Session XI

“Process Capability”

Central Limit Theorem

If the population from which samples are taken is not

normal, the distribution of sample averages will tend

toward normality provided that the sample size, n, is at

least 4. This tendency gets better and better as the

sample size gets larger. The standardized normal can be

used for the distribution averages with the

modification.

X

X XZ

n

Illustration of central limit theorem

Central Limit Theorem

Dice illustration of central limit theorem

Central Limit Theorem

Relationship of limits, specifications, and distributions

Control Limits & Specifications

Control Limits & Specifications

The control limits are established as a function of

the average

Specifications are the permissible variation in the

size of the part and are, therefore, for individual

values

The specifications or tolerance limits are established

by design engineers to meet a particular function

Process Capability & Tolerance

The process spread will be referred to as the

process capability and is equal to 6σ

The difference between specifications is called the

tolerance

When the tolerance is established by the design

engineer without regard to the spread of the

process, undesirable situations can result

Process Capability & Tolerance

Three situations are possible:

Case I: When the process capability is less than the

tolerance 6σ<USL-LSL

Case II: When the process capability is equal to the

tolerance 6σ=USL-LSL

Case III: When the process capability is greater

than the tolerance 6σ >USL-LSL

Process Capability & Tolerance

Case I: When the process capability is less than the

tolerance 6σ<USL-LSL

Case I: 6σ < USL-LSL

Case II: 6σ = USL-LSL

Process Capability & Tolerance

Case II: When the process capability is equal to the

tolerance 6σ=USL-LSL

Case III: 6σ > USL-LSL

Process Capability & Tolerance

Case III: When the process capability is greater than

the tolerance 6σ>USL-LSL

Process Capability The range over which the natural variation of a

process occurs as determined by the system of

common or random causes

Measured by the

proportion of output that

can be produced within

design specifications

This following method of calculating the process

capability assumes that the process is stable or in

statistical control:

• Take 25 (g) subgroups of size 4 for a total of 100

measurements

• Calculate the range, R, for each subgroup

• Calculate the average range, R bar= ΣR/g

• Calculate the estimate of the population standard

deviation

• Process capability will equal 6σ0

µ0

2

R

d

Process Capability

The process capability can also be obtained by using the standard deviation: • Take 25 (g) subgroups of size 4 for a total of 100

measurements• Calculate the sample standard deviation, s, for each

subgroup• Calculate the average sample standard deviation, s

bar = Σs/g• Calculate the estimate of the population standard

deviation • Process capability will equal 6σo µ

0

4

s

c

Process Capability

Capability Index

Process capability and tolerance are combined to form

the capability index.

0

0

6

6

p

p

USL LSLC

where C capabilityindex

USL LSL tolerance

process capability

Capability Index

The capability index does not measure process

performance in terms of the nominal or target

value. This measure is accomplished by Cpk.

0

{( ) ( )

3

6

pk

p

Min USL X or X LSLC

where C capabilityindex

USL LSL tolerance

process capability

Cp = USL - LSL

6 ơo

(USL- ¯X), (¯X-LSL)} Cpk = min{

The Capability Index does not measure process

performance in terms of the nominal or target

Capability Index

1. The Cp value does not change as the process center

changes

2. Cp=Cpk when the process is centered

3. Cpk is always equal to or less than Cp

4. A Cpk = 1 indicates that the process is producing

product that conforms to specifications

5. A Cpk < 1 indicates that the process is producing

product that does not conform to specifications

Capability Index

6. A Cp < 1 indicates that the process is not capable

7. A Cpk=0 indicates the average is equal to one of the

specification limits

8. A negative Cpk value indicates that the average is

outside the specifications

Capability Index

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

Cpk Measures