Statistics for Business and Economics Chapter 12 Methods for Quality Improvement

Learning Objectives Define Quality Describe Types of Variation Explain Control Charts x–Chart R–Chart p–Chart As a result of this class, you will be able to ...

What is Quality? Performance Features Reliability Primary operating characteristics of the product Features "bells and whistles" Reliability Probability product will function for a specified amount of time As a result of this class, you will be able to ...

What is Quality? Conformance Durability Serviceability Extent to which product meets preestablished standards Durability The life of the product Serviceability Ease and speed of repair As a result of this class, you will be able to ...

What is Quality? Aesthetics Other Perceptions Way product looks, feels, etc. Other Perceptions e.g. Company reputation As a result of this class, you will be able to ...

Process Process Operations Inputs Information Methods Energy Materials Machines People Outputs Finished Products Variability is present in the output of all processes

System Processes System Supplier Customer Feedback

Sources of Variation People Machines Materials Methods Measurement Environment As a result of this class, you will be able to ...

Types of Variation Common causes Special causes Methods, materials, people, environment Special causes Single worker, bad batch of material Only detectable when process is in–control (stable)

Time Series Plot (Run Chart) Graphically shows trends and changes in the data over time Time recorded on the horizontal axis Measurements recorded on the vertical axis Points connected by straight lines

Time Series Pattern: Random Behavior centerline Measurement Order of production

Time Series Pattern: Shift centerline Measurement Order of production

Time Series Pattern: Increased Variance centerline Measurement Order of production

Control Chart Uses Monitor process variation Differentiate between variation due to common causes v. special causes Evaluate past performance Monitor current performance

Sample Control Chart Upper control limit centerline Measurement Lower control limit Order of production

Control Limits 3–sigma limits Upper control limit: μ + 3σ Lower control limit: μ – 3σ .00135 .00135 Order of production

4 Possible Outcomes H0: Process is in control Ha: Process is out of control Reality Conclude process is in control H0 True Ha True Use 3–sigma limits because of small P(Type I Error)="False Alarm" Correct decision Type II Error H0 True Conclusion Type I Error Correct decision Ha True Conclude process is out of control

Types of Control Charts Type of Data Quantitative Data Qualitative Data R–chart x–chart p–chart 3

The x–Chart

Types of Control Charts Type of Data Quantitative Data Qualitative Data R–chart x–chart p–chart 3

x–Chart Monitors changes in the mean of samples Horizontal axis: Sample number Vertical axis: Mean of sample Control limits based on sampling distribution of x Standard deviation of x:

Sample x–Chart Upper control limit centerline x Lower control limit Sample Number

Determining the Centerline k = number of samples of size n (usually between 2 and 10) xi = sample mean of the ith sample

Estimating σ Determine the Range of each sample Range = Maximum – Minimum Determine the Average Range of the k samples Divide R by the constant d2 (based on sample size)

Determining the Control Limits where

x–Chart Summary Collect at least 20 samples of size 2 to 10 Calculate mean and range of each sample Determine the centerline and control limits where

x–Chart Example Samples from a machine filling 12oz soda cans

x–Chart Centerline Solution

x–Chart Control Limits Solution

x–Chart Control Limits Solution

x–Chart Solution UCL = 12.61 LCL = 11.38

Interpreting Control Charts Six zones Each zone is one standard deviation wide Upper A–B boundary UCL Zone A Upper B–C boundary Zone B Zone C centerline Lower B–C boundary Zone C Zone B Lower A–B boundary Zone A LCL Order of production

Zone Boundaries 3–sigma control limit zone boundaries: Upper A–B Boundary: Lower A–B Boundary: Upper B–C Boundary: Lower B–C Boundary:

Zone Boundaries Example Samples from a machine filling 12oz soda cans

Zone A–B Boundaries Solution Recall Upper A–B: Lower A–B:

Zone B–C Boundaries Solution Recall Upper B–C: Lower B–C:

x–Chart Solution UCL = 12.61 12.4 12.2 11.8 11.6 LCL = 11.38 A B C C B

Using JMP for Control Charts Graph >> Control Chart >> Xbar But for JMP…Data must be in vertical format…so we need to first stack the weight columns into a single column

Using JMP for Control Charts Tables >> Stack (now specify the columns Weight1 through Weight6 in the Stack Columns box) Result

Using JMP for Control Charts Now do chart (Graph >> Control Chart >> Xbar) Result

Pattern Analysis Rules Rule 1: One point beyond Zone A Either lower or upper half of control chart centerline LCL UCL A B C

Pattern Analysis Rules Rule 2: Nine points in a row in Zone C or beyond Either lower or upper half of control chart centerline LCL UCL A B C

Pattern Analysis Rules Rule 3: Six points in a row steadily increasing or decreasing centerline LCL UCL A B C

Pattern Analysis Rules Rule 4: Fourteen points in a row alternating up and down centerline LCL UCL A B C

Pattern Analysis Rules Rule 5: Two out of three points in Zone A or beyond Either lower or upper half of control chart centerline LCL UCL A B C

Pattern Analysis Rules Rule 6: Four out of five points in Zone B or beyond Either lower or upper half of control chart centerline LCL UCL A B C

Interpreting an x–Chart Process is considered out of control if any of the pattern analysis rules are detected Process is considered in control if none of the pattern analysis rules are detected

Interpreting x–Chart Example What does the chart suggest about the stability of the process? UCL = 12.61 A 12.4 B 12.2 C C 11.8 B 11.6 A LCL = 11.38

Interpreting x–Chart Solution Since none of the six pattern analysis rules are observed, the process is considered in control

Interpreting x–Chart Thinking Challenge Ten additional samples of size 5 are taken. What does the chart suggest about the stability of the process? UCL = 12.61 A 12.4 B 12.2 C C 11.8 B 11.6 A LCL = 11.38

Interpreting x–Chart Solution* Rule 5 and Rule 6 are violated. Process is out of control UCL = 12.61 A 12.4 B 12.2 C C 11.8 B 11.6 A LCL = 11.38

Using JMP for Pattern Tests JMP gives 8 tests---these six and two more. To get them, use the red triangle pull down menu and choose “Tests”

R–Chart

Types of Control Charts Type of Data Quantitative Data Qualitative Data x–chart R–chart p–chart 3

R–Chart Monitors changes in process variation Horizontal axis: Sample number Vertical axis: Sample ranges Control limits based on sampling distribution of R Mean of sampling distribution of R: μR Standard deviation of sampling distribution of R: σR

Estimating μR and σR Estimate of μR: k = number of samples of size n ≥ 2 Ri = sample range of the ith sample Estimate of σR:

Determining the Control Limits Note: If n ≤ 6, the LCL will be negative. Since the range can’t be negative the LCL is meaningless.

R–Chart Summary Collect at least 20 samples of size n ≥ 2 Calculate the range of each sample Determine the centerline and control limits where

Zone Boundaries Upper A–B Boundary: Lower A–B Boundary: Upper B–C Boundary: Lower B–C Boundary:

Interpreting an R–Chart Process is considered out of control if any of the pattern analysis rules 1 – 4 are detected: One point beyond Zone A Nine points in a row in Zone C or beyond Six points in a row steadily increasing or decreasing Fourteen points in a row alternating up and down Process is considered in control if none of the pattern analysis rules are detected

R–Chart Example Samples from a machine filling 12oz soda cans

R–Chart Solution Calculate the mean of the ranges:

R–Chart Solution Calculate the control limits. n = 5 D4 = 2.114 D3 = 0 (LCL will be zero)

R–Chart Solution Determine the A–B zone boundaries Upper A–B Boundary: Lower A–B Boundary:

R–Chart Solution Determine the B–C zone boundaries Upper B–C Boundary: Lower B–C Boundary:

R–Chart Solution The variation of the process is in control UCL = 2.3 1.9 B 1.5 C C .7 B .3 A LCL = 0 The variation of the process is in control

p–Chart

Types of Control Charts Type of Data Quantitative Data Qualitative Data R–chart x–chart p–chart 3

p–Chart Used for qualitative data Monitors variation in the process proportion Horizontal axis: Sample number Vertical axis: Sample proportion Control limits based on sampling distribution of p Mean of sampling distribution of p: μp Standard deviation of sampling distribution of p: σp ^ ^ ^ ^ ^

Estimating μp and σp ^ ^ Estimate of μp: Total number of defective units in all k samples Total number of units sampled p = Estimate of σp: ^

Determining the Control Limits Note: If the LCL is negative do not plot it on the control chart.

p–Chart Summary Collect at least 20 samples of size Calculate the proportion of defective units in each sample Determine the centerline and control limits p0 is an estimate of p Number of defective units in the sample Number of units in the sample p = ^

Zone Boundaries Upper A–B Boundary: Lower A–B Boundary: Upper B–C Boundary: Lower B–C Boundary:

Interpreting a p–Chart Process is considered out of control if any of the pattern analysis rules 1 – 4 are detected: One point beyond Zone A Nine points in a row in Zone C or beyond Six points in a row steadily increasing or decreasing Fourteen points in a row alternating up and down Process is considered in control if none of the pattern analysis rules are detected

p–Chart Example A manufacturer of pencils knows about 4% of pencils produced fail to meet specifications. How many pencils should be sampled for monitoring the process proportion?    Solution: Samples of size 216 or more should be selected.

p–Chart Example The pencil manufacturer has decided to select samples of size n = 225. The table shows the results for the past 20 samples. Construct a p–chart.   

p–Chart Solution Calculate the centerline:

p–Chart Solution Calculate the control limits: *Since LCL is negative, do not plot it on the control chart

p–Chart Solution Determine the A–B zone boundaries Upper A–B Boundary: .06407 Lower A–B Boundary: .01281

p–Chart Solution Determine the B–C zone boundaries Upper B–C Boundary: .05126 Lower B–C Boundary: .02562

p–Chart Solution The process is in control UCL = .07689 .06407 .05126 B .05126 C C .02562 B .01281 A The process is in control

Graph >> Control Charts >> P JMP p–Chart Solution Graph >> Control Charts >> P

Conclusion Defined Quality Described Types of Variation Explained Control Charts x–Chart R–Chart p–Chart As a result of this class, you will be able to ...