =
X +
A2R
= 6
+ 2
.2 =
8.2
LCL X
= X
-A2R
= 6
-2.
2 =
3.8
UC
L R=
D4R
= 2
.282
(3) =
6.8
LCL R
= D
3R =
0(3
) = 0
– – – –
––
– –– –
X a
nd R
Lim
its
n
A
2
D
3D
4d 2
5-32b
OPR
E 63
6448
Sum X R
1
2
3
4
2
54
7
6
7
6
3
9
6
5
8
8
6
5
6
9
5
20
24
32
24
28
Tota
l5
6
8
6
7
150
2
5
3
2
3
7
5
Sam
ple
Num
ber
Values
2 3 4
0 0 0
1.88
01.
023
0.72
9
3.26
72.
575
2.28
2
1.12
81.
693
2.05
9
– – X =
150
/ 25
= 6
R =
75
/ 25
= 3
A2R
= 0
.729
(3) =
2.2
UC
L X =
X +
A2R
= 6
+ 2
.2 =
8.2
LCL X
= X
-A2R
= 6
-2.
2 =
3.8
UC
L R=
D4R
= 2
.282
(3) =
6.8
LCL R
= D
3R =
0(3
) = 0
– – – –
––
– –– –
–LCL
X=
3.8
UC
LX
= 8.
2–
X =
6.0
– –
–
UC
L R
= 6.
8
R =
3.0
LCL
R=
0
RangeMean
X a
nd R
Cha
rt P
lots
n
A
2
D
3D
4d 2
5-32
OPR
E 63
6449
Exa
mple
: X
bar
chart
Co
ntr
ol
Lim
its
by σ
xb
ar
A q
ual
ity
contr
ol m
anag
er t
ook
five
sam
ple
s(S
1,
S2,
S3,
S4,
S5),
eac
h w
ith four
obse
rvat
ions,
of th
e dia
met
er o
f sh
afts
man
ufa
cture
d o
n a
lat
he
mac
hin
e. T
he
man
ager
com
pute
d t
he
mea
n o
f ea
ch s
ample
and t
hen
com
pute
d
the
gra
nd m
ean.
All
valu
es a
re in c
m.
Use
this
info
rmat
ion t
o
obta
in 3
-sig
ma
(i.e
., z
=3 )
contr
ol lim
its
for
mea
ns
of
futu
re
tim
es.
It is
know
n f
rom
pre
vious
exper
ience
that
the
stan
dard
d
evia
tio
n σ
xof
the
pro
cess
is
0.0
2 c
m.
12.1
212
.10
12.1
112
.12
12.1
0Xb
ar
12.0
912
.14
12.1
312
.12
12.1
212
.10
12.0
812
.10
12.0
912
.09
12.1
112
.15
12.1
512
.12
12.1
012
.11
12.1
112
.10
12.1
112
.08
1 2 3 4
S5S4
S3S2
S1O
bser
vatio
n
OPR
E 63
6450
Exa
mple
of
Contr
ol Li
mits
Cal
cula
tions
usi
ng σ
xb
ar
size
.
Sam
ple
an
dde
viat
ion
st
anda
rd
Proc
ess
mea
ns
sam
ple
of
on
dist
ribut
i
ofde
viat
ion
St
anda
rd
w
here
12.0
8
0.01
3
12
.11
LC
L
:lim
it
cont
rol
Low
er
12.1
4
0.01
3
12
.11
U
CL
:lim
it
cont
rol
Upp
er
0.01
40.
02
Hen
ce
4.
si
ze
sam
ple
that
N
ote
(g
iven
).
0.02
and
12.1
1
512
.12
12.1
0
12
.11
12.1
2
12
.10
x
==
==
=×
−=
−=
=×
+=
+=
==
=
==
=+
++
+=
n
x
nzx
zx
n
n
x
x
x
x
σ
σσ
σσ
σσ
σ
OPR
E 63
6451
Co
ntr
ol
Lim
it F
act
ors
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
1.72
1.69
1.67
1.65
1.64
1.62
1.61
1.60
1.59
0 0 0 0 00.
080.
140.
180.
220.
260.
280.
310.
330.
350.
360.
380.
360.
400.
41
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.43
0.31
0.29
0.27
0.25
0.24
0.22
0.21
0.20
0.19
0.19
0.18
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
D4
D3
A2
n
Fact
or fo
r R U
CL
Fact
or fo
r R L
CL
Fact
or fo
r Xba
rlim
itsSa
les
Size
OPR
E 63
6452
Xbar
Contr
ol Li
mits
by
Rb
ar
0.05
0.04
0.06
0.05
0.03
Ran
ge R
12.0
912
.14
12.1
312
.12
12.1
212
.10
12.0
812
.10
12.0
912
.09
12.1
112
.15
12.1
512
.12
12.1
012
.11
12.1
112
.10
12.1
112
.08
1 2 3 4
S5S4
S3S2
S1O
bser
vatio
n
08.12
046
.073.0
11.12
LCL
14.12
046
.073.0
11.12
UC
L
are
Lim
itsC
ontro
lr
Upp
er/L
owe
Hen
ce,
tabl
efr
om73.0
ther
efor
e,4
size
Sam
ple
046
.0
50.
05
0.
04
0.
06
0.
05
0.
03
ra
nges
sa
mpl
e
of A
vera
ge
22
2
=×
−=
−=
=×
+=
+=
==
=+
++
+==
RA
x
RA
x
An
RR
OPR
E 63
6453
Ran
ge (
R)
Ch
art
help
s co
ntr
ol
Pre
cisi
on
UCL
20 0510 Range15
12
34
56
78
910
1112
1314
1516
1718
1920
Sam
ple
#LC
L
●Av
erag
e R
ange
R=
10.1
kg
●St
anda
rd D
evia
tion
of R
ange
= 3
.5 k
g●
Con
trol L
imits
:10
.1 +
3 ×
3.5
= [2
0.6,
0]
Proc
ess
here
is “i
n co
ntro
l” (i.
e., p
reci
sion
is s
tabl
e)
OPR
E 63
6454
Ran
ge C
on
tro
l C
hart
Co
ntr
ol
Lim
its
tabl
e.Fa
ctor
sLi
mit
Con
trol
th
efr
om
obta
ined
ar
e
D an
d
D w
here
D
LC
L
limit,
co
ntro
lLo
wer
D
UC
Llim
it,
cont
rol
Upp
er
char
t.-
R th
eis
prec
isio
n or
disp
ersi
on
proc
ess
mon
itor
tous
edch
art
cont
rol
Th
e
43
3R
4R
RR
==
OPR
E 63
6455
12.1
212
.10
12.1
112
.12
12.1
0Xb
ar
0.05
0.04
0.06
0.05
0.03
Ran
ge R
12.0
912
.14
12.1
312
.12
12.1
212
.10
12.0
812
.10
12.0
912
.09
12.1
112
.15
12.1
512
.12
12.1
012
.11
12.1
112
.10
12.1
112
.08
1 2 3 4
S5S4
S3S2
S1O
bser
vatio
nExa
mple
:R c
har
t Li
mits
00.004
6.
000
.0LC
L
105
.004
6.
280
.2U
CL
are
/H
ence
,
.ta
ble
from
28.2an
d00.0
Ther
efor
e.4
046
.0
50.
05
0.
04
0.
06
0.
05
0.
03
ra
nges
sa
mpl
e
of A
vera
ge
34
43
=×
==
=×
==
==
=
=+
++
+==
RD
RD
Lim
itsC
ontr
olLo
wer
Upp
er
DD
nR
RR
OPR
E 63
6456
Perf
orm
an
ce V
ari
ati
on
Patt
ern
s
Stab
le
Uns
tabl
e
Tren
d
Cyc
lical
Shift
OPR
E 63
6457
Abno
rmal
Con
trol C
hart
Patte
rns
UC
LU
CL
LCL
LCL
Sam
ple
obse
rvat
ions
cons
iste
ntly
abo
ve th
ece
nter
line
Sam
ple
obse
rvat
ions
cons
iste
ntly
bel
ow th
ece
nter
line
OPR
E 63
6458
Abno
rmal
Con
trol C
hart
Patte
rns
UC
LU
CL
LCL
LCL
Sam
ple
obse
rvat
ions
cons
iste
ntly
incr
easi
ngSa
mpl
e ob
serv
atio
nsco
nsis
tent
ly d
ecre
asin
g
OPR
E 63
6459
Abno
rmal
Con
trol C
hart
Patte
rns
UC
LU
CL
LCL
LCL
Sam
ple
obse
rvat
ions
cons
iste
ntly
abo
ve th
ece
nter
line
Sam
ple
obse
rvat
ions
cons
iste
ntly
bel
ow th
ece
nter
line
OPR
E 63
6460
Zone
s Fo
r Non
-Ran
dom
Pat
tern
Te
sts
UC
L
LCL
Zone
A
Zone
B
Zone
C
Zone
C
Zone
B
Zone
A
3si
gma=
x +
2A
R
2si
gma=
x +
2 32A
R
()
1si
gma
=x +
1 32
AR
(
)
x 1sig
ma=
x −
1 32A
R
()
2si
gma=
x −
2 32A
R
()
3si
gma=
x −
2A
R
OPR
E 63
6461
Abno
rmal
Con
trol C
hart
Patte
rns
1. 8
con
secu
tive
poin
ts o
n on
e si
de o
f the
cen
ter l
ine.
2. 8
con
secu
tive
poin
ts u
p or
dow
n ac
ross
zon
es.
3. 1
4 po
ints
alte
rnat
ing
up o
r dow
n.4.
2 o
ut o
f 3 c
onse
cutiv
e po
ints
in z
one
A bu
t stil
l ins
ide
the
cont
rol l
imits
.5.
4 o
ut o
f 5 c
onse
cutiv
e po
ints
in z
one
A or
B.
OPR
E 63
6462
From
Contr
ol to
Im
pro
vem
ent
LCLµ
UCL
Out
of C
ontr
olIn
Con
trol
Impr
oved
Wei
ght
Tim
e
Targ
et
OPR
E 63
6463
Def
ect C
ontro
l For
Attr
ibut
es
•p
Cha
rts–C
alcu
late
per
cent
def
ectiv
es in
sam
ple
•c
Cha
rts–C
ount
num
ber o
f def
ects
in it
em
OPR
E 63
6464
Use
of p-C
har
ts
•W
hen
obse
rvat
ions
can
be
plac
ed in
to
two
cate
gorie
s.–
Goo
d or
bad
–Pa
ss o
r fai
l–
Ope
rate
or d
on’t
oper
ate
•W
hen
the
data
con
sist
s of
mul
tiple
sa
mpl
es o
f sev
eral
obs
erva
tions
eac
h
OPR
E 63
6465
Co
ntr
ol
Lim
its
forp–
Ch
art
0.
LCL
U
sefo
rmul
a. e
appr
oxim
at
todu
e
nega
tive
is LC
L
Som
etim
es.
re
plac
es ,
es
timat
e,
The
hi
stor
y.
from
as es
timat
ed
beca
n it
unkn
own,
is
Ifpr
oces
s.
in th
e
defe
ctiv
es
offr
actio
n
nom
inal
th
eis
an
d
)1(
on,
dist
ribut
i
Bin
omia
l
from
w
here
LCL
lim
it,
cont
rol
Low
er
UC
Llim
it,
cont
rol
Upp
er
proc
ess.
ain
defe
ctiv
es
of pr
opor
tion
em
onito
r th
to
used
,at
tribu
tes
for
char
t
Con
trol
p
pp
=
−=
=
+=
pp
ppp
np
p
p - z
σ z σ
p
p
p
σ
OPR
E 63
6466
The
Nor
mal
Dis
tribu
tion
still
appl
ies
95%
99.7
4%
-1σ
-3σ
-2σ
µ=0
1σ2σ
3σ
OPR
E 63
6467
p C
hart
Dat
a
2 50 4 .08
1 50 2 .04
3 50 0 0
4 50 3 .06
25 50 2 .04
Sam
ple
num
ber
Tota
l
1250 50 1.00
n
#def p
Sam
ple
size
Num
ber o
f def
ectiv
e ite
ms
foun
d in
sam
ple
Frac
tion
defe
ctiv
e in
sa
mpl
e
5-33a
OPR
E 63
6468
p C
hart
Cal
cula
tions
2 50 4 .08
Sam
ple
num
ber
1 50 2 .04
3 50 0 0
4 50 3 .06
25 50 2 .04
Tota
l
1250 50 1.00
.04(
.96)
p =
–
p(1-
p)n
3
= 3
50=
0.08
3
UC
L P
= p
+ 3
P–
UC
L P
= p
-3
P–
= .0
4 +
.083
= .1
23
= .0
4 -.
083
= 0
can'
t be
nega
tive
••
••
UC
L P
= 0.
123
LCL
P =
0
p =
0.04
–
= 3
σ σ
#def n
ΣΣ
n=
p
Pσn
#def p
5-33
OPR
E 63
6469
Exa
mple
:p
char
t dat
a:
120
Tota
l
10 9 8 11
12 8 13
11 9 10 8 11
1 2 3 4 5 6 7 8 9 10
11
12
Nu
mb
er
of
Defe
ctiv
es
Sam
ple
#
A Q
C m
anag
er c
ounte
d
the
nu
mb
er
of
defe
ctiv
e n
uts
p
rod
uce
dby
an
auto
mat
ic m
achin
e in
12
sam
ple
s. U
sing t
he
dat
a sh
ow
n,
const
ruct
a
contr
ol ch
art
that
will
des
crib
e 99.7
4 %
of
the
chan
ce v
aria
tion in t
he
pro
cess
when
the
pro
cess
in
contr
ol. E
ach s
ample
co
nta
ined
200 n
uts
.
OPR
E 63
6470
p C
har
t S
olu
tion
005
.015.0
305.0
LC
L
lim
it,
cont
rol
Low
er
095
.015.0
305.0
UC
L
limit,
co
ntro
lU
pper
3z
015
.020
0)
05.01(
05.0)
1(
05.020
012
120 pp
p
=×
−=
−=
=×
+=
+=
=
=−
=−
=
=×
=
p
p
z σ
p
z σ
p
np
p
p σ
OPR
E 63
6471
Exam
ple
of p
-Cha
rt
..
00.
020.
040.
060.
080.1
0.12
0.14
0.16
0.180.
20
2
4
6
8
10
12
14
16
18
20
Proportion defective
Sam
ple
num
ber
OPR
E 63
6472
Nu
mb
er
of
Defe
cts/
Un
it:
c-C
hart
s
Use
onl
y w
hen
the
num
ber o
f oc
curre
nces
per
uni
t of m
easu
re c
an
be c
ount
ed; n
on-o
ccur
renc
es c
anno
t be
cou
nted
.–
Scra
tche
s, c
hips
, den
ts, o
r erro
rs p
er it
em–
Cra
cks
or fa
ults
per
uni
t of d
ista
nce
–Br
eaks
or T
ears
per
uni
t of a
rea
–Ba
cter
ia o
r pol
luta
nts
per u
nit o
f vol
ume
–C
alls
, com
plai
nts,
failu
res
per u
nit o
f tim
e
OPR
E 63
6473
c-Char
t Contr
ols
Def
ects
/Unit
Dis
cret
e Q
ualit
y M
easu
rem
ent:
D =
Num
ber o
f “de
fect
s” (e
rrors
) per
uni
t of w
ork
Exam
ples
of D
efec
ts:
Num
ber o
f typ
os/p
age,
erro
rs/th
ousa
nd tr
ansa
ctio
ns,
equi
pmen
t bre
akdo
wns
/shi
ft, b
ags
lost
/thou
sand
flow
n, p
ower
ou
tage
s/ye
ar, c
usto
mer
com
plai
nts/
mon
th, d
efec
ts/c
ar...
If
n
= N
o. o
f opp
ortu
nitie
s fo
r def
ects
to o
ccur
, and
p =
Prob
abilit
y of
a d
efec
t/erro
r occ
urre
nce
in e
ach
then
D
~ B
inom
ial (
n, p
) with
mea
n np
, var
ianc
e np
(1-p
)≅
Pois
son
(np)
with
mea
n =
varia
nce
= np
, if
n is
larg
e (≥
20)
and
p is
sm
all (≤
0.05
)
With
c=
np=
aver
age
num
ber o
f def
ects
per
uni
t,
Con
trol l
imits
= c
+3 √c
OPR
E 63
6474
c-Char
t Contr
ol Li
mits
used
. is
Pois
son
io
n to
appr
oxim
aton
di
strib
uti
no
rmal
th
e
reas
ons
pr
actic
alfo
r B
ut
on
.di
strib
uti
Pois
son
a ha
sac
tual
ly
c
devi
atio
n.
stan
dard
th
eis c
and
un
it,pe
r
defe
cts
of
num
ber
an
dm
ean
th
eis c
whe
re
cz
c
LCL
lim
it,
cont
rol
Low
er
cz
c
UC
Llim
it,
cont
rol
Upp
er
cc
−=
+=
OPR
E 63
6475
c-C
hart
Exa
mpl
e: H
otel
Sui
te In
spec
tion-
-Def
ects
D
isco
vere
d/ro
om
Day
Def
ects
Day
Def
ects
Day
Def
ects
4 2 1 2 3 1 3 2 0
2 0 3 1 2 3 1 0 0
10 11 12 13 14 15 16 17 18
1 2 3 4 5 6 7 8 9
19 20 21 22 23 24 25 26
1 1 2 1 0 3 0 1 39To
tal
OPR
E 63
6476
Rec
all c
-Cha
rt Li
mits
Proc
ess
aver
age=
c =
Tota
l # d
efec
ts#
sam
ples
Sam
ple
stan
dard
dev
iatio
n=
cσ
=c
UC
L=
c +
zc
σLC
L=
c -z
cσ
OPR
E 63
6477
c C
hart
for
Hot
el S
uite
Insp
ectio
n
51
c =
39/2
6 =
1.50
UC
L =
5.16
LCL
= 0
0123
Number of defects5 4
015
2025
Day
OPR
E 63
6478
Exa
mple
of
c C
hart
42
Tota
l
3 6 4 5 4 0 2 5 6 0 3 1 0 3
1 2 3 4 5 6 7 8 9 10
11
12
13
14
Num
ber
of
com
pla
ints
Day
A b
ank
man
ager
re
ceiv
es a
cer
tain
n
um
ber
of
com
pla
ints
each
d
ay
about
the
ban
k’s
se
rvic
e. C
om
pla
ints
fo
r 14 d
ays
are
giv
en
in t
he
table
show
n.
Const
ruct
a c
ontr
ol
char
t usi
ng t
hre
e-si
gm
a lim
its.
OPR
E 63
6479
c Char
t Solu
tion
used
. is
Pois
son
io
n to
appr
oxim
aton
di
strib
uti
no
rmal
re
ason
s,
prac
tical
For
devi
atio
n.
stan
dard
th
eis c
unit.
per
de
fect
s
ofnu
mbe
r
and
mea
n
the
is c w
here
0.073.1
33
cz
c
LCL
lim
it,
cont
rol
Low
er
2.873.1
33
cz
c
UC
Llim
it,
cont
rol
Upp
er 73.1
31442
c
cc
=×
−=
−=
=×
+=
+=
=
== c
OPR
E 63
6480
Contr
ol Char
ts S
um
mar
y
–X-
bar a
nd R
cha
rts•
Varia
bles
dat
a•
Appl
icat
ion
of n
orm
al d
istr
ibut
ion
(by
Cen
tral L
imit
Theo
rem
)–
p ch
arts
•At
tribu
tes
data
(def
ects
per
n o
bser
vatio
ns)
•Ap
plic
atio
n of
bin
omia
l dis
trib
utio
n–
c ch
arts
•At
tribu
tes
data
(def
ects
per
insp
ectio
n)•
Appl
icat
ion
of P
oiss
on d
istr
ibut
ion
OPR
E 63
6481
Whic
h C
har
t to
Use
?
OPR
E 63
6482
OPR
E 63
6483
Sum
mar
y of
SPC
●St
atis
tical
pro
cess
con
trol p
rovi
des
sim
ple,
yet
po
wer
ful,
for m
anag
ing
proc
ess
whi
le a
void
ing
proc
ess
tam
perin
g●
A pr
oces
s 'in
con
trol'
(i.e.
; exh
ibiti
ng n
o sp
ecia
l ca
use
varia
tion)
is ri
pe fo
r the
nex
t sta
ge--
brea
kthr
ough
pro
cess
impr
ovem
ent
●A
proc
ess
still
burd
ened
with
spe
cial
cau
se
varia
tion
is s
till i
n th
e pr
oble
m-s
olvi
ng s
tage
OPR
E 63
6484
Pro
cess
Im
pro
vem
en
t
•M
easu
rem
ent
–Ext
ernal
and
Inte
rnal
•Anal
ysis
–Anal
yze
Var
iation
•Contr
ol
–Adju
st P
roce
ss•
Impro
vem
ent
–Red
uce
Var
iation
•In
nova
tion
–Red
esig
n
Product
/Pro
cess
D ACP
D ACP
Control
Improve
InnovateIm
prove
OPR
E 63
6485
Proc
ess
capa
bilit
y:Th
e in
here
nt
varia
bilit
y of
pro
cess
ou
tput
rela
tive
to th
e va
riatio
n al
low
ed b
y th
e de
sign
or
cust
omer
sp
ecifi
catio
nSp
ec
Lim
itsProc
ess
OPR
E 63
6486
Proc
ess
Cap
abilit
y An
alys
is●
Diffe
rs Fundamentally
from
Contr
ol Char
ting
Focu
ses
on im
pro
vem
ent,
not
contr
ol
Var
iable
s, n
ot
attr
ibute
s, d
ata
invo
lved
Cap
abili
ty s
tudie
s ad
dre
ss r
ange
of
ind
ivid
ualoutp
uts
Contr
ol ch
arting a
ddre
sses
ran
ge
of
sam
ple
mea
sure
s●
Ass
um
es N
orm
al D
istr
ibution
Rem
ember
the
Em
piric
al R
ule
?In
her
ent
capab
ility
(6s x
) is
com
par
ed t
o
speci
fica
tio
ns
●Req
uires
pro
cess
first
to b
e in
Co
ntr
ol
OPR
E 63
6487
Why
mea
sure
Pro
cess
Cap
abili
ty?
Proc
ess
varia
bilit
y ca
n gr
eatly
impa
ct c
usto
mer
sa
tisfa
ctio
nTh
ree
com
mon
term
s fo
r var
iabi
lity:
1.To
lera
nces
:Sp
ecifi
catio
ns fo
r ran
ge o
f ac
cept
able
val
ues
esta
blis
hed
by
engi
neer
ing
desi
gn o
r cus
tom
er
requ
irem
ents
2.
Proc
ess
varia
bilit
y:N
atur
al o
r inh
eren
t va
riabi
lity
in a
pro
cess
3.C
ontr
ol li
mits
:St
atis
tical
lim
its th
at re
flect
th
e in
here
nt v
aria
tion
of s
ampl
e st
atis
tics
OPR
E 63
6488
Proc
ess C
apab
ility
is b
ased
on
Nor
mal
Cur
ve
4
(95.
5%)
6
(99.
7%)
2 (68%
)
µσ
σσ
OPR
E 63
6489
The
Ran
ge o
f Pro
cess
Out
put
The
rang
e in
whi
ch "
all"
out
put c
an b
e pr
oduc
ed.
6
(99.
7%)
Proc
ess
rang
e =
6
µσ
σ
OPR
E 63
6490
Proce
ss C
apab
ility
Conce
pt
X4.
904.
955.
005.
055.
105.
15cm
Tole
ranc
e ba
nd
Inhe
rent
var
iabi
lity
(6
)
LSL
USL
Out
put
Out
put
out o
f spe
cou
t of s
pec
Out
put
Out
put
out o
f spe
cou
t of s
pec
Proc
ess
outp
utdi
strib
utio
n
5.01
0
σ
OPR
E 63
6491
Tw
o P
roce
ss C
apab
ilities
This
pro
cess
isC
APA
BLE
CA
PAB
LEof
pr
oduc
ing
all g
ood
outp
ut.
Con
trol
the
proc
ess.
Low
erSp
ecLi
mit
Upp
erSp
ecLi
mit
This
pro
cess
isN
OT
CA
PAB
LEN
OT
CA
PAB
LE.
INSP
ECT
-Sor
t out
the
defe
ctiv
es
××
OPR
E 63
6492
Cap
abili
ty A
nal
ysis
LSU
Sµ
Cap
abili
ty a
nal
ysis
det
erm
ines
whet
her
the
inher
ent
variab
ility
of
the
pro
cess
outp
ut
falls
within
the
acce
pta
ble
ran
ge
of
the
variab
ility
allo
wed
by
the
des
ign s
pec
ific
atio
ns
for
the
pro
cess
outp
ut.
The
range
of
poss
ible
solu
tions:
1.
Red
esi
gn
the
pro
cess
so t
hat
it
can a
chie
ve t
he
des
ired
outp
ut
2.
Use
an a
ltern
ate
pro
cess
that
can
ach
ieve
the
des
ired
outp
ut
3.
Ret
ain t
he
curr
ent
pro
cess
but
atte
mpt
to e
limin
ate
unac
cepta
ble
outp
ut
usi
ng 1
00
perc
en
t in
spect
ion
4.
Exa
min
e th
e sp
eci
fica
tio
nto
see
whet
her
they
are
nec
essa
ry o
r co
uld
be
rela
xed
without
adve
rsel
y af
fect
ing c
ust
om
er
satisf
action.
OPR
E 63
6493
Proce
ss C
apab
ility
Rat
io C
p
2
C M
otor
ola,
For
.m
anag
emen
t
Sigm
aSi
x
uses
n C
orpo
ratio
M
otor
ola6
C
wid
thPr
oces
sion
wid
thSp
ecifi
cat
C
C
ratio
ca
pabi
lity
Pr
oces
s
p
pp
p
=
−==
=
σLS
LU
SL
OPR
E 63
6494
Proce
ss C
apab
ility
Index
Cpk
Inde
x C
pkco
mpa
res
the
spre
ad a
nd lo
catio
nof
the
proc
ess,
rela
tive
to th
e sp
ecifi
catio
ns.
3U
pper
Spe
c Li
mit
-X– –
X -L
ower
Spe
c Li
mit
– –O
Rth
e sm
alle
r of:
Cpk
=
{σ 3 σ
Upp
er S
pec
Lim
it -X– –
X -L
ower
Spe
c Li
mit
– –O
RW
here
Zm
inis
the
smal
ler o
f:
Cpk
= Z m
in3
{σ σ
Alte
rnat
e Fo
rm
5-40
OPR
E 63
6495
Proce
ss C
apab
ility
Rat
io C
pk
Nor
mal
dis
tribu
tion
=> 9
9.73
% o
f out
put f
alls
in (µ
+3σ
) whe
n th
e pr
oces
s is c
ente
red.
If
the
proc
ess i
s not
cen
tere
d, w
e us
e
Cpk
= M
in [(
US
-µ) /
3σ,
(µ
-LS)
/ 3σ
]
Exam
ple.
M
BPF
: C
pk=
Min
[0.1
894,
0.5
952]
= 0
.198
4
With
cen
tere
d pr
oces
s (U
S -µ
) = (µ
-LS)
. T
hen
Cpk
= C
p=
(US
-LS)
/ 6σ
=V
oice
of t
he C
usto
mer
Voi
ce o
f the
Pro
cess
=0.
3968
Cp
=0.
861
1.1
1.3
1.47
1.63
2.0
Def
ects
/m =
10K
3K1K
100
101p
pm2
ppm
OPR
E 63
6496
Proc
ess C
apab
ility
: C
exa
mpl
espk
Cpk
= 1.
0C
pk=
1.33
Cpk
= 3.
0
LSL
USL
LSL
USL
Cpk
= 1.
0
LSL
USL
LSL
USL
LSL
USL
(a)
(f)(e
)(d
)
(c)
(b)
Cpk
= 0.
60C
pk=
0.80
LSL
USL
5-42
OPR
E 63
6497
Cap
abili
ty I
mpro
vem
ent
by
Mea
n Sh
ift
LS
= 7
5U
S =
85
Cpk=
0.76
60
80
Cp
k=
0.6
99
0
82.5
Gar
age
Door
Wei
ght
(kg)
Probab
ility
den
sity
of outp
ut
(wei
ght)
(bef
ore
shift)
(aft
er
shif
t)
Off
sp
ec
Off
sp
ec
Proc
ess
Adju
stm
ent
OPR
E 63
6498
Cap
abili
ty I
mpro
vem
ent
by
Var
ianc
e Red
uctio
n
LS
= 7
5U
S =
85
befo
reC
pk
= 0
.76
60
80
Cp
k=
0
.95
44
Gar
age
Door
Wei
ght
(kg)
Probab
ility
den
sity
of outp
ut
(wei
ght)
Aft
er
red
uct
ion
OPR
E 63
6499
Proce
ss C
ontr
ol an
d C
apab
ility
: Rev
iew
●Ev
ery
proc
ess
disp
lays
som
e va
riabi
lity—
norm
al o
r ab
norm
al●
Con
trol c
harts
can
iden
tify
abno
rmal
var
iabi
lity
●C
ontro
l cha
rts m
ay g
ive
fals
e (o
r mis
sed)
ala
rms
by
mis
taki
ng n
orm
al (a
bnor
mal
) for
abn
orm
al (n
orm
al)
varia
bilit
y●
On-
line
cont
rol l
eads
to e
arly
det
ectio
n an
d co
rrect
ion
●A
proc
ess
“in c
ontro
l” in
dica
tes
only
its
inte
rnal
sta
bilit
y●
Impr
ovin
g pr
oces
s ca
pabi
lity
invo
lves
cha
ngin
g th
e m
ean
and/
or re
duci
ng n
orm
al v
aria
bilit
y re
quiri
ng a
long
te
rm in
vest
men
t
OPR
E 63
6410
0
Des
ign for
Cap
able
Pro
cess
ing
•Si
mpl
ify
–Few
er p
arts
, ste
ps–M
odul
ar d
esig
n•
Stan
dard
ize
–Les
s va
riety
–Sta
ndar
d, p
rove
n pa
rts, a
nd p
roce
dure
s•
Mis
take
-pro
of–C
lear
spe
cs–E
ase
of a
ssem
bly,
dis
asse
mbl
y, s
ervi
cing
OPR
E 63
6410
1
Tag
uch
i Q
ual
ity
Philo
sophy
Loss
= k
(P -
T)2
not
0 if
with
in s
pecs
and
1 if
out
side
On
Targ
etis
mor
e im
porta
nt th
an
With
in S
pecs
LST
US
Conve
ntional
vie
wTag
uch
i’s v
iew
LS
TU
S
OPR
E 63
6410
2
Robust
Des
ign
Targ
et
Perfo
rman
ce (T
)Ac
tual
Pe
rform
ance
(P)
Des
ign
Par
amet
ers
(D)
Noi
se F
acto
rs (N
): In
tern
al &
Ext
erna
l
Prod
uct /
Pro
cess
•Id
entify
Pro
duct
/Pro
cess
Des
ign P
aram
eter
s th
at–
Hav
e si
gnific
ant
/ lit
tle
influen
ce o
n P
erfo
rman
ce–
Min
imiz
e per
form
ance
var
iation d
ue
to N
ois
e fa
ctors
–M
inim
ize
the
pro
cess
ing c
ost
•
Met
hodolo
gy:
Des
ign o
f Exp
erim
ents
(D
OE)
•Exa
mple
s -
Choco
late
mix
, In
a Tile
Co.,
Sony
TV
OPR
E 63
6410
3
The
Des
ign P
roce
ss
•G
oal
–Dev
elop
hig
h qu
ality
, low
cos
t pro
duct
s, fa
st•
Impo
rtanc
e–8
0% p
rodu
ct c
ost,
70%
qua
lity,
65%
suc
cess
•C
onve
ntio
nal
–Tec
hnol
ogy-
driv
en, I
sola
ted,
Seq
uent
ial,
Itera
tive
•D
iffic
ultie
s –R
evis
ions
, cos
t ove
rruns
, del
ays,
retu
rns,
reca
lls•
Solu
tion
–Cus
tom
er-d
riven
(QFD
), jo
intly
pla
nned
, pr
oduc
ible
OPR
E 63
6410
4
Concu
rren
t D
esig
n
•O
bjec
tive
–Int
erfu
nctio
nalc
oord
inat
ion
to s
atis
fy c
usto
mer
–Inv
olve
man
ufac
turin
g, s
uppl
iers
, R&D
•Pr
ereq
uisi
tes
–Bre
ak d
own
barri
ers
–Cro
ss fu
nctio
nal t
rain
ing
–Com
mun
icat
ion,
team
wor
k, g
roup
dec
isio
ns•
Res
ult
–Few
er re
visi
ons,
mis
com
mun
icat
ion,
del
ays
•D
iffic
ultie
s –T
ime
cons
umin
g, c
ompl
ex, o
rgan
izat
iona
l
OPR
E 63
6410
5
Ref
eren
ces
http
://de
min
g.en
g.cl
emso
n.ed
u/pu
b/tu
toria
ls/q
ctoo
ls/c
so.h
tmC
ontro
l cha
rt ca
se s
tudy
http
://w
ww
.qua
litya
mer
ica.
com
/kno
wle
dgec
ente
/kno
wct
rSPC
_Arti
cles
.htm
SPC
arti
cles