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About this lesson

The X-bar and Range chart is the most commonly used variable data control chart. When discussing SPC, this is always the example. This lesson explains how the data is recorded and interpreted on the chart. The lesson describes how to create this control chart in both Microsoft Excel and using Minitab. The lesson will include practice creating the chart.

Exercise files

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Xbar-R Chart.xlsx
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Quick reference

XbarR Chart

The Xbar-R chart (Subgroup Mean and Range) are the variable data control chart to be used when working with a subgroup sample of two to ten units.   It can be easily created in either Microsoft Excel or Minitab.

When to use

Use the Xbar-R Chart when seeking to control a product or process variable parameter.  It is the most commonly used variable data control chart.  It can be used with both high volume and low volume processes.  It is more sensitive to shifts in the mean than the I-MR chart.  And it is quite easy to do the math associated with calculating the control limits.

Instructions

The Xbar-R is comprised of a pair of control charts.  These are normally aligned so that the Subgroup Mean (Xbar) chart is directly above the Range chart.  This aids in the recognition of special cause occurrences.  The Subgroup Mean chart plots the average value of the data points that were used to sample the subgroup.  The Range chart plots the value of the difference between the largest point and the smallest point in the subgroup data points.  Therefore the Range value is always a positive number – even if all the data values were negative numbers.

Xbar-R Charts can be created in Microsoft Excel or in Minitab.  Within Minitab, control charts are created by using the “Stat” pull down menu, then selecting “Control Charts.”  Within the Control Charts window, select “Variable Charts for Subgroups” and then finally select “Xbar-R.”  In the Minitab Xbar-R Chart panel, you will need to select the data columns with your data.  If all the data is in one column, you will need to tell Minitab how many data points are in each subgroup.

If creating the Xbar-R Chart in Excel:

  1. Measure the attribute for the first item in the subgroup sample and record the data in a column in Excel.  Then measure the next item in the subgroup sample and record that in the next column.  By doing this, each row in Excel represents a subgroup.
  2. In an adjacent column, calculate the Mean for each subgroup and then calculate the range from the smallest to the largest data point in the subgroup.  These will be the values that are plotted.
  3. Calculate the global mean for all the data points and the mean for the Range values.
  4. Calculate the upper and lower control limits for the Range plot.  The lower control limit is zero for this chart if the number of data points in the subgroup is seven or less.
  5. If the Range plot shows an out of control condition, it must be corrected before proceeding to the Subgroup Mean plot, since that plot relies on the mean of the Range in its control limit calculations.
  6. Now calculate the control limits for the Subgroup Mean data points.
  7. Plot the Subgroup mean, Range, the calculated Means and the control limits.
  8. Take appropriate actions to remove special causes or to center your data within the customer spec limits. 

Hints & tips

  • Remember you are plotting the mean of the data, not the actual data.
  • Be sure to use the subgroup sample size when selecting the A2, d3 and d4 constant values.
  • Don’t select arbitrary subgroups, they should relate to the physical process or calendar groupings.
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  • 00:04 Hi, I'm Ray Sheen.
  • 00:05 Let's now look at the most commonly found control chart, the Xbar-R chart.
  • 00:12 >> The X-bar and range or Xbar-R chart, is also a pair of charts.
  • 00:17 Historically, this has been the most commonly found SPC chart.
  • 00:21 And for a good reason, variable data is richer than attribute data for analysis.
  • 00:25 Second, in the days when we had to do all these measurements and
  • 00:29 calculations by hand, it took far less effort to create
  • 00:32 a chart with only a sample of the data, rather than every single data point.
  • 00:37 Third, the calculations for
  • 00:39 control limits are very easy from a math perspective with this chart.
  • 00:42 And finally, even though it was less effort, it gave excellent results, and
  • 00:46 in some cases, even better results than the individual and moving range chart.
  • 00:51 This chart relies on variable data, meaning data measured on a scale with
  • 00:55 the possibility of even more precise data values,
  • 00:58 the scale with better discrimination is used.
  • 01:01 But this chart only needs small subgroup or
  • 01:04 sample of data, rather than every point.
  • 01:07 The two plots of the Xbar-R chart, are first the Xbar,
  • 01:11 which is the mean of the subgroup sample points.
  • 01:14 And the second is the R plot, which is the range from the lowest value to
  • 01:18 the highest value of the points within the subgroup sample.
  • 01:21 In both plots, we will have a sense of the central tendency with the calculation of
  • 01:25 the mean, and the level of variation with the calculation of the control points.
  • 01:30 By using multiple points for the sample, this technique employs a central limit
  • 01:34 theorem, to increase the normalcy of the data.
  • 01:37 And this chart was specifically designed to work with subgroup sizes of ten or
  • 01:43 less.
  • 01:44 Because of the design,
  • 01:45 this chart is more sensitive to process shifts than the I-MR chart.
  • 01:50 >> So let's look at an example of an Xbar-R control chart.
  • 01:54 As you can see, it is a pair of charts.
  • 01:56 The Xbar chart is the plot of the mean value, of each of the subgroup samples.
  • 02:01 The R chart plots the difference between the max and
  • 02:05 the min value in the subgroup samples.
  • 02:07 The plots are aligned over each other, so
  • 02:09 that you can see the impact of both, if a process change occurs.
  • 02:14 Based upon the statistical design and derivation of these charts,
  • 02:18 the lower control limit of the range chart will always be 0,
  • 02:22 until the subgroup sample size reaches at least 7 points.
  • 02:26 Well, we went through the steps of creating a control chart in our previous
  • 02:29 lesson, but
  • 02:30 let's talk through the unique elements of creating an Xbar-R control chart.
  • 02:35 The first step is to determine the subgroup size, and
  • 02:38 how you will sample the data.
  • 02:39 We typically look for a logical physical grouping, or a calendar-based grouping.
  • 02:44 Then follow the sample plan to collect the data points for the subgroup.
  • 02:48 Each subgroup will have ten or less data points.
  • 02:51 However, the subgroup size is constant.
  • 02:54 You'll always need the same number of data points for a subgroup entry.
  • 02:58 Now, calculate the average for each subgroup, and
  • 03:01 determine the range between the highest and lowest values within that subgroup.
  • 03:06 At this point, you can plot the data lines of your control charts.
  • 03:10 So next, we need to calculate the mean and the control limits,
  • 03:13 to complete the charts.
  • 03:15 Once you have 30 subgroups, you can calculate the mean and control limits for
  • 03:19 the range chart.
  • 03:20 We need these to be stable, before we should calculate the limits for
  • 03:23 the Xbar chart.
  • 03:25 If the range chart is not in statistical control,
  • 03:28 investigate, find the special cause, and take appropriate action.
  • 03:32 We've discussed that in earlier lessons.
  • 03:35 Once the range chart is in control, calculate the mean and
  • 03:38 control limits for the Xbar chart.
  • 03:40 I'll review the formulas on the next slide.
  • 03:43 And if the Xbar chart is not in statistical control, investigate and
  • 03:47 take appropriate action.
  • 03:49 Now, I want to go through with you, how we do the calculations manually or in Excel.
  • 03:55 The actual formulas are shown on the right side of the screen, and
  • 03:58 the calculation steps will be discussed on the left.
  • 04:01 Just like with I-MR, you need to collect and store your data.
  • 04:05 But there are some differences.
  • 04:07 You need to segregate the order of your subgroups, but
  • 04:10 not the order of the sample data points within the subgroups.
  • 04:14 I find it easiest to create a column for each of the subgroups, and
  • 04:18 then let the rows be the actual sample data points.
  • 04:21 Then as I get a data element for
  • 04:22 a subgroup, I drop it in the next open row on the column for that subgroup.
  • 04:27 This makes it easy to then calculate the subgroup mean or the Xbar value.
  • 04:31 And put that in the next row, followed by a row with the min,
  • 04:35 max calculation for the range.
  • 04:38 Now, the next step will be a little confusing.
  • 04:40 You need to calculate the mean value for the control charts.
  • 04:44 The mean value for
  • 04:44 the range chart is easy, it's the average of all the range values.
  • 04:48 The mean value for the Xbar chart is a little tricky.
  • 04:51 It's the average value of the subgroup means,
  • 04:55 not the average of all the data points.
  • 04:58 To calculate the upper and
  • 04:59 lower control limits of the range chart, is also quite easy.
  • 05:03 Upper control limit is the average of the range value times the constant d4.
  • 05:07 And the lower control limit is the average of the range value times the constant d3.
  • 05:13 Of course, if there are less than seven data points in the subgroup sample,
  • 05:17 d3 is zero, so the lower control limit is zero.
  • 05:21 The calculation for the control limits for
  • 05:23 the Xbar chart, are only a little more tricky.
  • 05:26 The upper control limit is the average of the Xbar values,
  • 05:29 plus the constant A2 times the average from the range values.
  • 05:34 And the lower control limit is the average of the Xbar value minus the constant A2,
  • 05:39 times the average for the range values.
  • 05:42 You can create the actual graphs in Excel,
  • 05:45 by using the line chart option in the graphics tool.
  • 05:49 >> Now, let's look at creating this chart in Minitab.
  • 05:52 You go to the Stat menu, select Control Charts, then select Variable Charts for
  • 05:57 Subgroups.
  • 05:58 Finally, select the Xbar-R chart.
  • 06:00 When you do that, you should get a panel that looks like this on your screen.
  • 06:05 You need to tell Minitab how to read your data, or
  • 06:08 the data points for a subgroup and adjacent columns, or all in one column.
  • 06:12 And again, I normally do adjacent columns.
  • 06:15 Now, place your cursor in the Variables window, to activate the columns display.
  • 06:19 Highlight the column or
  • 06:20 columns where your data is located, then click the Select button.
  • 06:24 Your data column should now move to the Variables window.
  • 06:28 If all your data is in just one column, you will need to tell Minitab how
  • 06:32 many data points in the column make up the subgroup.
  • 06:34 That window will appear on this panel if you need to do that.
  • 06:37 It will then divide the data into the subgroups for you.
  • 06:41 Now, click the OK button in the bottom of the panel, and
  • 06:43 Minitab will generate your control chart.
  • 06:46 >> So that's the Xbar-R chart.
  • 06:49 It was a favorite of Shewhart and those who did SPC manually, because it's so
  • 06:55 powerful and sensitive, yet relatively easy to complete the calculations.

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