# Variable sample size - Xbar-s charts - How to select A3 and B4

D

#### D.Salman

Dear Experts,
Good morning.
I want to develop Xbar-s charts for the attached data. Time format data (hh:mm).
The subgroups size is variable (12-15) because we have missing values. See yellow cells.
If I want to calculate UCL and LCL of X bar, how to select A3? And the same for B4 in the s chart?
Omar.

#### Attachments

• X bar & s charts.xls
71.5 KB · Views: 575

Bumping post.

#### antoine.dias

##### Quite Involved in Discussions
Dear Experts,
Good morning.
I want to develop Xbar-s charts for the attached data. Time format data (hh:mm).
The subgroups size is variable (12-15) because we have missing values. See yellow cells.
If I want to calculate UCL and LCL of X bar, how to select A3? And the same for B4 in the s chart?
Omar.

Look at the attached matrix for the different values.

Best regards,

Antoine

#### Attachments

• Table of constants for control charts.xls
26 KB · Views: 501
J

#### Jim Shelor

Dear Experts,
Good morning.
I want to develop Xbar-s charts for the attached data. Time format data (hh:mm).
The subgroups size is variable (12-15) because we have missing values. See yellow cells.
If I want to calculate UCL and LCL of X bar, how to select A3? And the same for B4 in the s chart?
Omar.
D.Salman,

For an X-bar,S chart with variable group sizes, you use the A3 and B4 for the specific subgroup using the number of samples in that subgroup. This procedure can be involved if your subgroup size varies significantly.

If you have Minitab, Minitab will take care of this automatically.

The result will be the control limits will vary with the subgroup size.

Looking at your data, sample 13 you are only missing values for 3 subgroups. For samples 14 and 15, you are missing samples for > 50% of the subgroups.

If I were you, I would construct the control chart using a subgroup size of 13 and force Minitab to use 13 rather than changing the A3 and B4 values for the 3 subgroups for which sample 13 is missing. This method should produce little or no effect on your control chart and is much easier to construct and analyze than using 15 samples and variable subgroups.

I have provided a finished spreadsheet of what using 13 samples per subgroup looks like.

I hope this helps.

Best regards,

Jim Shelor

#### Attachments

• DSalmam X-bar, S - Shelor Comments.xls
74 KB · Views: 311
Last edited by a moderator:
S

#### supergugi

Dear Experts
in the book "Introduction to statistical quality control"(3rd edition,D.C. Montgomery, McGraw Hill, 1996) the author suggests that "R chart is not used with variable sample size, because the center line changes continuously"; where the center line, like you obviously know, is the average sample range. I do not understand this statement. Can you help me?
Best Regard

#### Steve Prevette

##### Deming Disciple
Super Moderator
One thing to note is that the "s" chart uses the statistical sigma formula (stdev in Excel) The formula takes into account the number of data points in the grouping. When you use the "R" chart, you are using the Range from the highest value to the lowest value in the grouping. You must apply a correction factor for the number of data values in the grouping, thus becomes very difficult to work with variable grouping sizes.

In an xbar-s system of charts, you should be able to use the standard deviation formula (stdev in Excel) directly with no conversion. The baseline sigma value in the s chart should be able to be directly used on the xbar chart, after dividing by the square root of the number of values in the grouping. You should be able to do these calculations directly without using A3 and B4 tables, in my opinion.

S

#### supergugi

Thank you very much.
Best regards

#### Tim Folkerts

Trusted Information Resource
You can come pretty close if you just use the st dev directly, but it is not quite right. (The problem is due to the fact that you should average the variance, not the standard deviation.)

If you could just use the average of the st dev, then the limits would be 3*(sigma-bar)/n^0.5 as Steve says - we will call this A'3. The standard values is called A3 For a few values of n, this becomes

n.....A3.....A'3
2....2.659..2.121
3....1.994..1.732
4....1.628..1.500
6....1.287..1.225
10..0.975...0.949

You would consistently underestimate the control limits (and thus get extra points out of control) if you used 3*(sigma-bar)/n^0.5 instead of the correct A3 value

The error becomes smaller as n becomes bigger. For n=2, the error is 20%. For n=10, the error is only 3%.

Tim F

P.S. a quick set of calculations using random data exactly confirms these numbers. 10,000 pairs of standard normal data shows an average st dev of 0.797 +/- 0.006, which is the 20% error described above.

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