1 Dr. Jerrell T. Stracener, SAE Fellow EMIS 7370/5370 STAT 5340 Probability and Statistics for Scientists and Engineers UPDATED 11/20/06 Department of Engineering Management, Information and Systems An Application of Probability & Statistics Statistical Quality Control Leadership in Engineering

2 Statistical Quality Control is an application of probabilitistic and statistical techniques to quality control Statistical Quality Control

3 Statistical Quality Control - Elements Analysis of process capability Statistical process control Process improvement Acceptance sampling

4 Quality begins with customer requirements Quality must be designed in. It cannot be inspected in! Quality depends on: Parts selection and procurement Material Manufacturing/production processes Logistic processes. Statistical Quality Control - Basic Concepts

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6 Quality is meeting the customer’s needs over the life cycle of the product at the best value to the customer Quality has many dimensions Reliability Maintainability Performance Durability Conformance (to requirements and expectations). Statistical Quality Control - What is quality?

7 Making it happen - 99% agree that management is the problem not the workers - 35% of the problem is ‘not invented here’ syndrome getting their attention and education resistance to change - 15% gaining management commitment - 14% communication getting the word out within the company Failure of management to understand ‘variation’ Obstacles to Quality Improvement

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12 The application of statistical techniques is to understand and analyze the variation in a process. - Joseph Juran Quality Control Handbook Statistical Process Control - Definition

13 In many situations, our knowledge is limited to the information that can be obtained from data that has been obtained or that will be obtained The Situation

14 The challenge is to obtain the maximum information from the data and to arrive at the most accurate conclusions The Problem

15 Most data are characterized by variation, as opposed to deterministic, due to variation in Processes and materials Product/Manufacturing Inspection & Measurement Operation Environment etc Nature of Data

16 Methods and techniques are needed for analysis of data that account for Variation in the data Uncertainty in conclusion Need

17 Statistics is the science of analyzing data and drawing conclusions Statistical methods and techniques that provide tools for: - experimental design - analysis of data - making inferences Statistics

18 SPC is a powerful collection of problem-solving tools useful in achieving process stability and improving capability through the reduction of variability. SPC can be applied to any process Seven major tools 1. Histogram or stem and leaf display 2. Check sheet 3. Pareto chart 4. Cause and effect diagram 5. Defect concentration diagram 6. Scatter diagram 7. Control chart Statistical Process Control (SPC)

19 Causes of Variation Assignable (special) - Intermittent sources of variation that are unpredictable. Signaled by violation of Western Electric rules Common (natural) - Sources of variation always present affecting all output from a process Only management can affect common causes of variation Statistical Process Control

20 Histograms - Questions to ask What is the shape of distribution? What would you expect shape to be? If computer generated, is data really normal? Is variation acceptable? Is the centering acceptable? Did you generate a histogram with and without outlier points? Did you include specification limits and process limits on the histogram? Statistical Process Control - Histograms

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22 Statistical Process Control - Histograms The shape shows the nature of the distribution of the data The central tendency (average) and variability are easily seen Specification limits can be used to display the capability of the process LSL USL

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24 Refers to the uniformity of the process. Variability in the process is a measure of the uniformity of the output. - Instantaneous variability is the natural or inherent variability at a specified time - Variability over time Process Capability

25 A critical performance measure that addresses process results relative to process/product specifications. A capable process is one for which the process outputs meet or exceed expectation. Process Capability

26 Customary to use the six sigma spread in the distribution of the product quality characteristic Measures of Process Capability

27 The proportion of the process output that will fall outside the natural tolerance limits. Is 0.27% (or 2700 nonconforming parts per million) if the distribution is normal May differ considerably from 0.27% if the distribution is not normal Key Points

28 Process capability indices are used to measure the process variability due to common causes present in the process The C p index Inherent or potential measure of capability specification spread process spread The C pK index Realized or actual measure of capability Other indices C pM, C pMK C p = Process Capability Measures or Indices

29 C p measures potential or inherent capability of the process, given that the process is stable C p is defined as, for two-sided specifications and, for lower specifications only, for upper specifications only Measure of Potential Process Capability, C p

30 C pK measures realized process capability relative to actual production, given a stable process C pK is defined as Measure of Potential Process Capability, C pK

31 ppm = parts per million Interpretation C pK < 1= process not capable 1  C pK < 1.5= process capable, monitor frequently C pK  1.5= process capable, monitor infrequently Pareto C pK ’s to attack worst problems Can only convert C pK, C p to ppm if distribution normal Statistical Process Control

32 is the percentage of the specification band used up by the process Interpretation of C p

33 Impact of special causes on process capability time process stable time process unstable Statistical Process Control

34 Expresses detailed knowledge of the process Identifies process flow and interaction among the process steps Identifies potential control points Statistical Process Flow Diagram

35 Statistical Process Control - Pareto Diagram Identifies the most significant problems to be worked first Historically 80% of the problems are due to 20% of the factors Shows the vital few Number of occurrences Cumulative percent

36 Temp. Pressure x x x x x x x x x x x x x x Statistical Process Control - Scatter Plot Identifies the relationship between two variables A positive, negative, or no relationship can be easily detected

37 Six Sigma is a business initiative first espoused by Motorola in the early 1990’s. Six Sigma strategy involves the use of statistical tools within a structured methodology for gaining the knowledge needed to achieve better, faster, and less expensive products and services than the competition. A Six Sigma initiative in a company is designed to change the culture through breakthrough improvement by focusing on out-of-the-box thinking in order to achieve aggressive, stretch goals. Background of Six Sigma

38 1.Prioritize opportunities for improvement 2.Select the appropriate team 3.Describe the total process 4.Perform measurement system analysis 5.Identify and describe the potential critical process 6.Isolate and verify the critical processes 7.Perform process and measurement system capability studies 8.Implement optimum operating conditions and control methodology 9. Monitor processes over time/continuous improvement 10.Reduce common cause variation toward achieving six sigma Motorola’s Six Sigma Ten Steps

39 Product Specification Lower Specification Limit Nominal Specification Upper Specification Limit Target (Ideal level for use in product) Tolerance x (Product characteristic) (Maximum range of variation of the product characteristic that will still work in the product.)

40 (Make it to specifications) Good TUSLLSL Loss ($) No-Good x Traditional US Approach to Quality

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42 Setting Specification Limits on Discrete Components

43 Variability reduction is a modern concept of design and manufacturing excellence Reducing variability around the target value leads to better performing, more uniform, defect-free product Virtually eliminates rework and waste Consistent with continuous improvement concept accept reject target Don’t just conform to specifications Reduce variability around the target Variability Reduction

44 Sources of loss - scrap - rework - warranty obligations - decline of reputation - forfeiture of market share Loss function - dollar loss due to deviation of product from ideal characteristic Loss characteristic is continuous - not a step function. True Impact of Product Variability

45 Representative Loss Function Characteristics x Loss $ X nominal is best L = k (x - T) 2 x Loss $ X smaller is better L = k (x 2 ) x Loss $ X larger is better L = k (1/x 2 ) 

46 Variability-Loss Relationship LSL USL Target $ savings due to reduced variability Maximum $ loss per item Loss

47 Loss Computation for Total Product Population X nominal is best L = k (x - T) 2 x Loss $ T x Loss $ T

48 Statistical Tolerancing - Convention Normal Probability Distribution LTL Nominal UTL +3 -3 

49 Statistical Tolerancing - Concept LTL UTLNominal x

50 For a normal distribution, the natural tolerance limits include 99.73% of the variable, or put another way, only 0.27% of the process output will fall outside the natural tolerance limits. Two points should be remembered: % outside the natural tolerances sounds small, but this corresponds to 2700 nonconforming parts per million. 2. If the distribution of process output is non normal, then the percentage of output falling outside   3  may differ considerably from 0.27%. Caution

51 The diameter of a metal shaft used in a disk-drive unit is normally distributed with mean inches and standard deviation inches. The specifications on the shaft have been established as  inches. We wish to determine what fraction of the shafts produced conform to specifications. Normal Distribution - Example

USL LSL f(x) x nominal Normal Distribution - Example Solution

53 Thus, we would expect the process yield to be approximately 91.92%; that is, about 91.92% of the shafts produced conform to specifications. Note that almost all of the nonconforming shafts are too large, because the process mean is located very near to the upper specification limit. Suppose we can recenter the manufacturing process, perhaps by adjusting the machine, so that the process mean is exactly equal to the nominal value of Then we have Normal Distribution - Example Solution

USL LSL f(x) x nominal Normal Distribution - Example Solution

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59 Using a normal probability distribution as a model for a quality characteristic with the specification limits at three standard deviations on either side of the mean. Now it turns out that in this situation the probability of producing a product within these specifications is , which corresponds to 2700 parts per million (ppm) defective. This is referred to as three-sigma quality performance, and it actually sounds pretty good. However, suppose we have a product that consists of an assembly of 100 components or parts and all 100 parts must be non-defective for the product to function satisfactorily. Normal Distribution - Example

60 The probability that any specific unit of product is non-defective is x x... x = (0.9973) 100 = That is, about 23.7% of the products produced under three sigma quality will be defective. This is not an acceptable situation, because many high technology products are made up of thousands of components. An automobile has about 200,000 components and an airplane has several million! Normal Distribution - Example

61 The random variable X can modeled by a Weibull distribution with  = ½ and  = The spec time limit is set at x = What is the proportion of items not meeting spec? Weibull Distribution - Example

62 The fraction of items not meeting spec is That is, all but about 13.53% of the items will not meet spec. Weibull Distribution - Example

63 Interpretation based on Western Electric rules 1. Analyze the chart by separating it into equal zones above and below the centerline UCL LCL Centerline A B C C B A Statistical Process Control - Control Charts

64 2. A process is out of statistical control if: (a) any point is above or below the control limits (b) two out of three points in a row in zone A or above (c) four out of five points in a row in zone B or above (d) eight in a row in zone C or above Statistical Process Control - Control Charts

65 In general specification limits should not be on control charts Data must be displayed in time sequence Management controls the natural variation between the control limits Do not tweak the process Statistical Process Control - Control Charts

66 Helps reduce variability Monitors performance over time Allows process corrections to prevent rejections Trends and out-of-control conditions are immediately detected x x x x x x x x x x x x x x UCL LCL CL Statistical Process Control - Control Charts

67 Upper Control Limit Center Line Process Average Lower Control Limit The Normal Distribution and the Control Charts       

68 UCL =  W + K W Center Line =  W LCL =  W - K W where W is a statistic that measures a quality characteristic  W is the mean of W  W is the standard deviation of W K is the distance of the control limits from the center line, in multiples of  W General Model for the Shewhart Control Chart

69 Types of Error that Can Occur When Using Control Charts Actual State of Process Only Common CausesSpecial Causes Out of Control Control A False Alarm C Correct Decision D Failure to Detect B Correct Decision Control Chart Indicates

70 Control charts are a proven technique for improving productivity Control charts are effective in defect prevention Control charts prevent unnecessary process adjustment Control charts provide diagnostic information. Control charts provide information about process capability Use of Control Chart

71 Types of Control Chart Data Measurement (variables) Counts (attributes)

72 One Measurement (variables) X Moving Range Multiple X R S Types of Control Chart

73 Defectives p np Defects Counts (attributes) c  Types of Control Chart