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# Control Charts

A control chart is a tool used to determine if a process is under statistical control or not. A process is under control if it remains within certain control limits. By assuming normality of our data, we can often use $\pm 3\sigma$ (3 sigma or 3 standard deviations) as our control limits. With this value if the process is under control, 99.7% of the observations should lie within the control limits.

There are many different kinds of control charts. Some use attribute data, others use variable data, and others use a combination of both. Before learning about the types of charts, we need to understand the difference between variable and attribute data:

Variable Data: numerical, usually on a continuous scale - measurements (weight, distance, time,...)

Attribute Data: characteristics and descriptors - good/bad, acceptable/unacceptable, number of defects or blemishes,...

The following terms are also commonly used when discussing control charts:

Defect: a specific item of a product that fails to meet a specification or expectation.

Defective: an entire product with one or more defects which would be deemed unfit for customer use.  This product should be scrapped or reworked.

Number of defectives: in a sample, the number of defectives ($d$ out of $n$).

Number of defects: in a sample, the total number of defects ($c$).

Fraction defective: the number of defective items/total number of items ($p = x$ out of $n$).

## Variable Control Charts

A variable control chart plots continuous data measures over time. There are two main types: individual measurement charts and charts for data collected in small samples, called subgroups.

#### Variable Control Charts for Subgroup Data

Each point on the graph represents a subgroup, that is, a group of units produced under the same set of conditions. For example, if you want to chart a particular measurement from your process and you collect and measure five parts every hour, your subgroup size would be 5.  Normally a subgroup would be of a fixed or constant size.

Variable control charts for subgroups include the Xbar, R, s, and Zone charts with some examples shown below.

Xbar chart - Plots the process mean over time. Used to track the process level and detect the presence of special causes affecting the mean.

R chart - Plots the process range over time. Used to track process variation and to detect unexpected variation.

s chart - Plots the process standard deviation over time. Used to track process variation and to detect unexpected variation.

Zone chart - Plots the cumulative scores based on "zones" at 1, 2, and 3 standard deviations from the center line. Used to detect unexpected variation.

#### Variable Control Charts for Individual Data

Each point on the graph represents an individual measurement, that is, the subgroup size is 1. Individual charts are used when measurements are expensive, production volume is low, or products have a long cycle time. For example, they are used for destructive testing, such as tests of the impact strength of parts. These charts include I charts and MR charts.

I chart - Plots individual observations over time. Used to track the process level and to detect the presence of special causes.

Moving range chart - Plots the moving range over time. Used to track process variation and to detect the presence of special causes.  The moving range chart is calculated using a subgroup size of 2 (a moving window of 2 observations).

## Attribute Control Charts

Attribute control charts plot nonconformities (defects) or nonconforming units (defectives). A nonconformity refers to a quality characteristic and a nonconforming unit refers to the overall product. A unit may have many nonconformities, but the unit itself is either conforming or nonconforming. For example, a scratch on a metal panel is a nonconformity, but if several scratches exist on a panel, the entire panel might be considered nonconforming or defective.

You select your attribute control chart based on whether the data follow a binomial (number or proportion defective or nonconforming) or a Poisson distribution (count data on number of defects or nonconformities).

### Attributes Control Charts for Binomial Data

Values for binomial data are classified into one of two categories such as pass / fail or go / no-go. Binomial data are often used to calculate a proportion or a percentage, such as the percentage of sampled parts that are defective.

You can use either the P chart or the NP chart to plot your nonconforming units. The main difference between P and NP charts is the vertical scale.

• P charts show the proportion of nonconforming units on the y-axis.
• NP charts show the whole (integer) number of nonconforming units on the y-axis.

The chart that you choose does not affect which points are out of control.

#### Attributes control charts with varying subgroup sizes

##### When the subgroup sizes are different:
• The control limits for both charts vary.
• The center line on the NP chart varies, but the center line on the P chart is straight.

The varying center line may make the chart more difficult to interpret. A rule of thumb is to use a P chart if the subgroup sizes are different. However, you can use either chart.

With a P chart, the center line is straight.

With an NP chart, the center line varies with subgroup size changes.

### Attribute Control Charts for Poisson Data

You can use either the U chart or the C chart to plot your number of nonconformities or defects. The main difference between U and C charts is the vertical scale.

• U charts show the number of nonconformities per single unit on the y-axis.

• C charts show the number of nonconformities per sample, which can include more than one unit on the y-axis.

The chart that you choose does not affect which points are out of control.

# How to Select a Control Chart

With so many chart types available you might be confused about which one to use. To help with this situation, the flowchart below explains which one to use.