1. 1 SMU EMIS 7364 NTU TO-570-N Control Charts Basic Concepts and Mathematical Basis Updated: 3/2/04 Statistical Quality Control Dr. Jerrell T. Stracener, SAE Fellow

2. 2 Chance or Common Causes of Variability Chance causes or common causes are numerous small causes of variability that are inherent to a system or process and operate randomly or by chance. –At any time, numerous factors affect a system or process, causing variability. Most of these are not readily identifiable and yet have very small to moderate effects that, individually and in interaction with each other, cause detectable variability in the process and its output.

3. 3 Chance or Common Causes of Variability A stable process is in a state of statistical control and has only chance or common causes of variability operating on it. –The system of chance causes that operates on a process is stable and constantly present. This stable system of chance causes produces patterns of variability that follow known statistical distributions. If a set of data is analyzed and the pattern of variation of the data is show to conform to statistical patterns that are characteristic of those produced by chance, we can assume that only chance or common causes are operating on the system. –Because a system is in a state of statistical control is considered to be stable we can predict the status of output for the process for the near future.

4. 4 Assignable or Special Causes of Variability Assignable or special causes of variability have relatively large effects on the process and are not inherent to it. The circumstances or factors that cause this kind of variability can be identified. –Assignable or special causes can be recognized and assigned or attributed to specific special circumstances or factors. Examples of special causes are –Differences between machinery –Differences in a machine over time –A change in raw materials –Differences between workers –Differences in an individual worker over time –Differences in the relationship among production equipment, materials, and workers –a change in manufacturing conditions...

5. 5 Assignable or Special Causes of Variability A process is said to be out of statistical control if one or more special causes are operating on it. –Special causes are not inherent to the process, so they are considered to be outside the system. When patterns of variability do not conform to the patterns we would expect if only chance or common causes are operating, we know that one or more special causes are operating on the system. If special causes are operating, the system is considered to be out of statistical control, and intervention is required to eliminate the special causes and reduce process variability.

6. 6 Types of Control Charts Control charts can be divided into two categories that are determined by the type of measurements used to monitor a process. If, in monitoring a process, we sample output and evaluate each member of the sample to see whether individual items or events are conforming or nonconforming, the frequency or proportion of nonconforming items are used to evaluate such attribute data. If our quality characteristic is measured on a continuous scale such as height, weight, temperature, or time, we employ a variable control chart.

7. 7 Types of Control Charts Attribute control charts include: 1. Counts of nonconforming Items Charts: Pieces or number nonconforming charts (np charts) Fraction or proportion nonconforming charts (p Charts) 2. Area of Opportunity Charts: Number of nonconformities charts (c charts) Nonconformities per unit charts (u charts)

8. 8 Types of Control Charts Variables control charts contain more information than attribute charts and are generally used in pairs. One member of the pair monitors process variability while other monitors central tendency or the average quality level or the output of the process. The major types of variables control charts include the following: 1. Charts based on means of samples: mean and range charts (X & R Charts) mean and standard deviation charts (X & s charts) exponentially weighted moving average charts (EWMA charts) cumulative sum charts (CUSUM charts) 2. Charts based on individual measurements (X charts): individual measurements using the range individual measurements using the standard deviation

9. 9 Upper Control Limit Center Line Process Average Lower Control Limit The Normal Distribution and the Control Charts       

10. 10 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

11. 11 Testing Hypotheses There are two possible decision errors associated with testing a statistical hypothesis: A Type I error is made when a true hypothesis is rejected. A Type II error is made when a false hypothesis is accepted. True Situation DecisionH 0 trueH 0 false Accept H 0 correctType II error Reject H 0 Type I errorcorrect (Accept H 1 )

12. 12 Testing Hypotheses The decision risks are measured in terms of probability.  = P(Type I error) = P(reject H 0 |H 0 is true) = Producers risk  = P(Type II error) = P(accept H 0 |H 1 is true) = Consumers risk Remark: 100% ·  is commonly referred to as the significance level of a test. Note: For fixed n,  increases as  decreases, and vice versa, as n increases, both  and  decrease.

13. 13 Power Function Before applying a test procedure, i.e., a decision rule, we need to analyze its discriminating power, i.e., how good the test is. A function called the power function enables us to make this analysis. Power Function = P(rejecting H 0 |true parameter value) OC Function= P(accepting H 0 |true parameter value) = 1 - Power Function where OC is Operating Characteristic.

14. 14 Power Function A plot of the power function vs the test parameter value is called the power curve and 1 - power curve is the OC curve. 1 0 PR()PR() ideal power curve H0H0 H1H1 

15. 15 Power Function The power function of a statistical test of hypothesis is the probability of rejecting H0 as a function of the true value of the parameter being tested, say , i.e., PF() = PR() = P(reject H 0 |) = P(test statistic falls in C A |)

16. 16 Operating Characteristic Function The operating characteristic function of a statistical test of hypothesis is the probability of accepting H0 as a function of the true value of the parameter being tested, say , i.e., OC()= P A () = P(accept H 0 |) = P(test statistic falls in C R |)

17. 17 Runs One way to study the probability of false alarm is through the run length. A run is a series of consecutive points which all increases in value or all decrease in value. Another kind of runs occurs when a consecutive series of points all fall above the center line or all fall below the center line. Run length is the number of charts points that occur after a shift in parameter takes place before a signal is given by the chart that the shift has occurred. The average run length (ARL) is the average number of points that must be plotted before a point indicates that the process is out-of-control or a shift has occurred.

18. 18 Use of Control Chart 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

19. 19 General Model for the Shewhart Control Chart 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

20. 20 Types of Control Chart Data Measurement (variables) Counts (attributes)

21. 21 Types of Control Chart One Measurement (variables) X Moving Range Multiple X R S

22. 22 Types of Control Chart Defectives p np Defects Counts (attributes) c 