Chapter 5: Control Charts for Attributes
- P-Chart (Proportion Nonconforming): Tracks the fraction of defective items in samples, suitable for varying sample sizes.
- np-Chart (Number Nonconforming): Monitors the actual count of defective items when sample sizes are constant.
- C-Chart (Count of Nonconformities): Used for the total number of defects per unit (e.g., scratches on a part) when the sample size (unit) is constant.
- U-Chart (Average Nonconformities per Unit): Tracks the average number of defects per unit when the unit size or sample size varies.
- Collect Data: Take samples at regular intervals, noting attributes (e.g., number of broken items).
- Calculate Centerline (Average): Determine the overall average or proportion of defects from historical data.
- Calculate Control Limits: Establish Upper Control Limits (UCL) and Lower Control Limits (LCL) (typically ±3 standard deviations from the centerline).
- Plot Data: Plot sample results over time on the chart.
- Analyze: Points outside the limits or specific patterns (runs, trends) signal special causes requiring investigation, indicating the process is out of statistical control.
- When quality characteristics can't be measured (e.g., good/bad, yes/no).
- Monitoring defect rates, failure counts, or percentage of nonconforming products.
So let's dive into the Chapter 5 Control Charts for Attributes
5.1 Introduction
In many practical situations, quality characteristics cannot be measured numerically but are instead classified or counted. Such data are known as attribute data. Examples include the number of defective items, presence or absence of defects, and count of defects per unit.
Statistical Process Control uses attribute control charts to monitor and control processes when variable measurement is not feasible. This chapter explains the different types of attribute control charts, their construction, interpretation, and applications.
5.2 Attribute Data and Attribute Control Charts
Attribute data are discrete in nature and usually fall into one of the following categories:
Defectives: Items classified as conforming or non-conforming
Defects: Number of flaws or imperfections on a unit
Attribute control charts are used to:
Monitor fraction or number of defectives
Monitor number of defects per unit
5.3 Classification of Attribute Control Charts
Attribute control charts are classified into two main groups:
A. Charts for Defectives
p-chart – Fraction defective
np-chart – Number of defectives
B. Charts for Defects
c-chart – Number of defects
u-chart – Defects per unit
5.4 p-Chart (Fraction Defective Chart)
5.4.1 Purpose
The p-chart is used to monitor the proportion of defective items in a process when sample sizes may vary.
5.4.2 Construction of p-Chart
Step 1: Collect samples of size n
Step 2: Count number of defectives (d)
Step 3: Calculate fraction defective
[
p = \frac{d}{n}
]
Step 4: Compute average fraction defective ((\bar{p}))
5.4.3 Control Limits for p-Chart
[
UCL_p = \bar{p} + 3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}
]
[
CL_p = \bar{p}
]
[
LCL_p = \bar{p} - 3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}
]
If LCL is negative, it is taken as zero.
5.5 np-Chart (Number of Defectives Chart)
5.5.1 Purpose
The np-chart monitors the number of defectives when sample size remains constant.
5.5.2 Control Limits for np-Chart
[
CL_{np} = n\bar{p}
]
[
UCL_{np} = n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})}
]
[
LCL_{np} = n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})}
]
5.6 c-Chart (Defects Chart)
5.6.1 Purpose
The c-chart is used to monitor the number of defects per unit when:
Inspection area is constant
Defects follow a Poisson distribution
5.6.2 Control Limits for c-Chart
[
CL_c = \bar{c}
]
[
UCL_c = \bar{c} + 3\sqrt{\bar{c}}
]
[
LCL_c = \bar{c} - 3\sqrt{\bar{c}}
]
If LCL is negative, it is set to zero.
5.7 u-Chart (Defects per Unit Chart)
5.7.1 Purpose
The u-chart is used when:
Sample sizes vary
Defects per unit need to be monitored
5.7.2 Control Limits for u-Chart
[
u = \frac{c}{n}
]
[
UCL_u = \bar{u} + 3\sqrt{\frac{\bar{u}}{n}}
]
[
CL_u = \bar{u}
]
[
LCL_u = \bar{u} - 3\sqrt{\frac{\bar{u}}{n}}
]
5.8 Interpretation of Attribute Control Charts
A process is considered out of control if:
Any point lies outside control limits
Sudden increase or decrease in defect rate
Non-random patterns appear
Investigation should focus on assignable causes.
5.9 Comparison of Attribute Control Charts
| Chart | Monitors | Sample Size |
|---|---|---|
| p-chart | Fraction defective | Variable |
| np-chart | Number of defectives | Constant |
| c-chart | Number of defects | Constant |
| u-chart | Defects per unit | Variable |
5.10 Advantages and Limitations
Advantages
Simple and easy to apply
Suitable for go/no-go inspection
Useful where measurement is difficult
Limitations
Less sensitive than variable charts
Requires larger sample sizes
Limited diagnostic capability
5.11 Learning Objectives
After studying this chapter, the learner will be able to:
Differentiate between defectives and defects
Select appropriate attribute control charts
Construct and interpret p, np, c, and u charts
Identify out-of-control conditions
5.12 Review Questions
What is attribute data?
Differentiate between defectives and defects.
Explain the construction of a p-chart.
When is a u-chart preferred over a c-chart?
State assumptions of attribute control charts.
5.13 Short Answer Questions (Exam Oriented)
Define p-chart.
What distribution is assumed in a c-chart?
When is an np-chart used?
What is (\bar{u})?
5.14 Summary
This chapter discussed control charts for attribute data, including p, np, c, and u charts. These charts are essential when quality characteristics cannot be measured numerically and provide effective monitoring of defectives and defects in a process.
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