Quality management: SPC III – control charts for

attributes

Presented by:

Dr. Husam Arman

Control charts for nonconforming (defective):p (percentage of defective units) np (number of defective units)

Control charts for nonconformities (defects):c (number of defects per sample) u (number of defects per unit)

Control charts for attributes

Many quality characteristics are not measured on a continuous scale or even a quantitative scale.

In these cases, we may judge each unit of product as either conforming or nonconforming (defective) - on the basis of whether or not it possesses certain attributes.

In general charts for attributes are less expensive than charts for variables, because measurements are less complex to be performed.

Control charts for attributes

However, attributes charts are generally not as informative as variables charts because there is typically more information in a numerical measurement than in merely classifying a unit as conforming or nonconforming.

For that reason sample size is typically larger than for variable charts (at least n=10)

Control charts for attributes

Control charts for attributes

“Defect”: nonconformity that is serious enough to significantly affect the safe or effective use of the product

“Defective”: product unit with one or more defects.

We will see the following charts for attributes:

(p) control chart for fraction nonconformingDefectives

(np) control chart for number nonconforming

(u) control charts for nonconformities per unitDefects (for every unit of product we count defects on

it)

(c) control charts for nonconformities

Control charts for attributes

The fraction nonconforming (p) is defined as the ratio of the number of defective items in a population to the total number of items in that population:

p-CHARTS (Control Charts for Fraction Nonconforming)

n

ppp n...21p

p-CHARTS (Control limits )

Since p represents a probability, negative values of LCL are senseless→ If calculated LCL<0, then we just use LCL=0→ for that reason, control limits can be often asymmetric

npLCL

npUCL

p

p

)p1(p3

)p1(p3

p-CHARTS example

In some applications, the sample is a 100% inspection of process output over some period of time.

Since different numbers of units could be produced in each period, the control chart would then have a variable sample size.

In these cases, we have m samples with different sizes (ni).

m

ii 1

p̂p

m

p-CHARTS (variabe sample size)

There are 3 possible strategies:1) Determine different control limits for each individual

sample, depending on the sample size (ni):

… as a result, each point of the control chart is associated to 2 individual control limits

i

i

p (1 p)UCL p 3

n

p (1 p)LCL p 3

n

p-CHARTS (variable sample size)

2) Constant control limit calculated considering the average sample size. This approach can be reasonable if sample sizes do not differ greatly.

3) Use of the Standardized Control Chart.

m

ii 1

nn

m

p-CHARTS (variable sample size)

np-CHARTS (number nonconforming control charts)

n·p

p

np-CHARTS (number nonconforming control charts)

np-CHARTS (number nonconforming control charts)

Control charts for nonconformities (DEFECTS) – ‘c’ and ‘u’ charts

A nonconforming item is a unit of product that does not satisfy one or more of the specification for that product.

Each specific point at which a specification is not satisfied results in a defect or nonconformity.

Consequently, a nonconforming item (defective) will contain at least one nonconformity (defect).

c - charts

It is possible to develop control charts for the number of nonconformities in an inspection unit (n = 1).

The inspection unit is simply an entity for which it is convenient to keep records.

Examples: bubbles in a sheet of glass, flaws in a bolt of fabric, stains in a ream of paper

c - charts

1) c1

2) c2

......

m) cm

c

u – charts: constant sample size

Same as c-chart, but here n > 1. Example: circuit boards, c-chart or u-chart?

n

cu

m

i

m

i

n

cuCenterline

1

1)(

u is plotted on the chart

u – charts: constant sample size

u – charts: variable sample size

Same as with p-charts, we either use average n or we calculate individual limit for each sample

u – charts: variable sample size

u – charts: variable sample size

Other variable charts: individual chart

In many situations the sample size used for process monitoring consists of an individual unit (n=1). For example:

When the production rate is very slow and it is inconvenient to allow sample sizes of n>1 to accumulate before analysis (f.e. aircraft wings).

Destructive tests (f.e. to test the hermetic sealing of a plastic container, or to test a match just striking it).

In general, the larger the sample size (n), the more sensitive is the chart in detecting process shifts.

The drawback is that – for each point of the chart – more units are needed → more expenses to collect data

→ charts for individual measurements are less sensitive than “conventional” control charts

Individual chart

Considering (sample average) shifts of the same size, in the second case a larger portion of the distribution is outside the control ( ). So the probability to detect a shift is higher

x3

When the sample size consists of an individual unit (n=1), sample variability cannot be directly evaluated:

However, the moving range of two (or more) consecutive observations can be used to estimate the process variability

MRi = |xi – (xi -1)| In this case, we can build an R-chart,

considering samples made of n=2 successive units

R = xmax – xmin

Individual chart

… in this way we artificially create a variability to associate to each individual measurement

n = 2

Individual chart (moving-range chart)

It is essential to check the normality assumption when using the control chart for individuals.

For the moving-range (MR) chart we have:

MR 4

MR

MR 3

UCL D MR 3.267 0.5726 1.871

CL MR 0.5726

LCL D MR 0 0.5726 0

D3 and D4 are functions of the moving range sample size (n=2 in this case)

Individual chart (moving-range chart)

The construction of individual measurements charts with moving range, is similar to the -R charts’ construction.

For the control chart for individual measurements (x), the parameters are:

x2

x

x2

MRUCL x 3 x 3

d

CL x

MRLCL x 3 x 3

d

d2 is a function of the moving range sample size (n=2 in this case)

x

Individual chart (moving-range chart)

For individual measurements charts, we estimated 2

MRˆ

d

Individual chart (X-chart)