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About this lesson

The X-bar and Standard Deviation chart is the variable data control chart used when the subgroup is large. This lesson explains how the data is recorded and interpreted on the pair of control charts. The lesson describes how to create this control chart in both Microsoft Excel and using Minitab. The lesson will include practice creating the charts.

Exercise files

Download this lesson’s related exercise files.

Xbar-S Chart
12.6 KB Xbar-S Chart - Solution
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Quick reference

XbarS Charts

The Xbar-S chart (Subgroup Mean and Standard Deviation) are the variable data control chart to be used when working with a subgroup sample that has ten or more units.   It can be easily created in either Microsoft Excel or Minitab.

When to use

Use the Xbar-S Chart when seeking to control a product or process variable parameter with a high volume process.  With high volume and a large number of data points, the standard deviation is a better gauge of the subgroup variation than is the range from max value to min value.  The standard deviation uses all the data points and is not as easily impacted by an outlier.

Instructions

The Xbar-S is comprised of a pair of control charts.  These are normally aligned so that the Subgroup Mean (Xbar) chart is directly above the Standard Deviation 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 Standard Deviation chart plots the value of the standard deviation for all the subgroups.   The standard deviation is always a positive number. 

One other advantage of the Xbar-S chart is that it can work with subgroup sizes that vary slightly.  However, they should always be greater than 10 data points.  When the subgroup sizes are not equal, the control limit lines will not be straight lines but rather will be stair-stepped.  This is because the constants used in the equations change with changing subgroup sample size.

Xbar-S 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-S.”  In the Minitab Xbar-S 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 all the data is in one column you the data subgroups must have the same number of points.  If they don’t you must use the different column approach.

If creating the Xbar-S 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 standard deviation for each subgroup.  These will be the values that are plotted.
  3. Calculate the global mean for all the data points and the mean for the standard deviation values.
  4. Calculate the upper and lower control limits for the standard deviation plot. 
  5. If the Standard Deviation 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 Standard Deviation in its control limit calculations.
  6. Now calculate the control limits for the Subgroup Mean data points.
  7. Plot the Subgroup mean, Standard Deviation, 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 A3, B3 and B4 constant values.
  • Don’t select arbitrary subgroups, they should relate to the physical process or calendar groupings.
  • The popularity of the Xbar-S chart is growing.  When SPC was done manually, collecting the additional data points for an Xbar-S chart often required much more time.  But the improved process knowledge was minimal as compared to an Xbar-R chart.  In today’s business world with automation and the Internet of Things, there is plenty of data and processing it is essentially free.  So this chart is being used more often.
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  • 00:04 Hi, I'm Ray Sheen.
  • 00:06 Well now,
  • 00:06 let's take a look at the variable data control chart called the Xbar- S Chart.
  • 00:12 The Xbar and standard deviation, or Xbar- S Chart, is also a pair of charts.
  • 00:18 The Xbar and S chart are used when subgroups have a large sample size.
  • 00:23 And by large, I mean more than ten.
  • 00:25 With large sample size in the subgroup, the standard deviation is a better measure
  • 00:29 of variation than the range because it uses all the data, not just the extremes.
  • 00:35 Also a very significant advantage to the Xbar-S chart is that sample group size can
  • 00:39 vary as long as all the sample groups have at least ten data points.
  • 00:43 This chart relies on variable data meaning data measured on a scale
  • 00:48 with the possibility of even more precise values of
  • 00:50 using a scale with better discrimination.
  • 00:52 But this chart only needs a small subgroup sample of the data
  • 00:56 rather than every data point.
  • 00:58 The two plots of the Xbar and
  • 00:59 S chart are first the Xbar which is the mean of the subgroup sample points.
  • 01:04 This is the same thing that we had with the Xbar-R.
  • 01:07 But the second plot is different.
  • 01:09 The S is the standard deviation of the subgroup data points.
  • 01:13 In both parts, we'll have a sense of central tendency with a calculation of
  • 01:16 the mean and a level of variation with the calculation of the control limits.
  • 01:21 By using multiple points of the sample, this technique employs
  • 01:25 the Central Limit Theorem to increase the normality of the data.
  • 01:29 So let's take a look at an example of the Xbar-S Control Chart.
  • 01:33 Once again, it is a pair of charts.
  • 01:36 The Xbar chart is the plot of the mean value of each of the subgroup samples.
  • 01:41 The S chart plots the subgroup's standard deviation.
  • 01:45 The paths are aligned over each other, so
  • 01:47 that you can see the impact of both if a special cause occurs.
  • 01:51 If your subgroup size is less than 10, I recommend that you use the Xbar-R chart.
  • 01:56 With very few data points the subgroup standard deviation can be way off.
  • 02:00 The range is actually a better indication of variation with a small sample.
  • 02:05 Well we went through the steps of creating a control chart in previous modules.
  • 02:08 But let's create an Xbar-S chart now and look at anything that makes it special.
  • 02:13 The first step is to determine the subgroup size and
  • 02:15 how you'll sample the data.
  • 02:17 We typically look for logical, physical grouping or calender-base grouping.
  • 02:21 I mention that you can have different sample sizes for different sub-groups.
  • 02:26 Then, follow you sample plan to collect the data points for each subgroup.
  • 02:31 Each subgroup will have at least 10 data points.
  • 02:34 Now, calculate the average for each subgroup and
  • 02:37 determine the standard deviation for that subgroup.
  • 02:39 At this point, you can plot the data lines for your control charts.
  • 02:44 So now, we need to calculate the mean values and
  • 02:46 the control limits to complete those charts.
  • 02:49 Once you have at least 30 subgroups, you can calculate the mean and
  • 02:52 control limits for the standard deviation chart.
  • 02:55 We need this to be stable before we calculate the limits of the Xbar chart.
  • 02:59 If the standard deviation chart is not in statistical control, investigate,
  • 03:03 find the special cause, and take appropriate action.
  • 03:06 And of course, we've discussed how to do that in an earlier module.
  • 03:10 Once the standard deviation chart is in control, calculate the mean and
  • 03:13 control limits for the Xbar chart.
  • 03:16 I'll review the formulas on this on the next slide.
  • 03:18 And if the expert chart is not in statistical control, investigate and
  • 03:22 take action for that chart.
  • 03:24 Let's look at how to do the calculations manually or in Excel.
  • 03:28 The actual formulas shown on the right side of the screen and
  • 03:31 the calculation steps will be discussed on the left side.
  • 03:34 First, collect and store your data.
  • 03:36 I normally follow a similar approach I did with the Xbar-R chart,
  • 03:40 it's just that there's many more columns now for the additional data points.
  • 03:44 Then calculate the subgroup mean or Xbar value, you put that in
  • 03:47 the next free column, followed by a column with a standard deviation of the subgroup.
  • 03:53 Fortunately, Excel has functions which will do those calculations for you.
  • 03:57 Once again, you need to calculate the mean value of the mean values.
  • 04:01 That means that you want the average of all the Xbar values, and
  • 04:05 you'll need to do the same thing with standard deviation, calculate the mean or
  • 04:09 average value of the standard deviations.
  • 04:13 To calculate the upper lower control of the standard deviation chart,
  • 04:16 it's fairly easy.
  • 04:18 Upper control limit is the average of the standard deviation values
  • 04:21 times the constant B4 and the lower control is the average of
  • 04:25 the standard deviation values times the constant B3.
  • 04:29 The calculation for the control much for the Xbar chart are similar to what we did
  • 04:33 with the Xbar-R chart just with different constance.
  • 04:36 The upper control limit is the average of the Xbar values
  • 04:40 plus the constant A3 times the average standard deviation values.
  • 04:45 And the lower control limit is the average of the Xbar values minus
  • 04:49 the cost of A3 times the average for the standard deviation values.
  • 04:53 You can create the actual graphs on Excel by using the line chart option
  • 04:57 in the graphing tool.
  • 04:58 Let's look at creating this chart in Minitab.
  • 05:01 Go to the start menu, select control charts,
  • 05:04 then select variable chart for sub groups.
  • 05:07 And finally, select the Xbar-S chart.
  • 05:09 When you do that, you should get a pail that looks like this on your screen.
  • 05:14 You need to tell Minitab how to read your data.
  • 05:16 For all the data points in a subgroup in adjacent columns, or all in one column.
  • 05:22 I normally do adjacent columns.
  • 05:24 Next, place your cursor in the variables window to activate the call of display.
  • 05:29 Highlight the column, or
  • 05:30 columns, where your data is located, then click the select button.
  • 05:34 Your data column should now be in the variables window.
  • 05:36 And just like Xbar-R, if your data is all in just one column, you
  • 05:42 will need to tell minitab how many data points in the column make up one subgroup.
  • 05:47 Now, click on the OK button at the bottom of the panel and
  • 05:50 Minitab will generate your control chart.
  • 05:53 >> So that's the Xbar-S chart.
  • 05:56 Thanks to automation and the internet of things, it's gaining popularity with
  • 06:00 companies that are trying to use more data in their process control.

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