Control Charts for Attributes
TECH 4462
See also
ASTM E2587
ANSI/ASQ B1-B3
Attribute Data
“observed values or test results that indicate the presence or absence of specific characteristics or counts of occurrences of events in time or space.” ASTM E2587
Why Use Attributes
Where measurement is not possible
Where measurement is possible but is expensive or does not offer additional information.
Nonconforming Vs. Defective
Nonconforming Vs. Nonconformity
Nonconforming compares an object to a specification
Defective compares an object to an intended purpose
The whole item can be either conforming or nonconforming,
The object may have one or more nonconformities on it.
Limitations of Variables Control Charts
Cannot be applied to attribute data. Attributes cannot be converted to variables,
but
Variables can be converted to attributed data.
by
comparing to specification limits.
But
there is a loss of data that may have been helpful.
Types of Attribute Charts
Binary distribution
Poisson distribution
The p Chart
Used for subgroups consisting of the fraction occurrence of an event (nonconforming).
p = proportion or fraction nonconforming in the sample or subgroup
n = number in the sample or subgroup
np = number nonconforming in the sample or subgroup
Example
During first shift, 450 inspections are made of book shipments and 5 nonconforming units were found. Production during the shift was 15,000 units. What is the fraction nonconforming?
What about the 15,000 units?
Objectives of p Charts
Determine average quality level
Bring attention to changes in average
Improve quality level
Evaluate performance of operating and management personnel
Suggest places for Xbar and R charts
Determine acceptance criteria of a product before shipment to customer
Constructing a p-chart (Constant subgroup size)
Note: data can be “rolled up” into summary charts
Also, charts can be used to evaluate performance of:
An operator, work center, department, shift, plant, or whole corporation.
Constructing a p-chart (Constant subgroup size)
2. Determine the subgroup size and method.
Based on proportion nonconforming and confidence limit desired.� A small proportion of nonconforming units requires high inspection.� A minimum of 50 is suggested starting point.� The following formula offers a precise method for finding subgroup size:
Where:
n = Sample size
p = estimate of proportion nonconforming (use .5 if unknown).
Z⍺/2 = Z value for desired confidence limit (see table in book).
(using normal to approximate binomial)
E = Maximum allowable error in estimate of p
precision * p
Constructing a p-chart (Constant subgroup size)
Notice p is hiding here.
p(1-p)
n
Constructing a p-chart (Constant subgroup size)
3. Collect the Data
At least 25 subgroups (sometimes from historical records)
A good idea is to use a check sheet
Data can be plotted as a run chart until limits can be calculated
For each subgroup, p = np/n
Constructing a p-chart (Constant subgroup size)
4. Calculate Trial Central line and Control Limits
Central line is pbar
pbar is the average proportion nonconforming for all of the subgroups.
n is the size of the subgroups� (not the number of subgroups)
Constructing a p-chart (Constant subgroup size)
5. Calculate Revised Central line and Control Limits
If the process is stable, then pbar can be used as the best estimate of p0
If a subgroup is out of control with assignable cause it should be removed from the calculations.
p0 is the standard reference value for the proportion nonconforming.
Constructing a p-chart (Constant subgroup size)
6. Achieve the objective
Use the chart as a tool to improve quality.
Limits should be recalculated periodically and (hopefully) moved inward.
np/n
average p
First a run chart...
Now with trial limits...
Final chart with outlier (with assignable cause) removed
Notes
If the population nonconforming, ϕ, is known, it is not necessary to calculate trial control limits.
ONLY applies if the nonconformities appear randomly. They must be independent. In some processes, if a defect occurs, it is more likely that another will occur. In this case a p chart is not suitable.
Is the p clear?
p is the proportion nonconforming within a specific subgroup.
pbar is the average proportion nonconforming of many subgroups.
p0 is the standard or reference value of the proportion nonconforming based on the best estimate of pbar. It is used to calculate the revised control limits. It can be specified as a desired value.
ϕ is the population proportion nonconforming. If it is known, it can be used to calculate the limits.
p Chart for Variable Subgroup Size
Not the desired case. Use constant subgroups if possible.
But often used with 100% inspection (so subgroups will likely vary).
Limits have to be calculated for each subgroup.
Notice the UCL and LCL changing in response to the subgroup size.
Minimizing the Effect of Variable Subgroup Sizes
Changing control limits can be confusing. There are two methods to try to simplify the control of this data.
First, keep in mind that with a SMALLER subgroup, the limits are WIDER.
Number Nonconforming Chart (np Chart)
Easier to understand than p chart.
Subgroup size must be constant. Subgroup size should be shown on chart.
Process capability with attributes
Much simpler than for variables.
It is the percent nonconforming (central line of p-chart).
Control Charts for Count of Nonconformities
Based on poisson distribution:
c Chart - a count of nonconformities (subgroup size is ONE)
u Chart - count of nonconformities per unit (Subgroup size varies)
C-Chart
Count of nonconformities.
Two conditions must be met:
Constructing a c-Chart
Or c-charts can be used to evaluate an operator, work center, department, shift, plant or corporation.
Constructing a c-Chart
2. Determine the subgroup size and method.
The subgroup on a c-chart is one inspected unit.
One airplane, one case, one gross, one canoe.
one case
one canoe
Constructing a c-Chart
3. Collect the Data
Constructing a c-Chart
4. Calculate trial central line and control limits.
Constructing a c-Chart
5. Establish the revised central line and control limits.
excludes assignable cause
Constructing a c-Chart
6. Achieve the objective.
u Charts
c = count of nonconformities in a subgroup.
n = number inspected in a subgroup.
u = count of nonconformities/unit in a subgroup.
u = 5/9 = .556
u = 3/3 = 1
u = 3/6 = .5
u = ⅝= .625
u = 2/4 = .5