Statistical Process Control Copyright © 2010 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. HAWKES LEARNING SYSTEMS math courseware specialists Chapter 17 Statistical Process Control

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.3 Monitoring with an Chart Objectives: To learn the meaning of a mean chart using standard deviation. To learn the meaning of a mean chart using range. To find the center line of a mean chart. To find the upper and lower control limits.

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.3 Monitoring with an Chart Definitions: Process control relies on two assumptions: No matter what the specifications are for a product, the process that produces the product will create output that has variation. Improving a process requires removing variation from it. Though it would be ideal to remove all variation, this cannot be achieved. The goal then is to move towards the ideal. This notion is known as continuous improvement. Control chart – a graphical display of values of a process over time. This chart has an upper and lower control limit. If a value is above the upper control limit or below the lower control limit the process is out of control. The control chart also contains a centerline that represents the average value of the quality characteristic corresponding to the in-control state.

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.3 Monitoring with an Chart Definitions: Chart – a control chart that plots the means of small samples taken from the process at regular intervals over time. There are two ways to construct an chart: If the process mean and standard deviation are known the chart is constructed such that any sample average within ±3σ is considered in control and any sample average outside of ±3σ is considered out of control. It is defined by the formulas:

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.3 Monitoring with an Chart Definitions: If the process mean and standard deviation are not known the chart is constructed from factors in the table on the next slide. The table provides 3σ factors that relate to the sample ranges. It is defined by the formulas: where

3σ Factors for Computing Control Charts Limits HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.3 Monitoring with an Chart 3σ Factors for Computing Control Charts Limits Sample Size Mean Factor, A Lower Range, D3 Upper Range, D4 2 1.880 0.000 3.267 3 1.023 2.574 4 0.729 2.282 5 0.577 2.114 6 0.483 2.004 7 0.419 0.076 1.924 8 0.373 0.136 1.864 9 0.337 0.184 1.816 10 0.308 0.223 1.777 15 0.348 1.652 20 0.180 0.414 1.586 25 0.153 0.459 1.541 Over 25 155 – 0.0015n 0.45 + 0.001n

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.3 Monitoring with an Chart Process Controls: Suppose that a company produces bolts with an average width of 0.45 inches and a standard deviation of 0.11 inches. A total of 24 samples of size 25 are taken. Determine the upper and lower control limits for the process and determine if any of the sample values on the graph are considered out of control. Solution:

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.3 Monitoring with an Chart Process Controls: Suppose that a company produces bolts with an average width of 0.45 inches and a standard deviation of 0.11 inches. A total of 24 samples of size 25 are taken. Determine the upper and lower control limits for the process and determine if any of the sample values on the graph are considered out of control. Solution: Adding the Upper Control Limit and Lower Control Limit lines to the graph, we can see that one of the sample averages is considered out of control.

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.3 Monitoring with an Chart Process Controls: Suppose that a company produces bolts but the true mean and standard deviation are not known. A total of 24 samples of size 25 are taken. The mean of the sample means (grand mean) is 0.4525 and the mean of the sample ranges is 0.4029. Determine the upper and lower control limits for the process and determine if any of the sample values on the graph are considered out of control. Solution:

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.3 Monitoring with an Chart Process Controls: Suppose that a company produces bolts but the true mean and standard deviation are not known. A total of 24 samples of size 25 are taken. The mean of the sample means (grand mean) is 0.4525 and the mean of the sample ranges is 0.4029. Determine the upper and lower control limits for the process and determine if any of the sample values on the graph are considered out of control. Solution: Adding the Upper Control Limit and Lower Control Limit lines to the graph, we can see that one of the sample averages is considered out of control.

To learn the meaning of an R chart using standard deviation. HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.4 Monitoring with an R Chart Objectives: To learn the meaning of an R chart using standard deviation. To find the centerline of a R chart. To find the upper and lower control limits.

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.4 Monitoring with an R Chart Definitions: R Chart – a control chart that monitors the variation of the samples of process. The chart uses the average of the sample range as the center line and upper and lower level factors based on the sample size. A table of these factors is given on the next slide.

3σ Factors for Computing Control Charts Limits HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.4 Monitoring with an R Chart 3σ Factors for Computing Control Charts Limits Sample Size Mean Factor, A Lower Range, D3 Upper Range, D4 2 1.880 0.000 3.267 3 1.023 2.574 4 0.729 2.282 5 0.577 2.114 6 0.483 2.004 7 0.419 0.076 1.924 8 0.373 0.136 1.864 9 0.337 0.184 1.816 10 0.308 0.223 1.777 15 0.348 1.652 20 0.180 0.414 1.586 25 0.153 0.459 1.541 Over 25 155 – 0.0015n 0.45 + 0.001n

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.4 Monitoring with an R Chart Process Controls: Suppose that a company produces bolts but is concerned about the variation in the process. A total of 24 samples of size 25 are taken. The mean of the sample ranges is 0.4029. Determine the upper and lower control limits for the process using an R Chart and determine if any of the values of the sample ranges are considered out of control. Solution:

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.4 Monitoring with an R Chart Process Controls: Suppose that a company produces bolts but is concerned about the variation in the process. A total of 24 samples of size 25 are taken. The mean of the sample ranges is 0.4029. Determine the upper and lower control limits for the process using an R Chart and determine if any of the values of the sample ranges are considered out of control. Solution: Adding the Upper Control Limit and Lower Control Limit lines to the graph, we can see that all of the sample ranges are considered in control.

To learn the meaning of an p chart using standard deviation. HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.5 Monitoring with a p Chart Objectives: To learn the meaning of an p chart using standard deviation. To find the center line of a p chart. To find the upper and lower control limits.

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.5 Monitoring with a p Chart Definitions: p Chart – a control chart that monitors the proportions of a sample that possess a certain attribute. There are two ways to construct a p chart: If the true proportion of a population that possess a certain attribute is known, the centerline, Upper Control Limits, and Lower Control Limits are defined by the formulas: where p = true population proportion and n = the sample size.

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.5 Monitoring with a p Chart Definitions: If the true proportion is not known the chart is constructed from the sample proportion. It is defined by the formulas: where n = the sample size.

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.5 Monitoring with a p Chart Process Controls: Silvia Garcia is monitoring the operation of a machine that makes radar components. Historically, she expects 2 percent defectives and some chance variation. After studying the process, Silvia decides to construct a 3σ p chart. There were 10 samples that each had 200 observations. Solution: Note: Since a proportion cannot be negative, the Lower Control Level will be defined as 0.

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.5 Monitoring with a p Chart Process Controls: Silvia Garcia is monitoring the operation of a machine that makes radar components. Historically, she expects 2 percent defectives and some chance variation. After studying the process , Silvia decides to construct a 3σ p chart. There were 10 samples that each had 200 observations. Solution: Adding the Upper Control Limit and Lower Control Limit lines to the graph, we can see that two of the sample proportions are considered out of control.

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.5 Monitoring with a p Chart Process Controls: Silvia Garcia is monitoring the operation of a machine that makes radar components. She does not have historical data and does not know the true proportion of defectives. The sample proportion of defectives is 0.0445. After studying the process , Silvia decides to construct a 3σ p chart. There were 10 samples that each had 200 observations. Solution:

HAWKES LEARNING SYSTEMS math courseware specialists Statistical Process Control Section 17.5 Monitoring with a p Chart Process Controls: Silvia Garcia is monitoring the operation of a machine that makes radar components. She does not have historical data and does not know the true proportion of defectives. The sample proportion of defectives is 0.0445. After studying the process, Silvia decides to construct a 3σ p chart. There were 10 samples that each had 200 observations. Solution: Adding the Upper Control Limit and Lower Control Limit lines to the graph, we can see that none of the sample proportions are considered out of control.