Control Charts for Counts: Understanding p and u Charts in Statistical Quality Assurance -, Study notes of Engineering

Download Control Charts for Counts: Understanding p and u Charts in Statistical Quality Assurance - and more Study notes Engineering in PDF only on Docsity!

IE 361 Module 13

Control Charts for Counts ("Attributes

Data")

Prof.s Stephen B. Vardeman and Max D. Morris

Reading: Section 3.3,

Statistical Quality Assurance Methods for Engineers

In this module, we discuss the Shewhart control charts for so-called "fractionnonconforming" and "mean nonconformities per unit" contexts.

These are the

Shewhart

p

and

np

charts and the Shewhart

u

(and

c

) charts.

These tools

are easy enough to explain and use, but are typically really NOT very e

ff

ective

in modern applications where acceptable nonconformity rates are so small asto be stated in "parts per million." p

Charts (and

np

Charts)

The scenario under which a

p

chart or (

np

chart) is potentially appropriate

is one where periodically groups of

n

items (or outcomes) from a process are

In this kind of circumstance, the notation

p

X

n

the sample fraction nonconforming

is standard, and

control charts for

p

are called

p

charts

control charts for

X

n

p

) are called

np

charts

If the process producing items/outcomes is physically stable, a reasonable prob-ability model for

X

(met in Stat 231) is the

binomial

n, p

distribution, where

p

the current probability that any particular outcome is nonconforming

(the mental

fi

ction here is that the particular

n

outcomes observed are a random

sample of a huge pool of outcomes, a fraction

p

of which are nonconforming).

Stat 231 facts about the binomial distribution are that

μ

X

np

and

σ

X

q

np

p

so that (since

p

³

1 n

´

X

μ

ˆ p

p

and

σ

ˆ p

s

p

p

n

These facts in turn lead to

standards given (

p

chart) control limits for

p

LCL

ˆ p

p

s

p

p

n

and

UCL

ˆ p

p

s

p

p

n

Standards given control limits for

p

here are

LCL

ˆ p

p

s

p

p

n

s

and

UCL

ˆ p

s

A

JMP

plot of the corresponding control chart is in Figure 2 and shows clearly

that sample 9 produces an out-of-control signal.

That sample simply does not

fi

t the "stable process with

p

" model that stands behind the control

limits.

Figure 2: Standards Given (

p

p

Chart for the Arti

fi

cial Data

and retrospective control limits for

p

are

LCL

ˆ p

s

and

UCL

ˆ p

s

Figure 3 show that the retrospective limits do not produce a picture muchdi

ff

erent from the standards given ones used to make Figure 2.

The 20 samples

do not

fi

t with a "constant

p

" model.

Figure 3: Retrospective

p

Chart for the Arti

fi

cial Data

Example 13-

Below is a comparison between (

p

) standards given

limits for

p

for

n

and

n

n

CL

ˆ p

LCL

ˆ p

UCL

ˆ p

none

u

Charts (and

c

Charts)

The scenario under which a

u

chart or (

c

chart) is potentially appropriate is one

where periodically

k

inspection units of product or production from a process

are looked at and

X

the total number of "nonconformities" found across those

k

units

Figure 4: Cartoon of

k

Inspection Units of Process Output and

X

"Noncon-

formities"is observed.

This is illustrated in the cartoon in Figure 4, where "x"’s represent

nonconformities.

distribution (the Poisson distribution with mean

μ

X

kλ

A Stat 231 fact

about the Poisson distribution is that its mean and variance are the same, thatis that

σ

X

μ

X

So in the present context

μ

X

kλ

and

σ

X

kλ

and in turn that

μ

ˆ u

λ

and

σ

ˆ u

s

λ k

These facts lead to

standards given (

u

chart) control limits for

u

LCL

ˆ u

λ

s

λ k

and

UCL

ˆ u

λ

s

λ k

and for the case of

k

(so that

u

X

standards given (

c

chart) control

limits for

X

LCL

X

λ

λ

and

LCL

X

λ

λ

Example 13-

Below are some arti

fi

cially generated (using

λ

) Poisson

data,

X

We can think of these as either counts or rates for a constant number

of inspection units

k

Sample

X

Sample

X

Sample

k

X

u

X

k

UCL

ˆ u

q

2 k

LCL

ˆ u

none

none

none

none

none

Figure 5 is a

JMP

control chart for the whole data set, and there is no reason

in these data to doubt that the nonconformity rate is constant at

λ

per

inspection unit.

(Note that

JMP

in ambiguous fashion indicates that

LCL

It is better practice and provides clearer intent to say that there is no lower

control limit.)

Figure 5: Standards Given (

λ

u

Chart for the Arti

fi

cial Data