Control charts for Attributes and Variables Charts

Control charts are statistical tools used in quality control to monitor manufacturing and service processes. They help in identifying variations in processes and distinguishing between common causes (natural variations) and special causes (assignable variations). Control charts are broadly classified into Attribute control charts and Variable control charts based on the type of data being analyzed.

1. Attribute Control Charts

Attribute control charts are used when data can be categorized into discrete groups such as pass/fail, defective/non-defective, or good/bad. These charts help in monitoring quality characteristics that cannot be measured on a continuous scale but can be counted.

Types of Attribute Control Charts

1. p-Chart (Proportion Defective Chart)

   • Purpose: Monitors the proportion of defective items in a sample.
   • Application: Used when sample sizes vary, and each item can be classified as defective or non-defective.
   • Example: Monitoring the percentage of defective smartphones in a production batch.
   • Formula: p = x / np

 Where:

1. • • $p$ = proportion of defectives
     • x = number of defective units
     • n = sample size

2. np-Chart (Number of Defectives Chart)

   • Purpose: Tracks the number of defective items rather than the proportion.
   • Application: Used when the sample size remains constant.
   • Example: Counting the number of defective bulbs in a fixed sample of 100 bulbs per day.
   • Formula: np = n × p

Where:

1. • • np = number of defective items
     • n = sample size
     • p = proportion of defectives

2. c-Chart (Count of Defects Chart)

   • Purpose: Monitors the number of defects per unit, rather than defective items.
   • Application: Used when a single unit can have multiple defects (e.g., a car with multiple scratches or dents).
   • Example: Counting the number of surface defects in a sheet of glass.
   • Formula: c = ∑(number of defects)

3. u-Chart (Defects Per Unit Chart)

   • Purpose: Tracks the average number of defects per unit when sample sizes vary.
   • Application: Used when each sample has a different number of inspected units.
   • Example: Monitoring the number of defects per meter of fabric in textile production.
   • Formula: u = c / n

 Where:

• • • u = average defects per unit
    • c = total defects found
    • $n$ = total number of inspected units

Advantages of Attribute Control Charts

• Useful when measurement data is unavailable.
• Easy to implement for inspection processes.
• Provides insights into product quality trends.

Limitations of Attribute Control Charts

• Less precise compared to variable charts.
• Requires larger sample sizes for accurate conclusions.

Variable Control Charts

Variable control charts are used when data can be measured on a continuous scale such as weight, height, temperature, or time. These charts help in monitoring the variability and central tendency of a process.

Types of Variable Control Charts

1. X̄-Chart (Mean Chart)

   • Purpose: Monitors the average value of a process over time.
   • Application: Used when multiple observations are taken per sample.
   • Example: Monitoring the average weight of chocolate bars in a factory.
   • Formula: Xˉ=∑X / n

 Where:

1. • • Xˉ = sample mean
     • X = individual measurements
     • n = sample size

2. R-Chart (Range Chart)

   • Purpose: Measures process variability by tracking the range within a sample.
   • Application: Used alongside X̄-Charts to ensure consistent production quality.
   • Example: Monitoring variations in the thickness of metal sheets.
   • Formula: R = Xmax − Xmin
   •  Where:
     • R = range of sample
     • Xmax​ = largest observation
     • Xmin​ = smallest observation

3. s-Chart (Standard Deviation Chart)

   • Purpose: Tracks process variability using the standard deviation of sample data.
   • Application: Used when monitoring small variations in a stable production process.
   • Example: Controlling the uniformity of tablet weights in a pharmaceutical company.
   • Formula: s = √(∑(X−Xˉ)^2 / n−1)

​ Where:

1. • • s = standard deviation
     • X = individual observations
     • Xˉ = sample mean
     • $n$ = sample size

2. X̄-s Chart (Mean and Standard Deviation Chart)

   • Purpose: Combines X̄-Charts and s-Charts to analyze both central tendency and variability.
   • Application: Preferred when sample sizes are larger than 10.
   • Example: Ensuring precision in aerospace manufacturing processes.

Advantages of Variable Control Charts

• Provides greater accuracy than attribute charts.
• Helps detect both small and large variations.
• Effective for monitoring continuous improvement.

Limitations of Variable Control Charts

• More complex and expensive to implement.
• Requires trained personnel for accurate interpretation.

Key Differences Between Attribute Control Charts and Variable Control Charts