Xbar Chart and R Chart

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Xbar Chart


Purpose

An Xbar chart is used to observe and evaluate the behavior of a process over time and take corrective action if necessary. The chart plots the average values of each of a number of small sampled subgroups. It is usually plotted in conjunction with the R (Range) Chart or the s (Standard Deviation) Chart


The Xbar & Range Chart is the most common control chart used in measuring continuous data. It's a fundamental tool used to display the range of variability inherent in a process.

By monitoring an Xbar and Range Chart, you can determine whether a process is operating consistently or if a special cause has occurred that has changed the operating characteristics of your process.

By identifying and eliminating these special causes, you can improve the overall process, which in turn will reduce your scrap and rework numbers, increasing yields.

To interpret an Xbar & Range chart, you should first look at the Range Chart and note any special causes. If there are special causes in the Range chart, it is unwise to draw any conclusions about the Xbar chart.

Also look for positive or negative correlations between the Xbar and Range charts. Either of these conditions may affect your conclusions. To help judge the effects of skewed data, it is common to use a Histogram in conjunction with the Xbar and Range charts.

Anatomy

X Bar Chart

Reference: Statistical Process Control – Ford/GM/Chrysler

Terminology

A. Sample Mean – The means of the process subgroups as collected in sequential, or chronological, order from the process

B. Sample Number – The chronological index number for the sample, or subgroup, whose average value is being referenced

C. Lower Control Limit (LCL) – Line and numerical value representing the lower limit of the variation that could be expected if the process were in a state of statistical control, equal to the overall Mean minus the average Moving Range multiplied by a conversion factor.

D. Process Average – Overall average value of the individual process readings, over the period of inspection being referenced

E. Upper Control Limit (UCL) – Line and numerical value representing the upper limit of the variation that could be expected if the process were in a state of statistical control, equal to the overall Mean plus the average Moving Range multiplied by a conversion factor.

F. Plot of the individual sample Means vs sample number. Any excursion in this plot above the UCL or below the LCL represents an out-of-control condition and should be investigated Xbar Chart – The title "Xbar" Chart refers to the sample average value (X) being plotted

Out of Control Point – By definition, any point that exceeds either the UCL or the LCL is out of control. Minitab has a number of tests available for out of control conditions, and labels each point with a number corresponding to the test which the point fails


Major Considerations

The Xbar Chart, together with the R Chart, is a sensitive control chart for identifying assignable causes of product and process variation, and gives great insight into short-term variations

The Control Limits for the Xbar Chart are different, depending on whether it is being plotted for use with the R Chart or the s Chart


Application Cookbook

  1. Determine purpose of the chart
  2. Select data collection point
  3. Establish basis for sub-grouping
  4. Establish sampling interval and determine sample size n
  5. Set up forms for recording and charting data and write specific instructions on use of the chart
  6. Collect and record data. A minimum of 25 subgroups or samples of size n should be measured
  7. Compute the Process Average X
  8. If using the R Chart - Compute the Average Moving Range R (Ref. Tool "Control Chart – R Chart)
  9. If using the s Chart - Compute the Average Standard Deviation s (Ref. Tool "Control Chart – s Chart)
  10. Compute required Upper Control Limit UCL Xbar
  11. Compute required Lower Control Limit LCL Xbar
  12. Plot data points
  13. Interpret chart together with other pertinent sources of information on the process and take corrective action if necessary.

R (Range) Chart


Purpose

To observe and evaluate the variation of a process over time, and against control limits, and take corrective action if necessary. The R Chart plots the range values, or the difference between the highest and lowest values, for a series of subgroups. The R Chart is usually plotted in conjunction with the Xbar Chart.

Anatomy

Range Chart

Reference: Statistical Process Control – Ford/GM/Chrysler

Terminology

A. Sample Range – The Range values as calculated from sequential, or chronological, subgroups from the process

B. Sample Number – The chronological index number for the individual sample range value being referenced

C. Lower Control Limit (LCL) – Line and numerical value representing the lower limit of the variation that could be expected if the process were in a state of statistical control, equal to the average Range over the period, multiplied by a conversion factor.

D. Process Average Range – Average value of the individual sample ranges, over the period of inspection being plotted

E. Upper Control Limit (UCL) – Line and numerical value representing the upper limit of the variation that could be expected if the process were in a state of statistical control. It is equal to the average Range over the period, multiplied by a second conversion factor, different from the one used to calculate the LCL.

F. Plot of the Range values vs sample number.


Major Considerations

The Xbar Chart, together with the R Chart, is a sensitive control chart for identifying assignable causes of product and process variation, and gives great insight into short-term variations.


Application Cookbook

  1. Determine purpose of the chart
  2. Select data collection point
  3. Establish basis for sub-grouping
  4. Establish sampling interval and determine sample size n
  5. Set up forms for recording and charting data and write specific instructions on use of the chart
  6. Collect and record data.
  7. Compute the Average Range R
  8. Compute Upper Control Limit UCL R
  9. Compute Lower Control Limit LCLR
  10. Plot data points
  11. Interpret chart together with other pertinent sources of information on the process and take corrective action if necessary

Statistical Process Control Basics

Statistical Process Control

Control Charts

Control Chart Selection

Control Limits

Capability Indices - Cpk, Etc.

Process Capability Study

Statistical Process Control Control Charts

Xbar & R (Range) Chart

Xbar & s (Standard Deviation) Chart

I (Individuals) & MR (Moving Range) Chart

p Chart

np Chart

u Chart

From Xbar Chart to Statistical Quality Control.

From Xbar Chart to Free Six Sigma Tools.

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