As a quality analyst you are also responsible for controlling the weight of a box of cereal. The Operations Manager asks you to identify the ways in which statistical quality control methods can be applied to the weights of the boxes. Provide your recommendations to the Operations Manager in a two-three page report. Using the data provided in the **Doc Sharing** area labeled M4A2Data, create Xbar and R charts.

Your report should indicate the following along with valid justifications of your answers:

- The control limits of the weights of the boxes.
- Nonrandom patterns or trends, if any.
- If the process is in control.
- The appropriate action if the process is not in control.

>

## Xb

arR

Control Chart for Mean and Range [Title or Proce ss]

## ©

200

9Vertex

42 LLC

[Date] HELP Quality Characteristic Average Thic kness (mm),

X-bar SampleSize, n 5 k

3 Statistics from Data Table Process Capability R-bar 4.3 60 Upper Spec Limit, USL 40 Process Mean, m-hat 34

.

840 Lower Spec Limit, LSL 30 Process St.Dev., s-hat 1

.8

74 Cp 0.889 sX-bar 0.838 CPU 0.9

18 CPL 0.861 Cpk 0.861

Percent Yield
99.

% Control Limits for

X-bar Chart Control Limits for R Chart CLX-bar 34.8

40 CLR 4.

360 UCL

X-bar 37

.

355 CL+ksX-bar UCLR 9.2

19 LCLX-bar 32.3

25 CL-ksX-bar LCLR 0.000 a 0.0027 ARL 370.4 samples Data Table X-bar Chart R ChartSample X-bar Range CL UCL LCL CL UCL LCL

1 35.6

4

34.840 37.355 32.325 4.360 9.219 0.000

2

.8

5 34.840 37.355 32.325 4.360 9.219 0.000

3

6 34.840 37.355 32.325 4.360 9.219 0.000

4 35 3 34.840 37.355 32.325 4.360 9.219 0.000

5 35.6 6 34.840 37.355 32.325 4.360 9.219 0.000

6

5 34.840 37.355 32.325 4.360 9.219 0.000

7 33 3 34.840 37.355 32.325 4.360 9.219 0.000

8 34.4 6 34.840 37.355 32.325 4.360 9.219 0.000

9 36 5 34.840 37.355 32.325 4.360 9.219 0.000

10

34 3 34.840 37.355 32.325 4.360 9.219 0.000

11

35 4 34.840 37.355 32.325 4.360 9.219 0.000

12

35.6 4 34.840 37.355 32.325 4.360 9.219 0.000

13

33.4 5 34.840 37.355 32.325 4.360 9.219 0.000

14
35.2

3 34.840 37.355 32.325 4.360 9.219 0.000

15
36.8

5 34.840 37.355 32.325 4.360 9.219 0.000

16

35.2 5 34.840 37.355 32.325 4.360 9.219 0.000

17
33.2

3 34.840 37.355 32.325 4.360 9.219 0.000

18

5 34.840 37.355 32.325 4.360 9.219 0.000

19

5 34.840 37.355 32.325 4.360 9.219 0.000

20 36 4 34.840 37.355 32.325 4.360 9.219 0.000

21

4 34.840 37.355 32.325 4.360 9.219 0.000

22
32.6

5 34.840 37.355 32.325 4.360 9.219 0.000

23

37 4 34.840 37.355 32.325 4.360 9.219 0.000

24

34.8 3 34.840 37.355 32.325 4.360 9.219 0.000

25

4 34.840 37.355 32.325 4.360 9.219 0.000

Insert rows above the gray line

&L&8

© 2009 Vertex42 LLC&R&8Templates by Vertex42.com

Probability of a Type I Error (a):

If a sample value falls outside the control limits, we would conclude that the process is out of control. A Type I error is made when the process is concluded to be out of control when it is really in control. This probability is calculated assuming a normal distribution for the process.

The estimated process mean is calculated as the mean of the X-bar values from the data table. It is used as the Center Line for the X-bar Chart.

The estimated process standard deviation.

The number of measurements within each sample. For this chart, all samples are assumed to be the same size. In this spreadsheet, the sample size must be between 2 and 25.

The k-value is number of standard deviations (typically 3) that the upper and lower control limits are placed away from the center line.

In-Control Average Run Length:

If the process is in-control, the ARL is the number of samples, on average, you would observe before getting an out-of-control signal. In other words, you expect to get a false alarm (a point outside the control limits) every N samples, where N is the ARL.

R-bar is the mean of the Ranges in the data table and is used as the center line for the R-Chart.

The range for each sample is calculated as the Max value minus the Min value.

X-bar is the sample mean calculated as the sum of the observations divided by the number of observations in the sample (n).

This is the standard deviation of the sample mean, calculated as the process standard deviation divided by the square root of the sample size.

The Cp index is calculated as (USL-LSL)/(6*sigma) where sigma is the process standard deviation. You want Cp to be greater than 1.

The CPU index is the upper capability index for when you are only given an upper spec limit, USL. You want CPU > 1

The CPL index is the lower capability index for when you are only given a lower spec limit, LSL. You want a CPL > 1.

The Cpk index is used when the process mean is shifted away from the target value, or the point half way between the spec limits. It is the minimum of the CPU and CPL. You want a Cpk>1.

Percent Yield measures the proportion of the output that is within the spec limits, assuming a Normal population distribution.

Instructions:

This Control Chart template creates an X-bar Chart and R Chart with control limits calculated from values contained in the data table. All samples are assumed to be the same size.

– Enter the label and the sample size for the quality characteristic that you are monitoring.

– Choose a k-value (typically 3) for setting the control limits.

– Replace the X-bar and Range values in the Data Table with your own data set (use Paste Special – Values).

– You can delete unused rows in the data table.

– If you need to insert additional rows in the data table, insert rows above the gray line below the table so that series in the chart expand accordingly. Copy the formulas for CL, UCL, and LCL to fill in the blank spaces.

– The labels for CL, UCL, and LCL within the chart are created by selecting the last Data Point and formatting it so that the Data Labels include both the Series name and the Value.

XbarR

X-bar

CL

UCL

LCL

Sample #

## XbarS

Range

CL

UCL

LCL

Sample #

Range

## TermsOfUse

[Title or Process] © 2009 Vertex42 LLC [Date] HELP

Quality Characteristic Average Thickness (mm), X-bar

4

k 3

Process Capability

4.360 Upper Spec Limit, USL

Lower Spec Limit, LSL 25

Cp

Process Mean, m-hat 34.840 CPU sX-barCPL

ss Cpk 0.693

34.840

4.360

0.000

a 0.0027

ARL 370.4 samples

Data Table X-bar Chart R Chart

CL UCL LCL CL UCL LCL

2

5 34.840 41.939 27.741 4.360 9.880 0.000

4 35 3 34.840 41.939 27.741 4.360 9.880 0.000

5 35.6 6 34.840 41.939 27.741 4.360 9.880 0.000

6 33.4 5 34.840 41.939 27.741 4.360 9.880 0.000

7 33 3 34.840 41.939 27.741 4.360 9.880 0.000

8 34.4 6 34.840 41.939 27.741 4.360 9.880 0.000

9 36 5 34.840 41.939 27.741 4.360 9.880 0.000

10 34 3 34.840 41.939 27.741 4.360 9.880 0.000

11 35 4 34.840 41.939 27.741 4.360 9.880 0.000

12 35.6 4 34.840 41.939 27.741 4.360 9.880 0.000

13 33.4 5 34.840 41.939 27.741 4.360 9.880 0.000

14 35.2 3 34.840 41.939 27.741 4.360 9.880 0.000

15 36.8 5 34.840 41.939 27.741 4.360 9.880 0.000

16 35.2 5 34.840 41.939 27.741 4.360 9.880 0.000

17 33.2 3 34.840 41.939 27.741 4.360 9.880 0.000

18 36.4 5 34.840 41.939 27.741 4.360 9.880 0.000

19 34.2 5 34.840 41.939 27.741 4.360 9.880 0.000

20 36 4 34.840 41.939 27.741 4.360 9.880 0.000

21 35.8 4 34.840 41.939 27.741 4.360 9.880 0.000

22 32.6 5 34.840 41.939 27.741 4.360 9.880 0.000

23 37 4 34.840 41.939 27.741 4.360 9.880 0.000

24 34.8 3 34.840 41.939 27.741 4.360 9.880 0.000

25 34.6 4 34.840 41.939 27.741 4.360 9.880 0.000

Insert rows above the gray line

Control Chart for Mean and Standard Deviation | |||||||||||||||||||||||||||||||||||||||||||||||||||||

Sample Size, n | |||||||||||||||||||||||||||||||||||||||||||||||||||||

Statistics from Data Table | |||||||||||||||||||||||||||||||||||||||||||||||||||||

s-bar | 45 | ||||||||||||||||||||||||||||||||||||||||||||||||||||

c4 | 0.9213 | ||||||||||||||||||||||||||||||||||||||||||||||||||||

4.732 | 0.704 | ||||||||||||||||||||||||||||||||||||||||||||||||||||

0.716 | |||||||||||||||||||||||||||||||||||||||||||||||||||||

2.366 | 0.693 | ||||||||||||||||||||||||||||||||||||||||||||||||||||

1.840 | |||||||||||||||||||||||||||||||||||||||||||||||||||||

96.53% | |||||||||||||||||||||||||||||||||||||||||||||||||||||

Control Limits for X-bar Chart | Control Limits for S Chart | ||||||||||||||||||||||||||||||||||||||||||||||||||||

CLX-bar | CLS | ||||||||||||||||||||||||||||||||||||||||||||||||||||

UCLX-bar | 41.939 | UCLS | 9.880 | ||||||||||||||||||||||||||||||||||||||||||||||||||

LCLX-bar | 27.741 | LCLS | |||||||||||||||||||||||||||||||||||||||||||||||||||

St. Dev., s | |||||||||||||||||||||||||||||||||||||||||||||||||||||

33.8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||

&L&8© 2009 Vertex42 LLC&R&8Templates by Vertex42.com

Probability of a Type I Error (a):

If a sample value falls outside the control limits, we would conclude that the process is out of control. A Type I error is made when the process is concluded to be out of control when it is really in control. This probability is calculated assuming a normal distribution for the process.

The estimated process mean is calculated as the mean of the X-bar values from the data table. It is used as the Center Line for the X-bar Chart.

The estimated population standard deviation. This is calculated by dividing s-bar by c4.

The number of measurements within each sample. For this chart, all samples are assumed to be the same size. In this spreadsheet, the sample size must be between 2 and 25.

The k-value is number of standard deviations (typically 3) that the upper and lower control limits are placed away from the center line.

In-Control Average Run Length:

If the process is in-control, the ARL is the number of samples, on average, you would observe before getting an out-of-control signal. In other words, you expect to get a false alarm (a point outside the control limits) every N samples, where N is the ARL.

s-bar is the mean of the sample standard deviations from the data table. It is the center line for the S Chart.

The sample standard deviation for each sample.

X-bar is the sample mean calculated as the sum of the observations divided by the number of observations in the sample (n).

This is the standard deviation of the sample mean, calculated as the process standard deviation divided by the square root of the sample size.

This is the standard deviation of the sample mean, calculated from the estimated population standard deviation and c4.

c4 is a factor that depends on the sample size and can be found tabulated in most control chart factor tables. Assuming the population distribution is Normal, c4 is used to find the mean and standard deviation of the sample standard deviation.

The Cp index is calculated as (USL-LSL)/(6*sigma) where sigma is the process standard deviation. You want Cp to be greater than 1.

The CPU index is the upper capability index for when you are only given an upper spec limit, USL. You want CPU > 1

The CPL index is the lower capability index for when you are only given a lower spec limit, LSL. You want a CPL > 1.

The Cpk index is used when the process mean is shifted away from the target value, or the point half way between the spec limits. It is the minimum of the CPU and CPL. You want a Cpk>1.

Percent Yield measures the proportion of the output that is within the spec limits, assuming a Normal population distribution.

Instructions:

This Control Chart template creates an X-bar Chart and Standard Deviation Chart (s Chart) with control limits calculated from values contained in the data table. All samples are assumed to be the same size.

– Enter the label and the sample size for the quality characteristic that you are monitoring.

– Choose a k-value (typically 3) for setting the control limits.

– Replace the X-bar and St.Dev. values in the Data Table with your own data set (use Paste Special – Values).

– You can delete unused rows in the data table.

– If you need to insert additional rows in the data table, insert rows above the gray line below the table so that series in the chart expand accordingly. Copy the formulas for CL, UCL, and LCL to fill in the blank spaces.

– The labels for CL, UCL, and LCL within the chart are created by selecting the last Data Point and formatting it so that the Data Labels include both the Series name and the Value.

TermsOfUse

X-bar

CL

UCL

LCL

Sample #

Average Thickness (mm), X-bar

©

St. Dev., s

CL

UCL

LCL

Sample #

Standard Deviation

Terms of Use | |

© 2009 Vertex42 LLC. All rights reserved. | |

http://www.vertex42.com/ExcelTemplates/control-chart.html | |

This TermsOfUse worksheet may not be modified, removed, or deleted. | |

Limited Use Policy | |

You may make archival copies and customize this template (the “Software”) for personal use or for your | |

company use. The customized template (with your specific personal or company information) may be | |

used and shared within your company, but this template or any document including or derived from | |

this template may NOT be sold, distributed, or placed on a public server such as the internet. | |

No Warranties | |

THE SOFTWARE AND ANY RELATED DOCUMENTATION ARE PROVIDED TO YOU “AS IS.” | |

VERTEX42, LLC MAKES NO WARRANTIES, EXPRESS OR IMPLIED, AND EXPRESSLY DISCLAIMS ALL | |

REPRESENTATIONS, ORAL OR WRITTEN, TERMS, CONDITIONS, AND WARRANTIES, INCLUDING BUT NOT | |

LIMITED TO, IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, AND | |

NONINFRINGEMENT. WITHOUT LIMITING THE ABOVE YOU ACCEPT THAT THE SOFTWARE MAY NOT MEET | |

YOUR REQUIREMENTS, OPERATE ERROR FREE, OR IDENTIFY ANY OR ALL ERRORS OR PROBLEMS, OR DO | |

SO ACCURATELY. This Agreement does not affect any statutory rights you may have as a consumer. | |

Limitation of Liability | |

IN NO EVENT SHALL VERTEX42, LLC BE LIABLE TO YOU, FOR ANY DAMAGES, INCLUDING ANY LOST PROFITS, | |

LOST SAVINGS, OR ANY OTHER DIRECT, INDIRECT, SPECIAL, INCIDENTAL, OR CONSEQUENTIAL DAMAGES | |

ARISING FROM THE USE OR THE INABILITY TO USE THE SOFTWARE (EVEN IF WE OR AN AUTHORIZED DEALER | |

OR DISTRIBUTOR HAS BEEN ADVISED OF THE POSSIBILITY OF THESE DAMAGES), OR ANY MISTAKES AND | |

NEGLIGENCE IN DEVELOPING THIS SOFTWARE, OR FOR ANY CLAIM BY ANY OTHER PARTY. THE | |

ORGANIZATION, BUSINESS, OR PERSON USING THIS SOFTWARE BEARS ALL RISKS AND RESPONSIBILITY | |

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Somes states do not allow the limitation or exclusion of liability for incidental or consequential damages, | |

so the above limitation may not apply to you. |

http://www.vertex42.com/ExcelTemplates/control-chart.html

© 2009 Vertex42 LLC

http://www.vertex42.com/ExcelTemplates/control-chart.html

Control Chart Template |

http://www.vertex42.com/ExcelTemplates/control-chart.html

>Sheet

1ote: The following consists of

sets of three box weights in ounces

N 1.3

26.32

735027

3333333

36.33

88

58

4 6.3 6.29 5 66.33

76.323

8 96.28

116.39

126.4

6.339

N | 12 | ||||||||

#1 | #2 | # | 3 | StdDev | Xbar | ||||

6 | 6.2 | 8 | 6.26 | 0.02 | 6.28 | ||||

6.32 | 6.33 | 0.00 | 5 | 7 | 6.323 | ||||

6.2 | 9 | 6.36 | 0.035 | 11 | 4 | 6.3266666667 | |||

6.34 | 0.0264575131 | 6.31 | |||||||

6.295 | 6.315 | 6.39 | 0.050083264 | 6.3333333333 | |||||

6.292 | 6.319 | 0.0195533458 | 6.3136666667 | ||||||

6.289 | 6.4 | 0.0568711995 | 6.3373333333 | ||||||

6.286 | 6.327 | 6.471 | 0.0971613778 | 6.3613333333 | |||||

6.283 | 6.331 | 6.498 | 0.1128553647 | 6.3706666667 | |||||

10 | 6.335 | 6.525 | 0.1285496013 | 6.38 | |||||

6.277 | 6.339 | 0.0565891627 | 6.3353333333 | ||||||

6.274 | 6.343 | 0.0630951662 |

Control weight of Cereal Boxes