ENGM 720 Lecture 07 Xbar R S Control

ENGM 720 - Lecture 07 X-bar, R, & S Control Charts; ARL & OC Curves 2/21/2021 ENGM 720: Statistical Process Control 1

Assignment: l Reading: • • l Chapter 5 • Start & Finish reading Chapter 6 • Spring Break means reading statistics on the beach, right? Assignments: • • Obtain the draft Ctrl Chart Factors table from Materials Page Access Excel Template for X-bar, R, & S Control Charts: • • Download Assignment 5 for practice Use the Excel sheet to do the charting, and verify your hand calculations 2/21/2021 ENGM 720: Statistical Process Control 2

Process for Statistical Control Of Quality l Removing special causes of variation Statistical Quality Control and Improvement Improving Process Capability and Performance • Hypothesis Tests • Ishikawa’s Continually Improve the System Characterize Stable Process Capability Tools l Managing the process with control charts • Process Improvement • Process Stabilization • Confidence in Time Identify Special Causes - Bad (Remove) Identify Special Causes - Good (Incorporate) Reduce Variability “When to Act” Center the Process LSL 2/21/2021 Head Off Shifts in Location, Spread 0 USL ENGM 720: Statistical Process Control 3

Moving from Hypothesis Testing to Control Charts l A control chart is like a sideways hypothesis test • Detects a shift in the process • Heads-off costly errors by detecting trends 2 2 2 0 CL 0 2 2 -Sided Hypothesis Test Sideways Hypothesis Test 2/21/2021 UC L LC L Sample Number Shewhart Control Chart ENGM 720: Statistical Process Control 4

Test of Hypothesis l A statistical hypothesis is a statement about the value of a parameter from a probability distribution. l Ex. Test of Hypothesis on the Mean l • • Say that a process is in-control if its’ mean is 0. In a test of hypothesis, use a sample of data from the process to see if it has a mean of 0. Formally stated: • • H 0 : = 0 HA: ≠ 0 2/21/2021 (Process is in-control) (Process is out-of-control) ENGM 720: Statistical Process Control 5

Test of Hypothesis on Mean (Variance Known) l State the Hypothesis • • H 0 : = 0 H 1 : ≠ 0 l Take random sample from process and compute appropriate test statistic l Pick a Type I Error level (a) and find the critical value za/2 l Reject H 0 if |z 0| > za/2 2/21/2021 ENGM 720: Statistical Process Control 6

UCL and LCL are Equivalent to the Test of Hypothesis l l Reject H 0 if: • Case 1: • Case 2: For 3 -sigma limits za/2 = 3 2/21/2021 ENGM 720: Statistical Process Control 7

Two Types of Errors May Occur When Testing a Hypothesis l l Type I Error - a • • Reject H 0 when we shouldn't Analogous to false alarm on control chart, i. e. , • point lays outside control limits but process is truly in-control Type II Error - b • • Fail to reject H 0 when we should Analogous to insensitivity of control chart to problems, i. e. , • point does not lay outside control limits but process is never-theless out-of-control 2/21/2021 ENGM 720: Statistical Process Control 8

Choice of Control Limits: Trade-off Between Wide or Narrow Control Limits l l Moving limits further from the center line • Decreases risk of false alarm, BUT increases risk of insensitivity Moving limits closer to the center line • Decreases risk of insensitivity, BUT increases risk of false alarm 2/21/2021 ENGM 720: Statistical Process Control 9

Consequences of Incorrect Control Limits l l Bad Thing 1: • A control chart that never finds anything wrong with the process, but the process produces bad product Bad Thing 2: • Too many false alarms destroy the operating personnel’s confidence in the control chart, and they stop using it 2/21/2021 ENGM 720: Statistical Process Control 10

Differences in Viewpoint Between Test of Hypothesis & Control Charts Hypothesis Test Checks for the validity of assumptions. (ex. : is the actual process mean what we think it is? ) Tests for sustained shift (ex. : have we actually reduced the variation like we think we have? ) 2/21/2021 Control Chart Detect departures from assumed state of statistical control Detects shifts that are -lived Detects steady drifts Detects trends ENGM 720: Statistical Process Control 11 short

Example: Part Dimension l When a process is in-control, a dimension is normally distributed with mean 30 and std dev 1. Sample size is 5. Find the control limits for an x-bar chart with a false alarm rate of 0. 0027. • r. v. x - dimension of part • r. v. x - sample mean dimension of part 2/21/2021 ENGM 720: Statistical Process Control 12

Distribution of x vs. Distribution of x 2/21/2021 ENGM 720: Statistical Process Control 13

Ex. Part Dimension Cont'd l Find UCL: l The control limits are: 2/21/2021 ENGM 720: Statistical Process Control 14

Ex. Modified Part Limits l Consider an in-control process. A process measurement has mean 30 and std dev 1 and n = 5. • Design a control chart with prob. of false alarm = 0. 005 • If the control limits are not 3 -Sigma, they are called "probability limits". 2/21/2021 ENGM 720: Statistical Process Control 15

General Model: Shewhart Control Chart l Suppose x is some quality characteristic, and w is a sample statistic of x. l Suppose mean of w is μw and std dev of w is σw, then: • l • • UCL = μ w + Lσw • LCL = μ w – Lσw CL = μ w where L is the “distance” of the control limits from the center line, and expressed in multiples (units) of the standard deviation of the statistic, i. e. σw. This type of chart is called a Shewhart Control Chart 2/21/2021 ENGM 720: Statistical Process Control 16

Rational Subgroups / Samples should be selected so that if assignable causes are present: • • Chance for differences between samples is maximized Chance for differences within a sample is minimized l Use consecutive units of production l Keep sample size small so that: • • • New events won’t occur during sampling Inspection is not too expensive But size is large enough that x is normally distributed 2/21/2021 ENGM 720: Statistical Process Control 17

Why Monitor Both Process Mean and Process Variability? Process Over Time 2/21/2021 ENGM 720: Statistical Process Control Charts X-bar R 18

Teminology l Causes of Variation: • Assignable / Special Causes l Meaning of Control: • • • Common / Chance Causes • Meets customer constraints on product • Keep the process from operating predictably Things that we can do something about In Specification • In Statistical Control • No Assignable Causes of variation present in the process • Random, inherent variation in the process 2/21/2021 ENGM 720: Statistical Process Control 19

Statistical Basis of x Chart l Suppose a quality characteristic is x ~ N( , ) and we know and l If x 1, x 2, …, xn is a random sample of size n then: and l Recall that the probability is that either: or 2/21/2021 ENGM 720: Statistical Process Control 20

Statistical Basis of x Chart Cont'd l which is equivalent to: • l Where LCL and UCL are the lower and upper control limits, respectively In practice, one must estimate and from data coming from an in-control process 2/21/2021 ENGM 720: Statistical Process Control 21

Statistics of the Range l l R – the range – is a sample statistic If x 1, x 2, …, xn is a random sample of size n from a normal distribution then one can estimate using the range: • l where d 2 is a function of n and can be found in Appendix VI Can get a better estimate for if using more than one sample • • Compute Ri for each of m samples where i = 1, …, m Then use the sample average of Ri 2/21/2021 ENGM 720: Statistical Process Control 22

Computing Trial Control Limits for x Chart l Assume a quality characteristic x ~ N( , ) l Take m 20 samples of size n = 4, 5, or 6 l For each sample i, compute x and Ri for i = 1, …, m l Compute: x and R 2/21/2021 ENGM 720: Statistical Process Control 23

Computing Trial Control Limits for x Chart l General model for x chart l Substituting estimates for μx and σ x and using 3 -sigma limits: l Where A 2 comes from Appendix VI and depends on n 2/21/2021 ENGM 720: Statistical Process Control 24

Computing Trial Control Limits for R - Chart l x and R charts come as a pair l General model for R chart l Substituting estimates for R and using 3 -sigma limits 2/21/2021 ENGM 720: Statistical Process Control 25

Computing Trial Control Limits for R - Chart (continued) l where and l D 3 and D 4 are tabulated in Appendix VI and depend on n l NOTE: R chart is quite sensitive to departures from normality 2/21/2021 ENGM 720: Statistical Process Control 26

Control Chart Factors Table (Appendix VI – see Materials Page for Engineering Notebook Copy) l For a constant sample size (n) and 3σ limits: Table factors derived from Montgomery, D. C. , (2005) Statistical Quality Control, 5 th Ed. 2/21/2021 ENGM 720: Statistical Process Control 27

Trial Control Chart Limits: Guidelines for Sampling l Sample should be of size 3 to 8 (sizes 4 – 6 are more common) l Sample must be homogeneous • same time (consecutive units) • same raw materials • same operator • same machine l Time may pass between samples but not within samples 2/21/2021 ENGM 720: Statistical Process Control 28

Steps for Trial Control Limits l l Start with 20 to 25 samples Use all data to calculate initial control limits Plot each sample in time-order on chart. Check for out of control sample points • • If one (or more) found, then: 1. 2. 3. Investigate the process; Remove the special cause; and Remove the special cause point and recalculate control limits. If can’t find special cause - drop point & recalculate anyway 2/21/2021 ENGM 720: Statistical Process Control 29

Control Chart Sensitizing Rules l l Western Electric Rules: 1. One point plots outside three-sigma limits; 2. Eight consecutive points plot on one side of the center line; 3. Two out of three consecutive points plot beyond two-sigma warning limits on the same side of the center line; or 4. Four out of five consecutive points plot beyond one-sigma warning limits on the same side of the center line. If chart shows lack of control, investigate for special cause 2/21/2021 ENGM 720: Statistical Process Control 30

Control Chart Examples UC L x x LC L Rule 1 2/21/2021 Rule 2 Rule 3 ENGM 720: Statistical Process Control Rule 4 31

Control Chart Sensitizing Rules l Additional Sensitizing Rules: One or more points very near a control limit. Six points in a row steadily increasing or decreasing. Eight points in a row on both sides of the center line, but none in-between the one-sigma warning limits on both sides of the center line. 8. Fourteen points in a row alternating above and below the center line. 9. Fifteen points in a row anywhere between the one-sigma warning limits (including either side of the center line). 10. Any unusual or non-random pattern to the plotted points. 5. 6. 7. 2/21/2021 ENGM 720: Statistical Process Control 32

Charts Based on Standard Values, x Chart l If values for and are known (i. e. , do not need to estimate from data) l Quantity A is tabulated in Appendix VI 2/21/2021 ENGM 720: Statistical Process Control 33

R - Chart Based on Standard Values l If values for R and are known l l We define a random variable W = R / – called the relative range The parameters of the distribution of W are a function of sample size (n): l From the relative range we can compute the mean of R 2/21/2021 ENGM 720: Statistical Process Control 34

The Standard Deviation of R l The standard deviation of R is given as: (Text does not derive this) l therefore l where • l D 1 and D 2 are constants tabulated in Appendix VI Caution: Be careful when using standard values • make sure these values are representative of the actual process 2/21/2021 ENGM 720: Statistical Process Control 35

X-Bar & R-Charts l The X-Bar Chart l The R-Chart checks variability in for changes in location between sample variation samples UC L x R LC L Sample Number X-Bar ( Means ) Control Chart 2/21/2021 Sample Number R - ( Range ) Control Chart ENGM 720: Statistical Process Control 36

X-Bar & Sigma-Charts t. Used when sample size is greater than 10 Control Limits: l Sigma-Chart Control • Approximate 3 limits are Limits: found from S & table • Approximate, asymmetric l X-Bar 3 limits from S & table 2/21/2021 ENGM 720: Statistical Process Control 37

X-Bar & Sigma-Charts t. Limits can also be generated from data: l historical X-Bar Control Limits: l Sigma-Chart Control • Approximate 3 limits are Limits: found from known 0 & • Approximate, asymmetric table 2/21/2021 3 limits from 0 & table ENGM 720: Statistical Process Control 38

Operating Characteristic (OC) Curve l Ability of the x and R charts to detect shifts (sensitivity) is described by OC curves l For x chart; say we know l • Mean shifts from 0 (in-control value) to 1 = 0 +k (out-of-control value) The probability of NOT detecting the shift on the first sample after shift is 2/21/2021 ENGM 720: Statistical Process Control 39

OC Curve for x Chart l Plot of b vs. shift size (in std dev units) for various sample sizes n l x chart not effective for small shift sizes, i. e. , k 1. 5 l Performance gets better for larger n and larger shifts (k) 2/21/2021 ENGM 720: Statistical Process Control 40

OC curve for R Chart l Uses distribution of relative range r. v. , i. e. , l Suppose l OC curve for R chart plots b vs. ratio of in-control to out -of-control standard deviation for various sample sizes § 0 - in-control std dev § 1 - out-of-control std dev • l That is, plot β vs. l = 1/ 0 (we won’t plot that here, but … ) R chart not very effective for detecting shifts for small sample sizes (see Fig. 5 -14 in fifth edition of the text) 2/21/2021 ENGM 720: Statistical Process Control 41

Probability of Detecting Shift for Subsequent Samples l After the shift has occurred: • P(NOT detecting shift ON 1 st sample) • P(DETECTING shift ON 2 nd sample) • P(DETECTING shift ON rth sample) • P(DETECTING shift BY 2 nd sample) • P(DETECTING shift BY rth sample) 2/21/2021 ENGM 720: Statistical Process Control 42

Average Run Length (ARL) l Expected number of samples taken before shift is detected is called the Average Run Length (ARL) 2/21/2021 ENGM 720: Statistical Process Control 43

Performance of Any Shewhart Control Chart l l In-Control ARL: • Average number of points plotted on control chart before a false alarm occurs (ideally, should be large) Out-of-Control ARL: • Average number of points, after the process goes outof-control, before the control chart detects it (ideally, should be small) 2/21/2021 ENGM 720: Statistical Process Control 44

ARL Curve for x Chart l Plot of ARL 1 vs. shift size (in sd units) for various sample sizes n: l Average Time to Signal, (ATS): • Number of time periods that occur until signal is generated on control chart • h - time interval between samples 2/21/2021 ENGM 720: Statistical Process Control 45

Questions & Issues 2/21/2021 ENGM 720: Statistical Process Control 46