Minitab Technical Support Document Capability Analysis ...

Definitions

T LSL USL ? tol m

x

Within ^Within = StDev(Within) n ni c4(ni)

Overall ^ Overall = StDev(Overall) P(X) Z xi xij

Minitab Technical Support Document

Capability Analysis (Normal) Formulas ? Capability Statistics (Default)

= 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, ^Within is estimated using the average moving range by default. If the subgroup size is greater than 1, ^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.

Contact Tech Support: Tel:

E-mail: Web:

US Office ? Minitab Inc. +1-814-231-2682 techsupport@

UK Office ? Minitab Ltd. +44 (0) 24 7643 7507 techsupport@minitab.co.uk

Page 1

French Office ? Minitab SARL +33 (0) 1 55 33 12 60 support@minitab.fr

Formulas

Capability Statistics (Cp, Pp)

Cp =

(USL - LSL) ( ) 6 * Within

(USL - ? )

CPU = ( ) 3 * Within

CPL =

(? - LSL) ( ) 3 * Within

Cpk = minimum{CPU, CPL}

Minitab Technical Support Document

Capability Analysis (Normal) Formulas ? Capability Statistics (Default)

C^p

=

(USL - LSL) ( ) 6 *^Within

CP^U

=

(USL - x) ( ) 3 *^Within

CP^L

=

(x - LSL) ( ) 3 *^Within

{ } Cp^k = minimum CP^U ,CP^L

C^pm = (USL - LSL)

tol *

(xi - T )2

(n -1)

C^pm = min(T - LSL,USL - T )

tol * 2

(xi - T )2

(n -1)

C^pm = (USL - T )

tol * 2

(xi - T )2

(n -1)

C^pm = (T - LSL)

tol * 2

(xi - T )2

(n -1)

when LSL and USL are given, and T=m

when LSL and USL are given, and Tm

when USL and T are given

when LSL and T are given

Note: This formula is always used in Release 13. Cpm is not calculated in Release 13 if there is only one specification limit. Note: This formula only applies to Release 14.

Note: This formula only applies to Release 14.

Note: This formula only applies to Release 14.

Contact Tech Support: Tel:

E-mail: Web:

US Office ? Minitab Inc. +1-814-231-2682 techsupport@

UK Office ? Minitab Ltd. +44 (0) 24 7643 7507 techsupport@minitab.co.uk

Page 2

French Office ? Minitab SARL +33 (0) 1 55 33 12 60 support@minitab.fr

Minitab Technical Support Document

Capability Analysis (Normal) Formulas ? Capability Statistics (Default)

CC^pk = USL - ?^ 3^ within

when only USL is given

CC^pk = ?^ - LSL 3^ within

CC^pk = min{(USL - ?^ ), (?^ - LSL)}

3^ within

when only LSL is given when LSL and USL are given

where:

?^ = Target

when Target is given

?^ = 1 (LSL + USL) when both LSL and USL are given but no Target

2

?^ = x

otherwise

Pp = (USL - LSL)

6 * Overall

PPU = (USL - ? )

3 * Overall

PPL = (? - LSL)

3 * Overall

Ppk = minimum{PPU, PPL}

P^p = (USL - LSL)

6 *^ Overall

PP^U

=

(USL - x ) ( ) 3 *^ Overall

PP^ L

=

(x - LSL) ( ) 3*^Overall

{ } Pp^k = minimum PP^U , PP^L

Note: The capability statistics displayed in the MINITAB output are estimates, but for simplicity do not include the "hats". For example, Cp^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.

Contact Tech Support: Tel:

E-mail: Web:

US Office ? Minitab Inc. +1-814-231-2682 techsupport@

UK Office ? Minitab Ltd. +44 (0) 24 7643 7507 techsupport@minitab.co.uk

Page 3

French Office ? Minitab SARL +33 (0) 1 55 33 12 60 support@minitab.fr

Minitab Technical Support Document

Capability Analysis (Normal) Formulas ? Capability Statistics (Default)

Confidence Intervals for Capability Statistics

Definitions for Confidence Intervals

= f n df

df = k(n -1)

k = number of samples n = average sample size

fn

=

adjustment

for

method

used

to

estimate

2 within

For:

? Pooled standard deviation, fn = 1 ? Average and median moving range, = k - w +1 where w is the number of observations used to calculate the moving range

? Sbar, fn varies with n

n

fn

2

0.88

3

0.92

4

0.94

5

0.95

6-7

0.96

8-9

0.97

10-

0.98

17

18-

0.99

64

65+ 1.00

? Rbar, fn = 0.9 ? Square root of MSSD, = k -1

Contact Tech Support: Tel:

E-mail: Web:

US Office ? Minitab Inc. +1-814-231-2682 techsupport@

UK Office ? Minitab Ltd. +44 (0) 24 7643 7507 techsupport@minitab.co.uk

Page 4

French Office ? Minitab SARL +33 (0) 1 55 33 12 60 support@minitab.fr

Formulas for Confidence Intervals

Minitab Technical Support Document

Capability Analysis (Normal) Formulas ? Capability Statistics (Default)

Cp

LowerBound = C^p

2 1- 2,

, UpperBound = C^p

2 2,

where is defined under "Definitions of Confidence Intervals" above.

Cpk

LowerBound = Cp^k - Z 2

1 9kn

+ Cp^k 2 2

,

UpperBound

= Cp^k

+ Z 2

1 + Cp^k 2 9kn 2

where k, n, and are defined under "Definitions of Confidence Intervals" above.

Cpm

LowerBound = Cp^m

2 1- 2,

, UpperBound = Cp^m

2 2,

where:

( ) = kn 1 + a2 2 1 + 2a2

a = x - T arg et ^ overall

k and n are defined under "Definitions of Confidence Intervals" above

Pp

LowerBound = Pp^

2 1- 2,kn-1

,

UpperBound

= Pp^

2 2,kn-1

kn -1

kn -1

where k and n are defined under "Definitions of Confidence Intervals" above.

Ppk

LowerBound = Pp^k - Z 2

1 9kn

+

Pp^k 2

2(kn -1)

,

UpperBound

=

Pp^ k

+

Z 2

1 9kn

+

Pp^k 2

2(kn -1)

where k and n are defined under "Definitions of Confidence Intervals" above.

Contact Tech Support: Tel:

E-mail: Web:

US Office ? Minitab Inc. +1-814-231-2682 techsupport@

UK Office ? Minitab Ltd. +44 (0) 24 7643 7507 techsupport@minitab.co.uk

Page 5

French Office ? Minitab SARL +33 (0) 1 55 33 12 60 support@minitab.fr

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download