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Supplement 6

Supplement 6. Statistical Process Control. Statistical Process Control (SPC) Control Charts for Variables The Central Limit Theorem Setting Mean Chart Limits (x-Charts) Setting Range Chart Limits (R-Charts) Using Mean and Range Charts Control Charts for Attributes. Outline.

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Supplement 6

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  1. Supplement 6 Statistical Process Control

  2. Statistical Process Control (SPC) • Control Charts for Variables • The Central Limit Theorem • Setting Mean Chart Limits (x-Charts) • Setting Range Chart Limits (R-Charts) • Using Mean and Range Charts • Control Charts for Attributes Outline

  3. Statistical Process Control (SPC) • All process are subject to a certain degree of variability • Natural or common causes • Special or assignable causes • Provides a statistical signal when assignable causes are present • Detect and eliminate assignable causes of variation

  4. Natural Variations • Also called common causes • Affect virtually all production processes • Expected amount of variation • Output measures follow a probability distribution • For any distribution there is a measure of central tendency and dispersion • If the distribution of outputs falls within acceptable limits, the process is said to be “in control”

  5. Assignable Variations • Also called special causes of variation • Generally this is some change in the process • Variations that can be traced to a specific reason • The objective is to discover when assignable causes are present • Eliminate the bad causes • Incorporate the good causes

  6. Each of these represents one sample of five boxes of cereal # # # # # Frequency # # # # # # # # # # # # # # # # # # # # # Weight Samples To measure the process, we take samples and analyze the sample statistics following these steps (a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Figure S6.1

  7. The solid line represents the distribution Frequency Weight Samples To measure the process, we take samples and analyze the sample statistics following these steps (b) After enough samples are taken from a stable process, they form a pattern called a distribution Figure S6.1

  8. Central tendency Variation Shape Frequency Weight Weight Weight Samples To measure the process, we take samples and analyze the sample statistics following these steps (c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Figure S6.1

  9. Prediction Frequency Time Weight Samples To measure the process, we take samples and analyze the sample statistics following these steps (d) If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable Figure S6.1

  10. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Prediction Frequency Time Weight Samples To measure the process, we take samples and analyze the sample statistics following these steps (e) If assignable causes are present, the process output is not stable over time and is not predicable Figure S6.1

  11. Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes

  12. Types of Data Variables Attributes • Characteristics that can take any real value • May be in whole or in fractional numbers • Continuous random variables • Defect-related characteristics • Classify products as either good or bad or count defects • Categorical or discrete random variables

  13. The mean of the sampling distribution (x) will be the same as the population mean m x = m s n The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n sx = Central Limit Theorem Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve

  14. What is standard deviation?

  15. The standard deviation is a statistic that tells you how tightly all the various examples are clustered around the mean in a set of data.

  16. Why is this useful? Here's an example: • Let's say Springfield Elementary has a higher mean test score than Shelbyville Elementary. Your first reaction might be to say that the kids at Springfield are smarter. • But a bigger standard deviation for one school may tells you that there are relatively more kids at that school scoring toward one extreme or the other. By asking a few follow-up questions you might find that, say, Springfield's mean was skewed up because the school district sends all of the gifted education kids to Springfield. Or that Shelbyville's scores were dragged down because students who recently have been "mainstreamed" from special education classes have all been sent to Shelbyville.

  17. (a) In statistical control and capable of producing within control limits Frequency Upper control limit Lower control limit (b) In statistical control but not capable of producing within control limits (c) Out of control Size (weight, length, speed, etc.) Process Control Figure S6.2

  18. Three population distributions Mean of sample means = x Beta Standard deviation of the sample means Normal = sx = Uniform s n | | | | | | | -3sx -2sx -1sx x +1sx +2sx +3sx 95.45% fall within ± 2sx 99.73% of all x fall within ± 3sx Population and Sampling Distributions Distribution of sample means Figure S6.3

  19. Sampling distribution of means Process distribution of means x = m (mean) Sampling Distribution Figure S6.4

  20. Steps In Creating Control Charts Take samples from the population and compute the appropriate sample statistic Use the sample statistic to calculate control limits and draw the control chart Plot sample results on the control chart and determine the state of the process (in or out of control) Investigate possible assignable causes and take any indicated actions Continue sampling from the process and reset the control limits when necessary

  21. For variables that have continuous dimensions • Weight, speed, length, strength, etc. • x-charts are to control the central tendency of the process • R-charts are to control the dispersion of the process • These two charts must be used together Control Charts for Variables

  22. For x-Charts when we know s Upper control limit (UCL) = x + zsx Lower control limit (LCL) = x - zsx where x = mean of the sample means or a target value set for the process z = number of normal standard deviations sx = standard deviation of the sample means = s/ n s = population standard deviation n = sample size Setting Chart Limits

  23. Hour 1 Sample Weight of Number Oat Flakes 1 17 2 13 3 16 4 18 5 17 6 16 7 15 8 17 9 16 Mean 16.1 s = 1 Hour Mean Hour Mean 1 16.1 7 15.2 2 16.8 8 16.4 3 15.5 9 16.3 4 16.5 10 14.8 5 16.5 11 14.2 6 16.4 12 17.3 n = 9 UCLx = x + zsx= 16 + 3(1/3) = 17 ozs LCLx = x - zsx = 16 - 3(1/3) = 15 ozs Setting Control Limits For 99.73% control limits, z = 3

  24. Variation due to assignable causes Out of control 17 = UCL Variation due to natural causes 16 = Mean 15 = LCL Variation due to assignable causes | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 Out of control Sample number Setting Control Limits Control Chart for sample of 9 boxes

  25. For x-Charts when we don’t know s Upper control limit (UCL) = x + A2R Lower control limit (LCL) = x - A2R where R = average range of the samples A2 = control chart factor found in Table S6.1 x = mean of the sample means Setting Chart Limits

  26. Sample Size Mean Factor Upper Range Lower Range n A2D4D3 2 1.880 3.268 0 3 1.023 2.574 0 4 .729 2.282 0 5 .577 2.115 0 6 .483 2.004 0 7 .419 1.924 0.076 8 .373 1.864 0.136 9 .337 1.816 0.184 10 .308 1.777 0.223 12 .266 1.716 0.284 Control Chart Factors Table S6.1

  27. Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 Setting Control Limits

  28. Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 UCLx = x + A2R = 16.01 + (.577)(.25) = 16.01 + .144 = 16.154 ounces From Table S6.1 Setting Control Limits

  29. Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 UCLx = x + A2R = 16.01 + (.577)(.25) = 16.01 + .144 = 16.154 ounces UCL = 16.154 Mean = 16.01 LCLx = x - A2R = 16.01 - .144 = 15.866 ounces LCL = 15.866 Setting Control Limits

  30. R – Chart • Type of variables control chart • Shows sample ranges over time • Difference between smallest and largest values in sample • Monitors process variability • Independent from process mean

  31. Upper control limit (UCLR) = D4R Lower control limit (LCLR) = D3R where R = average range of the samples D3 and D4 = control chart factors from Table S6.1 Setting Chart Limits For R-Charts

  32. Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D4= 2.115, D3 = 0 UCLR = D4R = (2.115)(5.3) = 11.2 pounds UCL = 11.2 Mean = 5.3 LCLR = D3R = (0)(5.3) = 0 pounds LCL = 0 Setting Control Limits

  33. (a) These sampling distributions result in the charts below (Sampling mean is shifting upward but range is consistent) UCL (x-chart detects shift in central tendency) x-chart LCL UCL (R-chart does not detect change in mean) R-chart LCL Mean and Range Charts Figure S6.5

  34. (b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) UCL (x-chart does not detect the increase in dispersion) x-chart LCL UCL (R-chart detects increase in dispersion) R-chart LCL Mean and Range Charts Figure S6.5

  35. Steps In Creating Control Charts Take samples from the population and compute the appropriate sample statistic Use the sample statistic to calculate control limits and draw the control chart Plot sample results on the control chart and determine the state of the process (in or out of control) Investigate possible assignable causes and take any indicated actions Continue sampling from the process and reset the control limits when necessary

  36. Upper control limit Target Lower control limit Patterns in Control Charts Normal behavior. Process is “in control.” Figure S6.7

  37. Upper control limit Target Lower control limit Patterns in Control Charts One plot out above (or below). Investigate for cause. Process is “out of control.” Figure S6.7

  38. Upper control limit Target Lower control limit Patterns in Control Charts Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7

  39. Upper control limit Target Lower control limit Patterns in Control Charts Two plots very near lower (or upper) control. Investigate for cause. Figure S6.7

  40. Upper control limit Target Lower control limit Patterns in Control Charts Run of 5 above (or below) central line. Investigate for cause. Figure S6.7

  41. Upper control limit Target Lower control limit Patterns in Control Charts Erratic behavior. Investigate. Figure S6.7

  42. Using an x-chart and R-chart: • Observations are variables • Collect 20 - 25 samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart • Track samples of n observations each Which Control Chart to Use Variables Data

  43. Have a nice day!

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