Tool #6 - The Xbar and R Control Chart

The Control Chart Defined

Thus far in our training, we have learned that Histograms and Check sheets consolidate the data collected, to show the overall picture, while the Pareto diagram is used to indicate problem areas. However, for production purposes, we want to know more about the nature of changes that take place over a specified period of time, or as they occur in "real time".

Control charts are generally used in a production or manufacturing environment and are used to control, monitor and IMPROVE a process. Common causes are always present and generally attributed to machines, material and time vs. temperature. This normally takes a minor adjustme ent to the process to make the correction and return the process to a normal output. HOWEVER, when making a change to the process, it should always be a MINOR change. If a plot is observed that shows a slight deviation trend upward or downward, the "tweaking" adjustment should be a slight change, and then another observation should be made. Too often people will over-correct by making too big of an adjustment which then causes the process to dramatically shift in the other direction. For that reason, all changes to the process should be SLIGHT and GRADUAL!

A control chart is a graph or chart with limit lines, called control lines. There are basically three kinds of control lines:

  • the upper control limit (UCL),
  • the central line (actual nominal size of product),
  • the lower control limit (LCL).

The purpose of drawing a control chart is to detect any changes in the process that would be evident by any abnormal points listed on the graph from the data collected. If these points are plotted in "real time", the operator will immediately see that the point is exceeding one of the contol limits, or is heading in that direction, and can make an immediate adjustment. The operator should also record on the chart the cause of the drift, and what was done to correct the problem bringing the process back into a "state of control".

The method in which data is collected to be charted is as follows: A sampling plan is devised to measure parts and then to chart that measurement at a specified interval. The time interval and method of collection will vary. For our example, we will say that we collect data five times a day at specified time intervals. In making the control chart, the daily data is averaged out in order to obtain an average value for that day. Each of these values then becomes a point on the control chart that then represents the characteristics of that given day. To explain further, the five measurements made in one day constitute one sub group, or one plot point. In some manufacturing firms, measurements are taken every 15 minutes, and the four plots (a subgroup) are totaled and then an average value is calculated. This value then equals one plot for the hour, and that plot is placed on the chart; thus, one plot point on the chart every hour of the working day.

It is when these plot points should fall outside the UCL or LCL, that some form of change must occur on the assembly or manufacturing line. Further, the cause needs to be investigated and have proper action taken to prevent it from happening again………called preventative action, and continuous improvement in the Quality world. The use of control charts is called "process control." In reality, however, a trend will develop that indicates the process is leading away from the center line, and corrective action is usually taken prior to a point exceeding one of the control limits.

There are two main types of Control Charts. Certain data are based upon measurements, such as the measurement of unit parts. These are known as "indiscrete values" or "continuous data". Other types of data are based on counting, such as the number of defective articles or the number of defects. These are known as "discrete values" or "enumerated data".

The Xbar & R Control Chart

An Xbar & R Control Chart is one that shows both the mean value ( X ), and the range ( R ). The Xbar portion of the chart mainly shows any changes in the mean value of the process, while the R portion shows any changes in the dispersion of the process. This chart is particularly useful in that it shows changes in mean value and dispersion of the process at the same time, making it a very effective method for checking abnormalities within the process; and if charted while in progress, also points out a problem in the production flow in real time mode.

Steps In Making the Xbar and R Chart

  • STEP #1 - Collect the data. It is best to have at least 100 samples.
  • STEP #2 - Divide the data into sub groups, it is recommended the subgroups be of 4 or 5 data points each. The number of samples is represented by the letter " n " and the number of subgroups is represented by the letter " k ". The data should be divided into subgroups in keeping with the following conditions:
    1. The data obtained should be from the same grouping of products produced.
    2. A sub group should not include data from a different lot or different process.

  • STEP #3 - Record the data on a data sheet. Design the sheet so that it is easy to compute the values of X bar and R for each sub group (see the page in the class example).
  • STEP #4 - Find the mean value (Xbar). Use the following formula for each subgroup:
  • Xbar formula

  • STEP #5 - Find the range, R. Use the following formula for each subgroup.

    R = X (largest value) - X (smallest value) Example 14.0 - 12.1 = 1.9


It is now time for you to practice some of your learning. I have completed many of the Xbar and R values for you, however, you really should perform a few calculations to gain the experience. Using the attached Exercise Sheet, calculate the remaining Xbar and R values.

Click Here for the Excel Spreadsheet Version
Click Here for the Printable PDF Version

  • STEP #6 - Find the overall mean, or X double bar Xdouble bar.
    Total the mean values of Xbar, for each subgroup and divide by the number of subgroups (k).
  • STEP #7 - Compute the average value of the range (R). Total R for all the groups and divide by the number of subgroups (k).


On the same Work Sheet that you just computed the X double bar figures, now compute the R bar explained above.

  • STEP #8 - Compute the Control Limit Lines. Use the following formulas for Xbar and R Control Charts. The coefficients for calculating the control lines are A2, D4, and D3 are located on the bottom of the Work Sheet you are presently using, and presented here:
  • Xbar Control Chart:

    Central Line (CL) = X double bar figure you calculated.

    Upper Control Limit (UCL) = X double bar + A2 * R bar.

    Lower Control Limit (LCL) = X double bar - A2 * R bar.

    R Control Chart:

    Central Line (CL) = R bar figure you calculated.

    Upper Control Limit (UCL) = D4 * R bar.

    Lower Control Limit (LCL) = D3 * R bar.

    For our Class Exercise, the details are as follows:

    X Control Chart CL = X double bar = 12.94

    • UCL = 12.94 + .577 * 1.35 = 13.719 Note that we are using 5 subgroups, so on the chart n = 5, and under the A2 column, 5 = 0.577. 1.35 is the figure you calculated for R bar.
    • LCL = 12.94 - .577 * 1.35 = 12.161

    R Control Chart CL = R bar = 1.35

    • UCL = 2.115 * 1.35 = 2.86 Note that we are using 5 subgroups, so on the chart n = 5, and under the D4 column, 5 = 2.115.
    • LCL = Since our subgroups equal 5, if you look under the D3 column, there is no calculation coefficient to apply, thus there is no LCL.

  • STEP #9 - Construct the Control Chart. Using graph paper or Control Chart paper, set the index so that the upper and lower control limits will be separated by 20 to 30 mm (units). Draw in the Control lines CL, UCL and LCL, and label them with their appropriate numerical values. It is recommended that you use a blue or black line for the CL, and a red line for the UCL and LCL. The central line is a solid line. The Upper and Lower control limits are usually drawn as broken lines.
  • STEP #10 - Plot the Xbar and R values as computed for each subgroup. For the Xbar values, use a dot (.), and for the R values, use an (x). Circle any points that lie outside the control limit lines so that you can distinguish them from the others. The plotted points should be about 2 to 5 mm apart. Below is what our Xbar chart looks like when plotted.

Xbar chart

Below is what our Rbar chart looks like when plotted. R chart

  • STEP #11 - Write in the necessary information. On the top center of the control charts write the Xbar and R chart, and the R Chart so that you (and others) will know which chart is which. On the upper left hand corner of the Xbar control chart, write the n value to indicate the subgroup size; in this case n = 5.


Now that we know how to make a control chart, it is even more important to understand how to interpret them and realize when there is a problem. All processes have some kind of variation, and this process variation can be partitioned into two main components. First, there is natural process variation, frequently called "common cause" or system variation. These are common variations caused by machines, material and the natural flow of the process. Secondly is special cause variation, generally caused by some problem or extraordinary occurrence in the system. It is our job to work at trying to eliminate or minimize both of these types of variation. Below is an example of a few different process variations, and how to recognize a potential problem.

Xbar analysis

In the above chart, there are three divided sections. The first section is termed "out of statistical control" for several reasons. Notice the inconsistent plot points, and that one point is outside of the control limits. This means that a source of special cause variation is present, it needs to be analyzed and resolved. Having a point outside the control limits is usually the most easily detectable condition. There is almost always an associated cause that can be easily traced to some malfunction in the process.

In the second section, even though the process is now in control, it is not really a smooth flowing process. All the points lie within the control limits, and thus exhibits only common cause variations.

In the third section, you will notice that the trending is more predictable and smoother flowing. It is in this section that there is evidence of process improvement and the variation has been reduced.

Therefore, to summarize, eliminating special cause variation keeps the process in control; process improvement reduces the process variation, and moves the control limits in toward the centerline of the process. At the beginning of this process run, it was in need of adjustment as the product output was sporadic. An adjustment was made, and while the plotted points were now within the boundaries, it is still not centered around the process specification. Finally, the process was tweaked a little more and in the third section, the process seems to center around the CL.

There are a few more terms listed below that you need to become familiar with when analyzing a Xbar Chart and the process:

    RUN - When several plotted points line up consecutively on one side of a Central Line (CL), whether it is located above or below the CL, it is called a "run". If there are 7 points in a row on one side of the CL, there is an abnormality in the process and it requires an adjustment.

    TREND - If there is a continued rise or fall in a series of points (like an upward or downward slant), it is considered a "trend" and usually indicates a process is drifting out of control. This usually requires a machine adjustment.

    PERIODICITY - If the plotted points show the same pattern of change over equal intervals, it is called "periodicity". It looks much like a uniform roller coaster of the same size ups and downs around the centerline. This process should be watched closely as something is causing a defined uniform drift to both sides of the centerline.

    HUGGING - When the points on the control chart seem to stick close to the center line or to a control limit line, it is called "hugging of the control line". This usually indicates that a different type of data, or data from different factors (or lines) have been mixed into the sub groupings. To determine if you are experiencing "hugging" of the control line, perform the following exercise. Draw a line equal distance between the centerline and the upper control limit. Then draw another line equal distance between the center line and the lower control limit. If the points remain inside of these new lines, there is an abnormality, and the process needs closer analysis.


Now it is time for the final test to see if you can make a Control Chart. Below is link for a completely filled out data sheet, and a blank variable control chart form. Your challenge is to calculate the subgroups Xbar and Rbar numbers; calculate the CL, UCL and LCL for the data and the Range Chart, and place those limit lines and numbers on the chart. Last, of course, plot the points and indicate if there are any abnormalities observed in the process. Below your forms to work on, you will also find the completed results so you can check your work. Good Luck!

The Blank Variable Control Chart is available in two formats:

For Microsoft Word Format CLICK THIS LINK
For Adobe Acrobat Format (.pdf) CLICK THIS LINK

Your Completed Data Sheet for this Exercise is available in two formats:

For Microsoft Excel Format CHOOSE THIS LINK
For Adobe Acrobat Format (.pdf) CHOOSE THIS LINK

FINAL PRODUCT COMPARISON: Your Final Xbar and R Chart should look THIS CHART.

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