Controls

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steps_in_constructing_xbar_and_r_charts.docx
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Steps in Constructing the Xbar Chart

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1. Find the mean of each subgroup Xbar(1), Xbar(2), Xbar(3)... Xbar(k) and the grand mean of all subgroups using: C:\Users\adam\Desktop\mgt340_m8_l3_g1.jpg

2. Find the UCL and LCL using the following equations:

C:\Users\adam\Desktop\mgt340_m8_l3_g2.jpg

                               A(3) can be found in the following table:                                n   A(3)      n   A(3)                                2   2.659    6   1.287                                3   1.954    7   1.182                                4   1.628    8   1.099                                5   1.427    9   1.032

3. Plot the LCL, UCL, centerline, and subgroup means

4. Interpret the data using the following guidelines to determine if the process is in control:

· One point outside the 3 sigma control limits

· Eight successive points on the same side of the centerline

· Six successive points that increase or decrease

· Two out of three points that are on the same side of the centerline, both at a distance exceeding 2 sigmas from the centerline

· Four out of five points that are on the same side of the centerline, four at a distance exceeding 1 sigma from the centerline f. Using an average run length (ARL) for determining process anomalies

Example: The following data consists of 20 sets of three measurements of the diameter of an engine shaft.

n

 

#1

 

#2

 

#3

 

StdDev

 

Xbar

1

 

2.0000

 

1.9998

 

2.0002

 

0.0002

 

2.0000

2

 

1.9998

 

2.0003

 

2.0002

 

0.0003

 

2.0001

3

 

1.9998

 

2.0001

 

2.0005

 

0.0004

 

2.0001

4

 

1.9997

 

2.0000

 

2.0004

 

0.0004

 

2.0000

5

 

2.0003

 

2.0003

 

2.0002

 

0.0001

 

2.0003

6

 

2.0004

 

2.0003

 

2.0000

 

0.0002

 

2.0002

7

 

1.9998

 

1.9998

 

1.9998

 

0.0000

 

1.9998

8

 

2.0000

 

2.0001

 

2.0001

 

0.0001

 

2.0001

9

 

2.0005

 

2.0000

 

1.9999

 

0.0003

 

2.0001

10

 

1.9995

 

1.9998

 

2.0001

 

0.0003

 

1.9998

11

 

2.0002

 

1.9999

 

2.0001

 

0.0002

 

2.0001

12

 

2.0002

 

1.9998

 

2.0005

 

0.0004

 

2.0002

13

 

2.0000

 

2.0001

 

1.9998

 

0.0002

 

2.0000

14

 

2.0000

 

2.0002

 

2.0004

 

0.0002

 

2.0002

15

 

1.9994

 

2.0001

 

1.9996

 

0.0004

 

1.9997

16

 

1.9999

 

2.0003

 

1.9993

 

0.0005

 

1.9998

17

 

2.0002

 

1.9998

 

2.0004

 

0.0003

 

2.0001

18

 

2.0000

 

2.0001

 

2.0001

 

0.0001

 

2.0001

19

 

1.9997

 

1.9994

 

1.9998

 

0.0002

 

1.9996

20

 

2.0003

 

2.0007

 

1.9999

 

0.0004

 

2.0003

Sbar chart limits: SBAR = 0.0002

UCL = B(4) x SBAR = 2.568 x .0002 = 0.0005136 LCL =  B(3) x SBAR = 0 x .0002 = 0.00

Xbar chart limits: XDBLBAR = 2.0000

UCL = XDBLBAR + A(3) x SBAR   =   2.000+1.954 x .0002  = 2.0003908 LCL =  XDBLBAR - A(3) x SBAR   =    2.000-1.954 x 0002  = 1.9996092

S-Chart:

Xbar Chart:

C:\Users\adam\Desktop\AU_MGT340_W8_G5.jpg

 

Xbar and R Charts (1 of 2)

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Theoretical Control Limits for Xbar Charts:

C:\Users\adam\Desktop\mgt340_m8_l4_g1.jpg

Although theoretically possible, since we do not know either the population process mean or standard deviation, these formulas cannot be used directly and both must be estimated from the process itself. First the R chart is constructed. If the R chart validates that the process variation is in statistical control, the XBAR chart is constructed.

1. Find the mean of each subgroup Xbar(1), Xbar(2), Xbar(3)... Xbar(k) and the grand mean of all subgroups using: C:\Users\adam\Desktop\mgt340_m8_l4_g2.jpg

2. Find the UCL and LCL using the following equations:

· UCLX-bar  =  X-double-bar  +  A2 (R-bar)

· LCLX-bar  =  X-double-bar  -  A2 (R-bar)

A(2) can be found in the following table:

n   A(2)    

n   A(2)

2   1.880   

6   .483

3   1.023   

7   .419

4    .729   

8   .373

5    .577   

9   .337

3. Plot the LCL, UCL, centerline, and subgroup means

4. Interpret the data using the following guidelines to determine if the process is in control:

· One point outside the 3 sigma control limits

· eight successive points on the same side of the centerline

· Six successive points that increase or decrease

· Two out of three points that are on the same side of the centerline,  both at a distance exceeding 2 sigmas from the centerline

· Four out of five points that are on the same side of the centerline, four at a distance exceeding 1 sigma from the centerline

· Using an average run length (ARL) for determining process anomalies.

 

Using the navigation on the left, please proceed to the next page.

 

 

Xbar and R Charts (2 of 2)

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Example: The following data consists of 20 sets of three measurements of the diameter of an engine shaft.

n

 

1

 

2

 

3

 

Range

 

Xbar

1

 

2.0000

 

1.9998

 

2.0002

 

0.0004

 

2.0000

2

 

1.9998

 

2.0003

 

2.0002

 

0.0005

 

2.0001

3

 

1.9998

 

2.0001

 

2.0005

 

0.0007

 

2.0001

4

 

1.9997

 

2.0000

 

2.0004

 

0.0007

 

2.0000

5

 

2.0003

 

2.0003

 

2.0002

 

0.0001

 

2.0003

6

 

2.0004

 

2.0003

 

2.0000

 

0.0004

 

2.0002

7

 

1.9998

 

1.9998

 

1.9998

 

0.0000

 

1.9998

8

 

2.0000

 

2.0001

 

2.0001

 

0.0001

 

2.0001

9

 

2.0005

 

2.0000

 

1.9999

 

0.0006

 

2.0001

10

 

1.9995

 

1.9998

 

2.0001

 

0.0006

 

1.9998

11

 

2.0002

 

1.9999

 

2.0001

 

0.0003

 

2.0001

12

 

2.0002

 

1.9998

 

2.0005

 

0.0007

 

2.0002

13

 

2.0000

 

2.0001

 

1.9998

 

0.0003

 

2.0000

14

 

2.0000

 

2.0002

 

2.0004

 

0.0004

 

2.0002

15

 

1.9994

 

2.0001

 

1.9996

 

0.0007

 

1.9997

16

 

1.9999

 

2.0003

 

1.9993

 

0.0010

 

1.9998

17

 

2.0002

 

1.9998

 

2.0004

 

0.0006

 

2.0001

18

 

2.0000

 

2.0001

 

2.0001

 

0.0001

 

2.0001

19

 

1.9997

 

1.9994

 

1.9998

 

0.0004

 

1.9996

20

 

2.0003

 

2.0007

 

1.9999

 

0.0008

 

2.0003

Rbar chart limits:

RBAR = 0.0005

UCL=D(4)*RBAR = 2.574 * .0005 = 0.001287 LCL=D(3)*RBAR = 0.000 * .0005 = 0.000

Xbar chart limits:

XDBLBAR = 2.0000

UCL = XDBLBAR + A(2) x RBAR = 2.000+1.023 x .0005 = 2.0005115 LCL = XDBLBAR - A(2) x RBAR = 2.000-1.023 x.0005 = 1.9994885

R-Chart:

C:\Users\adam\Desktop\AU_MGT340_W8_G6.jpg

Xbar-Chart:

C:\Users\adam\Desktop\AU_MGT340_W8_G7.jpg

 

Using the navigation on the left, please proceed to the next page.