An XbarR chart is a specific member of a family of control charts. A control chart is a tool used in quality control, specifically SPC or statistical process control, as originally developed by Walter A. Shewhart at Western Electric in 1924 to improve the quality of telephones. The control chart, also known as the Shewhart chart or process-behaviour chart is a statistical tool intended to assess the nature of variation in a process and to facilitate forecasting and management. ... In engineering and manufacturing, quality control and quality engineering are involved in developing systems to ensure products or services are designed and produced to meet or exceed customer requirements and expectations. ... Statistical process control (SPC) is a method for achieving quality control in manufacturing processes. ... Walter Andrew Shewhart (March 18, 1891 - March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control. ... Western Electric (sometimes abbreviated WE and WECo) was a US electrical engineering company, the manufacturing arm of AT&T from 1881 to 1995 . ... 1924 (MCMXXIV) was a leap year starting on Tuesday (link will take you to calendar). ...

A control chart is a plot of measurements of a product on two special scales, usually located above and below each other and running horizontally. For the specific case of the XbarR chart, the top chart has a centerline represented by Xbar, which is simply the sample average. The top and bottom borders are also known as the upper control limit or UCL and the lower control limit or LCL, which are represented by plus and minus three standard deviations from the mean. The bottom chart has the range of each subgroup plotted in a similar manner. The significance of subgroups is explained below. In probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. ...

The purpose of any control chart is to help determine if variations in measurements of a product are caused by small, normal variations that cannot be acted upon, or by some larger special cause that can be acted upon or fixed. The type of chart to be used is based on the nature of the data.

The XbarR chart is normally used for numerical data that is grouped in subgroups in some logical manner, for example 3 games of bowling that occur on one night. This allows the night to be considered as a unit, so a special cause such as a slippery floor or a sick bowler will be more obvious as a point on the chart.

External links

• Tutorial article at SPCforExcel website

• Free calculators for XbarR and other control charts

Introduction to Xbar-R Charts Suppose you are a member of a bowling team. You bowl three games a night once a week in a bowling league. You are interested in determining if you are improving your bowling game. What are some different approaches you could use? One idea is that you could plot the score from each game. However, you are more interested in what your average score is on a given night. So another idea is to plot the average of the three games each night. You definitely would like to increase that average over time. You are also interested in being more consistent, i.e., not having one great game followed by a poor one. Thus, another idea is to keep track of the range in scores for the three games each night. In situations such as this (when you want to monitor averages over time but still keep track of the variation between individual results), the Xbar-R chart is very useful.

The Xbar-R chart is a method of looking at two different sources of variation. One source is the variation in subgroup averages. The other source is the variation within a subgroup. Consider the bowling example above. You have data available on a fairly frequent basis (three games each week). You can also rationally subgroup the data. The three individual games you bowl on one night can be used to form a subgroup.

Continuing with the bowling example, suppose that one night your three bowling scores are 169, 155, and 189. These three scores form a subgroup. You can calculate the range of this subgroup by subtracting the minimum score from the maximum score. Thus the range is:

Range = Maximum - Minimum = 189 - 155 = 34

You can plot this value on a range (R) chart. This is done for each subgroup (one night of bowling three games). The range chart shows how much variation there is within each subgroup, i.e., the amount of variation in your bowling scores on one night. You would like this variation to be small and be consistent over time. The chart for averages (Xbar) presents a different variation than the range chart. Using the three scores above, you can calculate an average score for the night by taking the average of the three individual scores. The subgroup average is:

Xbar = (169+155+189)/3 = 171

You can plot this value on the Xbar chart. This is done for each subgroup. The Xbar chart shows how much week to week variation there is in your weekly average bowling score. You would like this variation to be small and be consistent over time. This permits you to predict what your average score will be on any night, within certain limits. The figure below is an example of the Xbar-R chart for this bowling example. The top part of the figure is the Xbar chart. Each weekly average bowling score (i.e., the average of the three individual games) is plotted. The overall average (Xdbar = X double bar) has been calculated and plotted as a solid line. Xdbar is the average of all the subgroup averages. Upper and lower control limits have also been calculated and plotted. The Xbar chart is in statistical control. The lower part of the figure is the range (R) chart. The range is plotted for each week. The average range and control limits have been calculated and plotted. The range is also in statistical control.

What does it mean when the Xbar-R chart is in statistical control? It means that the subgroup average is consistent over time and the variation within a subgroup is consistent over time. We can predict what the process will do in the near future. In the bowling example, this means that you can predict what the average of your three games on any given night will be. Your average will be between 140 and 181with a long term average of about 161. You can also predict what your range in bowling scores will be on any given night. The range can be anywhere from 0 to 52 with an average range of about 20. As long as the process stays in control (your bowling), the results will continue to the same.