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X-Bar and R Charts

Definition of X-Bar and R Charts:

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All processes vary. The data you collect from your process will vary. But is the variation common to the elements of your process or something special? If your data is continuous or variable data, the X-bar and R charts will tell you. 

In this article, we will discuss what the X-bar/R charts are, how they are constructed, their benefits, and best practices for using them.

Overview: What are X-bar and R charts? 

The control chart was developed by Dr. Walter A. Shewhart, an American physicist, engineer, and statistician, who is sometimes referred to as the father of statistical quality control. 

While employed at Western Electric and Bell Laboratories, Shewhart became aware that operators were frequently adjusted their processes when the process appeared to vary from the nominal value. This constant tampering with the process without knowing whether the variation was due to an assignable cause or the natural random variation existing in all processes was a problem.

Shewhart defined his concern in terms of assignable or special cause and random or common cause variation. The control chart was developed to help distinguish between the two. The actions one takes will depend on the type of variation. For common cause variation, the action is to work on improving the process by reducing the variation or moving the process average. For special cause variation, the correct response is to try and find the root cause of the unexpected variation and either incorporate or eliminate the cause depending on whether the special variation improved or degraded the process.

X-bar/R charts are a pair of control charts where continuous or variable data is collected in rational subgroups. The X-bar chart measures between-sample variation (signal), while the R chart measures within-sample variation (noise). Here is some further information about the charts.

  • The Xbar & R chart is the most commonly used control chart
  • Consists of two charts displaying central tendency and variability
  • Xbar chart:
    • Plots the mean (average) of each subgroup to see between-sample variation
    • Useful for identifying special cause changes to the process mean (Xbar)
    • Control limits, related to +/- 3 standard deviations, are calculated using the Range chart (Rbar)
  • R chart:
    • Displays changes in the within subgroup variation

Below are some further descriptions and formulas for calculating the values on the X-bar/R charts.

In the formulas above, you see mention of the constants A2, D4, and D3. Here is the table where the value of the constants are taken from based on the size of the subgroup or sample. For example, if your sample size was 7, you would use 0.419, 1.924 and 0.076, respectively, for A2, D4, and D3.

Here are the steps for constructing the X-bar/R charts.

  1. Determine an appropriate subgroup size and sampling plan.
  2. Collect your data (take a set of readings at each specified interval of time).
  3. Calculate the average and range for each subgroup.
  4. Plot the data (both the averages and the ranges).
  5. After 20 or more sets of measurements, calculate control limits for the R chart.
  6. If the R chart is not in control, take appropriate action and investigate.
  7. If the R chart is in control, calculate control limits for the X-bar chart.
  8. If the X-bar chart is not in control, take appropriate action and investigate.
  9. If both charts are now in control, extend the control limits for ongoing monitoring.

An important consideration in using X-bar/R charts is the selection of an appropriate subgroup or sample size. 

Rational subgrouping is the process of selecting a subgroup based upon logical grouping criteria or statistical considerations. The purpose of subgrouping is to minimize the within-sample variation so as to maximize the ability to see between-sample variation. 

Do not create subgroups for convenience. There must be some logic and rationale for the subgroups. Do not artificially subgroup data to dampen or reduce apparent variation.

Once you have constructed your X-bar/R charts, you’ll need to interpret what they are telling you about your process. 

Here are the guidelines for interpreting them:

  • To correctly interpret X-bar/R charts, always examine the R chart first.
  • The X-bar chart control limits are derived from the R bar (average range) values. If the values are out of control in the R chart, then the X-bar chart control limits may not be accurate.
  • If the points are out of control in the R chart, then stop the process, identify the special cause, and address the issue. Remove those subgroups from the calculations.
  • Once the R chart is in control, then review the X-bar chart and interpret the points against the control limits and the Western Electric Rules.
  • All the points should be interpreted against control limits, not specification limits. Specification limits are provided by the customer or management, whereas control limits are calculated from the average and range values of the subgroups. Spec limits should not appear on your control chart.
  • If any point is out of control in the X-bar chart, identify the special cause and address the issue.
  • Process capability studies can be performed only after both the X-bar and R chart values are signaling common cause variation. 

3 benefits using of X-bar/R charts 

Tampering with a process in statistical control will increase your variation and degrade your quality. The X-bar/R charts will provide the following benefits and keep you from tampering.

1. Know when to intervene in the process and when not to 

If your X-bar/R charts give you a signal of special cause variation, you should investigate and react accordingly. Otherwise, don’t chase individual values that seem to be different. 

2. Provides a video of the process 

The X-bar/R charts show the process change over time. It will indicate what has changed and what has not changed.

3. Easy to understand and construct 

The concept of the control chart was first introduced in the early 1920s and was intended to help people understand when to react to a data point and when not to react. In many cases, the calculations and construction of the control chart was done at the operating level. This required the control chart to be simple to create and maintain. 

Why are X-bar/R charts important to understand? 

X-bar/R charts provide valuable information about your process. Understanding what they are telling you is important for taking the appropriate action.

Tells you what has changed and what has not changed in your process

If the X-bar/R charts exhibit common cause variation, then there has been no significant change in the process. If nothing has changed, don’t chase reasons for a single data point that seems different. Focus on improving the process by reducing variation or shifting the average as appropriate. If there are signals of special cause variation, then investigate to try and find the root cause of the change.

Definition of out of control 

X-bar/R charts are considered in control if all the data points are within the control limits and exhibiting random variation. While you might have data points within the control limits, they may be defined as out of control if they violate one or more of the Western Electric Rules for non-random patterns.

Rational subgroups

How you select your sample and how big of a sample you use will have an impact on the value of your X-bar/R charts. Take care to properly define your rational subgroup and collect your data in a manner to optimize the signals your charts will give you about your process.

An industry example of X-bar/R charts 

The Master Black Belt (MBB) of a steel manufacturing plant was reviewing the data from the mold coating process. He was perplexed because almost all of the values were 150 microns in thickness, which happened to be the nominal value for the process.

He knew data was collected and recorded by the operator, so he went to speak to one. He was told operators often put down the 150 micron value because any other value would have their supervisor coming over and asking, “What happened?” Apparently, the supervisor didn’t understand the concept of process variation. 

The MBB convinced the plant manager to start using X-bar/R charts to monitor the process. They chose to sample the process 5 times an hour for one week. He also got assurance the operators could record the true values without any repercussions from their supervisor. 

Below are the X-bar/R charts from the process data. 

As you can see, except for a single data point on the R chart, the process looks to be exhibiting common cause variation. Chasing a high or low value as exists for point 15 and 26 is chasing the natural variation of the process. 

The next step would be to examine the process capability. Using a nominal of 150 plus/minus 5 micros as the specifications you can see the process, while stable and in control, is not capable of meeting the required specs. While the process is centered, an effort should be made to reduce the variation of the process.

A team was formed to analyze the mold coating process, and it was determined that the flow speed of the powder, air pressure, and nozzle setting needed to be adjusted to reduce the variation. 

Collecting the data for an additional week showed the improvement in the process capability.

3 best practices when thinking about X-bar/R charts

There are many ways to improperly construct and use a control chart. Use these tips to get the most value out of your X-bar/R charts. 

1. Don’t control chart just anything 

Be sure you use X-bar/R charts to monitor the performance of key process characteristics. Don’t make a control chart for the sake of control charting.

2. Sampling strategy

One of the best sampling strategies is to use a consecutive periodic approach, where you collect consecutive units from the process (minimizes the within sample variation) on a fixed schedule. For example, grab five consecutive bottles from the line every 30 minutes, and test them for carbonation. 

3. Use an X-bar/S chart for larger samples 

The use of the R chart for monitoring the range of small samples (<8) is common. The standard deviation (S) is a more accurate representation of the process variation when samples get larger. Use X-bar/S charts where the S is the standard deviation within a sample rather than the R or range.

Frequently Asked Questions (FAQ) about X-bar/R charts

1. What are X-bar and R charts used for? 

They are used to distinguish between common and special cause variation in a process. Using subgroup averages (X-bar) and subgroup ranges (R) of continuous data allow you to monitor your process over time to understand whether there has been a shift in the process average or variation.

2. Why are there two charts? 

The X-bar chart measures the central tendency or average of your process variable. It shows the between-sample variation and whether the process average has changed. The R chart measures the within-sample variation and whether the overall variation of your process has changed.

3. Does the data need to be normally distributed for the X-bar/R charts to work? 

No, that is a common misconception. The X-bar/R charts are robust to normality and are designed to be effective over a wide range of different statistical distributions.

Wrapping up X-bar/R charts 

X-bar/R charts are one of the most powerful tools in the Lean Six Sigma tool box. The charts provide important information regarding the variation of your process. They are designed to distinguish between common and special cause variation over time. This enables you to properly react to any changes in your process average or variation.

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