Minitab Technical Support Document
Capability Analysis (Normal) Formulas –
Capability Statistics (Default)
Definitions
T
LSL
USL
µ
tol
m
x
σ Within
σˆ Within = StDev(Within)
n
ni
c4(ni)
σ Overall
σˆ Overall = StDev(Overall)
P(X)
Z
xi
xij
= target
= lower specification limit
= upper specification limit
= process mean
= sigma tolerance
= midpoint of LSL and USL
= estimate of process mean
= within subgroup process standard deviation
= estimate of within subgroup process standard deviation
= total number of (nonmissing) observations
= number of (nonmissing) observations in subgroup i
= unbiasing constant for subgroups of size ni (for use with sample standard
deviations)
= overall process standard deviation
= estimate of overall process standard deviation
= probability of event X
= standard normal variable
= the ith observation
= the jth observation of the ith subgroup
Note: If the subgroup size is 1, σˆ Within is estimated using the average moving range by default. If the subgroup
size is greater than 1, σˆ Within is estimated using a pooled standard deviation by default. See Methods of
Estimating Sigma below.
Note: c4(ni) is an unbiasing constant whether or not the sample size is small or large. The bias is negligible for
large samples (see, e.g. Farnum), as with larger samples, c4(ni) approaches the value of one.
In Releases 13 and 14 you can estimate sigma either using unbiasing constants or not. The default is to use
unbiasing constants. To not use unbiasing constants, choose Stat > Quality Tools > Capability Analysis >
Normal > Estimate (In Release 13, Stat > Quality Tools > Capability Analysis (Normal) > Estimate) and
uncheck Use unbiasing constants.
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Capability Analysis (Normal) Formulas –
Capability Statistics (Default)
Formulas
Capability Statistics (Cp, Pp)
(USL − LSL )
(6 * σ Within )
(USL − µ )
CPU =
(3 * σ Within )
(µ − LSL )
CPL =
(3 * σ Within )
(USL − LSL )
(6 * σˆ Within )
(USL − x )
CPˆ U =
(3 * σˆ Within )
(x − LSL )
CPˆ L =
(3 * σˆ Within )
Cp =
Ĉp =
Cpk = minimum{CPU, CPL}
Cpˆ k = minimum CPˆ U , CPˆ L
Ĉpm =
(USL − LSL)
∑ (x
tol *
Ĉpm =
2
min (T − LSL, USL − T )
∑ (x
i
−T)
Note: This formula is always used in Release 13.
Cpm is not calculated in Release 13 if there is
only one specification limit.
∑ (x
i
−T)
when LSL and USL are
given, and T≠m
Note: This formula only applies to Release 14.
when USL and T are given
Note: This formula only applies to Release 14.
when LSL and T are given
Note: This formula only applies to Release 14.
2
(n − 1)
(T − LSL)
tol
*
2
2
(n − 1)
(USL − T )
tol
*
2
Ĉpm =
−T)
}
when LSL and USL are
given, and T=m
(n − 1)
tol
*
2
Ĉpm =
i
{
∑ (x
i
−T)
2
(n − 1)
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Minitab Technical Support Document
Capability Analysis (Normal) Formulas –
Capability Statistics (Default)
USL − µˆ
CCˆ pk =
3σˆ within
µˆ − LSL
CCˆ pk =
3σˆ within
when only USL is given
when only LSL is given
min{(USL − µˆ ), (µˆ − LSL )} when LSL and USL are given
CCˆ pk =
3σˆ within
where:
when Target is given
µ̂ = Target
when both LSL and USL are given but no Target
1
µ̂ = (LSL + USL )
2
µ̂ = x
otherwise
Pp =
(USL − LSL )
P̂p =
6 * σ Overall
(USL − µ )
PPU =
3 * σ Overall
(µ − LSL )
PPL =
3 * σ Overall
Ppk = minimum{PPU, PPL}
(USL − LSL )
6 * σˆ Overall
(USL − x )
PPˆ U =
(3 * σˆ Overall )
(x − LSL )
PPˆ L =
(3 * σˆ Overall )
Ppˆ k = minimum PPˆ U , PPˆ L
{
}
Note: The capability statistics displayed in the MINITAB output are estimates, but for simplicity do not include
the “hats”. For example, Cpˆ k is displayed as Cpk.
Note: In Release 14, you can display confidence intervals for the capability statistics. To see the formulas, see
Confidence Intervals below.
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Capability Analysis (Normal) Formulas –
Capability Statistics (Default)
Confidence Intervals for Capability Statistics
Definitions for Confidence Intervals
ν = f n ∗ df
df = k (n − 1)
k = number of samples
n = average sample size
2
fn = adjustment for method used to estimate σ within
For:
• Pooled standard deviation, fn = 1
• Average and median moving range, ν = k − w + 1 where w is the number of observations used to
calculate the moving range
•
Sbar, fn varies with n
n
2
3
4
5
6-7
8-9
1017
1864
65+
•
•
fn
0.88
0.92
0.94
0.95
0.96
0.97
0.98
0.99
1.00
Rbar, fn = 0.9
Square root of MSSD, ν = k − 1
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Capability Analysis (Normal) Formulas –
Capability Statistics (Default)
Formulas for Confidence Intervals
Cp
LowerBound = Cˆ p
χ 12−α
2 ,ν
ν
χ α2 ,ν
, UpperBound = Cˆ p
2
ν
where ν is defined under “Definitions of Confidence Intervals” above.
Cpk
Cpˆ k 2
Cpˆ k 2
1
1
+
, UpperBound = Cpˆ k + Z α 2
+
9kn
2ν
9kn
2ν
where k, n, and ν are defined under “Definitions of Confidence Intervals” above.
LowerBound = Cpˆ k − Z α 2
Cpm
LowerBound = Cpˆ m
χ 12−α
2 ,ν
ν
, UpperBound = Cpˆ m
χ α2 ,ν
2
ν
where:
(1 + a )
ν = kn
2 2
1 + 2a 2
x − T arg et
a=
σˆ overall
k and n are defined under “Definitions of Confidence Intervals” above
Pp
LowerBound = Ppˆ
χ 12−α
2 , kn −1
, UpperBound = Ppˆ
χ α2 ,kn−1
2
kn − 1
kn − 1
where k and n are defined under “Definitions of Confidence Intervals” above.
Ppk
1
Ppˆ k 2
1
Ppˆ k 2
+
+
, UpperBound = Ppˆ k + Z α 2
9kn 2(kn − 1)
9kn 2(kn − 1)
where k and n are defined under “Definitions of Confidence Intervals” above.
LowerBound = Ppˆ k − Z α 2
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Capability Analysis (Normal) Formulas –
Capability Statistics (Default)
PPM
i) Observed Performance PPM < LSL = 1,000,000 * ((number of observations < LSL) / n)
ii) Observed Performance PPM > USL = 1,000,000 * ((number of observations > USL) / n)
Observed Performance Total = sum of (i) and (ii)
⎛
(LSL − x ) ⎞⎟
iii) Exp. “Within” Performance PPM < LSL = 1,000,000 * P ⎜⎜ Z >
σˆWithin ⎟⎠
⎝
⎛
(USL − x ) ⎞⎟
iv) Exp. “Within” Performance PPM > USL = 1,000,000 * P⎜⎜ Z >
σˆWithin ⎟⎠
⎝
Exp. “Within” Performance Total = sum of (iii) and (iv)
⎛
(LSL − x ) ⎞⎟
v) Exp. “Overall” Performance PPM < LSL = 1,000,000 * P⎜⎜ Z >
σ̂ Overall ⎟⎠
⎝
⎛
(USL − x ) ⎞⎟
vi) Exp. “Overall” Performance PPM > USL = 1,000,000 * P⎜⎜ Z >
σˆ Overall ⎟⎠
⎝
Exp. “Overall” Performance Total = sum of (v) and (vi)
Note: % figures are obtained by dividing the formulas above by 10,000.
Note: If you use a historical mean, that value will be used in place of X for CPˆ U , CPˆ L , Cpˆ k , PPˆ U , PPˆ L ,
Ppˆ k , and all PPM (Expected).
If you use a historical sigma, that value will be used in place of σˆ Within for Ĉp CPˆ U , CPˆ L , and all PPM
(Expected “Within”).
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Capability Analysis (Normal) Formulas –
Capability Statistics (Default)
Methods of Estimating σ
Overall standard deviation
σˆ Overall = StDev(Overall) =
s=
s
c 4 (n)
⎡
2⎤
⎢∑∑ (xij − x ) ⎥
⎦
⎣ i j
(n − 1)
Estimating σWithin when subgroup size > 1
Pooled standard deviation – Default
sp
σˆ Within = StDev(Within) =
c 4 (d )
⎡
2⎤
⎢∑∑ (xij − xi ) ⎥
i
j
⎦
sp = ⎣
⎡
⎤
⎢∑ (ni − 1)⎥
⎣ i
⎦
d = ∑ (ni − 1) + 1 degrees of freedom
Note: When the subgroup size is constant, the pooled standard deviation = (square root of the average of the
variances) / c4(∑(ni-1)+1):
sp
σˆ Within = StDev(Within) =
c 4 (d )
k
sp =
∑s
i =1
2
i
k
d = n − k + 1 degrees of freedom
k = number of subgroups
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Capability Analysis (Normal) Formulas –
Capability Statistics (Default)
Rbar - Average of subgroup ranges
⎛ fr ⎞
∑i ⎜⎜ d (i ni ) ⎟⎟
⎝ 2 i ⎠
σˆ Within = StDev(Within) =
∑ fi
i
[d 2 (n)]
[d 3 (n)]2
2
fi =
σˆ Within = StDev(Within) =
R
if all ni are the same
d 2 (ni )
Sbar - Average of subgroup standard deviations
⎛ hs ⎞
∑ ⎜⎜ c (i ni ) ⎟⎟
⎝ 4 i ⎠
σˆ Within = StDev(Within) =
∑ (hi )
⎡
2⎤
⎢∑ (xij − xi ) ⎥
⎦
⎣ j
(ni − 1)
si =
hi =
(c4 (ni ))2
2
1 − (c 4 (ni ) )
Note: When the subgroup size is constant, sbar = (average of the subgroup standard deviations) / c4(ni):
s
σˆ Within = StDev(Within) =
c 4 (ni )
k
s=
∑s
i =1
i
k
k = number of subgroups
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Capability Analysis (Normal) Formulas –
Capability Statistics (Default)
Estimating σWithin when subgroup size = 1
Average moving range – Default
MR
d 2 (w)
MRi = the ith moving range = max[xi, …, xi-w+1] – min[xi, …, xi-w+1], for i = w, …, n
(MRw + ... + MRn )
MR =
(n − w + 1)
w = the number of observations used in the moving range. w = 2 by default.
σˆ Within = StDev(Within) =
Median moving range
~
MR
σˆ Within = StDev(Within) =
d 4 (w)
MRi = the ith moving range = max[xi, …, xi-w+1] – min[xi, …, xi-w+1], for i = w, …, n
~
MR = the median of all MRi
w = the number of observation used in the moving range. w = 2 by default.
Square root of MSSD – Square root of half of the Mean of the Squared Successive Differences
(
σˆ Within = StDev(Within) =
1 ∑ di
*
2 (n − 1)
c 4 ( ni ) |
2
)
di = successive differences
c4(ni)| ≅ c4(ni)
References:
Montgomery, D.C. (2001). Introduction to Statistical Quality Control, 4th Edition. John Wiley & Sons.
Donald J. Wheeler(1995). "Advanced Topics in Statistical Process Control", SPC Press, Inc.
N. R. Farnum (1994). “Modern Statistical Quality Control and Improvement”, Wadsworth Publishing.
I.W. Burr (1976). Statistical Quality Control Methods, Marcel Dekker, Inc.
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