November 2012

sigmaOne of the purposes of control charts is to estimate the average and standard deviation of a process.  The average is easy to calculate and understand – it is just the average of all the results.  The standard deviation is a little more difficult to understand – and to complicate things, there are multiple ways that it can be determined – each giving a different answer. 

Sometimes people ask why some software packages give different values for the control limits. Which program is correct?  The answer is probably both.  The difference is simply how the standard deviation is estimated.  The objective of this newsletter is to show three different, but common, ways that the standard deviation may be estimated.  We will look at data that are formed into subgroups and the control limits on the X chart. 

In this issue:

As always, you can leave comments at the end of the newsletter.

Data Subgroups

The data we will use are shown in the table.  We have 10 subgroups, each containing 3 observations or results. 

Table 1: Subgroup Data

Subgroup

X1

X2

X3

Subgroup

Average

Subgroup

R

Subgroup

s

1

80.3

86.9

108.0

91.73

27.7

14.47

2

99.4

89.5

96.4

95.10

9.9

5.08

3

95.1

95.9

85.3

92.10

10.6

5.90

4

99.0

123.9

100.6

107.83

24.9

13.94

5

97.1

98.6

107.7

101.13

10.6

5.74

6

97.4

105.5

104.5

102.47

8.1

4.42

7

97.9

106.0

95.6

99.83

10.4

5.46

8

81.6

99.9

101.1

94.20

19.5

10.93

9

90.8

90.1

95.1

92.00

5.0

2.71

10

107.3

102.7

92.5

100.83

14.8

7.57

 

 

 

Sum

977.23

141.50

76.21

 

 

 

Average

97.72

14.15

7.62

 

So, our subgroup size is constant for each of the 10 subgroups.  The subgroup average, range and standard deviation have also been calculated for use below.  The overall sum and average are given for subgroup averages, subgroup ranges and subgroup standard deviations – again for use below.  There may be some minor differences due to rounding.

Three Ways to Estimate the Standard Deviation

We will look at three different ways to estimate the standard deviation.  These impact how control limits are calculated.  Control limits for the X chart are given by:

 control limits based on sigma

where UCL and LCL are the upper and lower control limits, n is the subgroup size, and σ is the estimated standard deviation of the individual values.  Remember: the standard deviation of the subgroup averages is equal to the standard deviation of the individual values divided by square root of the subgroup size.  These control limit equations may be different from the ones you normally use.  They are not, as will be shown below.

The value of σ depends on the method you use to estimate it.  We will look at three methods for estimating σ for subgroup data:

  1. Average of the subgroup ranges
  2. Average of the subgroup standard deviations
  3. Pooled standard deviation

 

Average of Subgroup Ranges

The average of the subgroup ranges is the classical way to estimate the standard deviation.  The average range is simply the average of the subgroup averages when the subgroup size is constant:

 rbar calculation

where Ri is the range of the ith subgroup and k is the number of subgroups.  The standard deviation is then estimated from the following equation:

 sigma from Rbar

where d2 is a constant that depends on subgroup size.   Table 2 shows the values of d2 based on subgroup sizes up to 20.   From the table, you can see that d2 for a subgroup size of 3 is 1.693.  

Table 2: Constants for Control Charts

n

d2

c4

2

1.128

0.7979

3

1.693

0.8862

4

2.059

0.9213

5

2.326

0.9400

6

2.534

0.9515

7

2.704

0.9594

8

2.847

0.9650

9

2.970

0.9693

10

3.078

0.9727

11

3.173

0.9754

12

3.258

0.9776

13

3.336

0.9794

14

3.407

0.9810

15

3.472

0.9823

16

3.532

0.9835

17

3.588

0.9845

18

3.640

0.9854

19

3.689

0.9862

20

3.735

0.9869

 

For the data in Table 1, the average range and σ are given by:

rab and sigma calculations

Using the estimate of the standard deviation from the average range, we can now calculate the control limits:

 control limits from estimated sigma from rbar

You may not be used to calculating control limits this way for the X chart.  You probably use the following equations:

normal control limit equations

where A2 is a constant that depends on subgroup size.   Consider just the UCL.  There are two different equations for the UCL above, which must give the same result.  So,

 ucl equality

This can be rearranged to the following:

 rearranging to find A2

Substituting for R and solving for A2 gives:

 a2 equation

Substituting in d2 and n for our example gives:

 calculating A2

This is the value of A2 for a subgroup size of 3 that you find in the tabulated control chart constants for A2.  For a table of these values, please see our newsletter our X-R control charts.  So, both methods for calculating the control limits are equivalent.  The X control chart for these data is shown in Figure 1.

Figure 1: X Based on Sigma from Average Range

 xbar control chart

Average of Subgroup Standard Deviations

The average of the subgroup standard deviations could also be used to estimate the standard deviation.  When the subgroup size is constant, the average of the subgroup standard deviations is given by:

 sbar calculatin

where si is the standard deviation of the ith subgroup and k is the number of subgroups.  The standard deviation is then estimated from the following equation:

 estimating sigma from sbar

where c4 is constant that depends on subgroup size.  The values of c4 are shown in Table 2 above.  For n = 3, the value of c4 is 0.8862.  For the data in Table 1, the average standard deviation and σ are given by:

 sbar and sigma calcuations

This value of σ is different than that estimated by the average range, which was 8.36.  Thus, the control limits will be different also.  The control limits based on the standard deviation estimated from the subgroup standard deviations are:

 control limit calculations from sbar sigma

 

The differences are not large, but there are differences.  This is why people wonder why the control limits can be slightly different.  It is usually the way the standard deviation is estimated.  The control chart with these limits will look about the same as in Figure 1 - just with the control limits a little wider in this example. 

 

Pooled Standard Deviation

The pooled standard deviation, sp, can also be used to estimate the standard deviation.  The standard deviation, σ, is equal to the pooled standard deviation divided by c4:

 sigam from pooled standard deviation

where:

 pooled standard deviation equation

where Xij is the jth observation in the ith subgroup, Xi is the average of the observations in the ith subgroup, ni is the number of observations in the ith subgroup, c4 is the constant defined above but this time it depends on the degrees of freedom (df); which is given by the sum of the ni -1 values (the denominator under the square root sign.   Thus,

sp calculation

For df = 20, the value of c4 is 0.9869.  The estimated standard deviation is then given by:

sigma from sp

This third method of estimating the standard deviation gives another value for σ.  The control limits will  be different as well:

control limits from sp

The control chart will look the same as Figure 1 - again with slightly different control limits.

 

Summary

This newsletter has looked at the three different methods of estimating the standard deviation from data that are in subgroups.  Each method gives a different value for the estimate standard deviation:

  • σ from the average range = 8.36
  • σ from the average standard deviation = 8.60
  • σ from the pooled standard deviation = 8.66

This leads to different values for the control limits.  Which method is correct?  All three are correct.  Dr. Donald Wheeler has suggested that the average range method is more robust than the pooled standard devaition - so there may be something there.  But in the end, the important thing is the story that the control chart is telling you about your process.  Minor changes in the estimate of the standard deviation will not change this in most cases.

 

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Thanks so much for reading our publication. We hope you find it informative and useful. Happy charting and may the data always support your position.

Sincerely,

Dr. Bill McNeese
BPI Consulting, LLC

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Comments (20)

  • anon

    Good morning, Bill -The newsletters are great but it would be nice to be able to print them and HTMLs do not print well. Any chance that you could either publish in PDF format or add a "printer friendly" option?Thanks.dave  

    Dec 10, 2012
  • anon

    Hi Dave,

    I will take a look at doing the PDF format.  You can always copy the newsletter and paste into Word in the meantime.  Thanks for the comment on the newsletters.

     

    Bill

    Jan 12, 2013
  • anon

    Very informative article , had observed the difference in control its but known the reason yet..thanks for sharing the underlying reason..

    Mar 11, 2016
  • anon

    How do you calculate the 1461.15?? From the initial data, how should I interpret the Xij???

    Feb 23, 2017
  • anon

    There should really be a second summation sign in the numerator.  Below are the calculations.  Xij is the ith observation in the jth subgroup.

    SubgroupX1X2X3Xbar(X1-Xbar)2
    (X2-Xbar)2(X3-Xbar)2Sum
    180.386.9108.091.73130.7223.36264.6418.68
    299.489.596.495.1018.4931.361.6951.54
    395.195.985.392.10914.4446.2469.68
    499.0123.9100.6107.8378.03258.1452.32388.49
    597.198.6107.7101.1316.276.4243.1265.81
    697.4105.5104.5102.4725.679.24.1339
    797.9106.095.699.833.7438.0317.9259.69
    881.699.9101.194.20158.7632.4947.61238.86
    990.890.195.192.001.443.619.6114.66
    10107.3102.792.5100.8341.823.4869.44114.74
    Sum1461.15

    Feb 23, 2017
  • anon

     How to calculate C4?? Is there a ful table available??

    Feb 28, 2017
  • anon

    There is a table in the article. What are you looking for?

    Feb 28, 2017
  • anon

    Dear Dr. Bill McNeeseI have a question about Control Limit UCL & LCL, Sigma = Rbar/d2 ,and Theory is Control Limit is 3Sigma (+/-) so, Can we just use UCL=Xbar+3*Rbar/d2 and LCL=Xbar-3*Rbar/d2 ( 3*Rbar/d2 = 3Sigma), Hope you can help me to have more clear about it ,I known that correct Formula shows ,Formula  ,UCL = Xbar + 3 Sigma/SQRT(n)  ,LCL = Xbar-3Sigma/SQRT(n) ,But it will not be 3Sigma because still devide to SQRT(n), Why is it? ,Thank you so much  

    Jun 18, 2017
  • anon

    Please see this link for information on where control limits come from:

    https://www.spcforexcel.com/knowledge/control-chart-basics/control-limits

    Control limits are +/- three standard deviations of what is being plotted.  So, with subgroup averages, it is +/- three standard devations of the subgroup averages.  The subgroup averages standard devaition comes from sigma/sqrt(n) where n is the subgroup size and sigma is the standard devaition of the individual values (estimated by Rbar/d2).

    Jun 18, 2017
  • anon

    Understood now, thanks so much Dr Bill

    Jun 19, 2017
  • anon

    Dear Sir,     Greetings,   I have a doubt, is it calculate the control limits for population method.I have 125 nos samples. i will not seperate this for sub groups. How to calculate the UCL & LCL. Is it possible. 

    Aug 21, 2017
  • anon

    Hi there, i was wondering if u could help me in the following problemI have been given 50 numbers in an excel sheet.. each number represents an average of 5 subsample observations. I have not been given the individual 5 numbers, just their average. and therefore am unable to calculate the range.Ive also been told the 50 average of subsamples follow a normal distribution following N(2,3), sample size n = 5 , the standard deviation = 3, and to follow a 2-sigma deviation.The average of the 50 averages i calculated =1.94How would i go about calculating the LCL and UCL, given the information i have.? many thanks in advance mate

    Oct 14, 2017
  • anon

    All the information you need in the article.  You have the average.  Are you saying the distribution of subgroup averages has a standard deviaiton of 3?  Or is the individual values?  What do you mean by a 2 sigma deviation?

    Oct 15, 2017
  • anon

    In X-R chart the value of A2R is 1.8 calculate the value of sigma i.e standard deviation?

    Nov 29, 2017
  • anon

    Not clear to me what you are asking.

    Nov 29, 2017
  • anon

    Why don't we estimate standrad deviation by using standard deviation of all samples?

    Dec 12, 2017
  • anon

    If you use the calculated standard deviation of all the range, it will inflated when the data are not in control.  The purpose of a control chart is to determine if the data are homogeneous.  Calculating the standard deviation assumes that the data are homogenous.

    Dec 12, 2017
  • anon

    For PpK calculation, overall standard deviation is used. Can you please explain how to calculate for above datas.

    May 30, 2018
  • anon

    The calculated standard deviation is the same as the STDEV function in excel.  It is the square root of the sum of the (Xi-Xavg)^2 divided by n - 1.

    May 30, 2018

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