١ Topics: Sources of Variation

Home , Bar chart

Islamic University, Gaza - Palestine Topics:

• Control Charts for Variables

• Some Special Difficulties of Estimating Indirect Costs in Economy Studies

• Did it Pay to Use Statistical Quality Control?

• Costs and Product Liability

• SQC Techniques in Relation to certain Economic Aspects of Product liability

• Control Charts for Variables

Islamic University, Gaza - Palestine Sources of Variation

• Variation exists in all processes. • Variation can be categorized as either; – Common or Random causes of variation, or • Random causes that we cannot identify • Unavoidable • e.g. slight differences in process variables like diameter, weight, service time, temperature Assignable causes of variation • Causes can be identified and eliminated • e.g. poor employee training, worn tool, machine needing repair

١ Islamic University, Gaza - Palestine Control charts identify variation

• Chance causes - “common cause” – inherent to the process or random and not controllable – if only common cause present, the process is considered stable or “in control” • Assignable causes - “special cause” – variation due to outside influences – if present, the process is “out of control”

Islamic University, Gaza - Palestine Control Charts for x and R

Notation and values

• Ri = range of the values in the ith sample

Ri = xmax - xmin • R = average range for all m samples •  is the true process mean •  is the true process standard deviation

٢ Islamic University, Gaza - Palestine Control Charts for Xbar and R

Statistical Basis of the Charts

• Assume the quality characteristic of interest is normally distributed with mean , and standard deviation, .

• If x1, x2, …, xn is a sample of size n, then he average of this sample is x x  x   x x  1 2  n • n • is normally distributed with mean, , and standard deviation,

x / n

Islamic University, Gaza - Palestine Control Charts for Xbar and R

Statistical Basis of the Charts

• The probability is 1 -  that any sample mean will fall between    Z     Z  2/ x  2/ n and    Z     Z  2/ x  2/ n • The above can be used as upper and lower control limits on a control chart for sample means, if the process parameters are known.

٣ Islamic University, Gaza - Palestine Control Charts for Xbar and R

Control Limits for the xbar chart

UCL  x  A 2R Center Line  x

LCL  x  A 2R

• A2 is found in Appendix VI for various values of n.

Islamic University, Gaza - Palestine Control Charts for Xbar and R

Control Limits for the R chart

UCL  D4 R Center Line  R

LCL  DR3

• D3 and D4 are found in Appendix VI for various values of n.

٤ Islamic University, Gaza - Palestine Control Charts for Xbar and R

Estimating the Process Standard Deviation

• The process standard deviation can be estimated using a function of the sample average range. R   d2

• This is an unbiased estimator of 

Islamic University, Gaza - Palestine Control Charts for Variables

• To create the centerline of the chart:

 Xi X  i  1 m

٥ Islamic University, Gaza - Palestine Control Charts for Variables

• To find the average range:

 R i R  i  1 m

Islamic University, Gaza - Palestine Control Charts for Variables

• Control limits for the R chart:

UCLR  D4 R

LCLR  D3R

٦ Islamic University, Gaza - Palestine Control Charts for Variables

• Centerline and Control limits for X-bar and S chart (if n  10):

m m Xi  si i  1 X  s  i  1 m m UCL  B s UCLX  X  A3 s s 4

LCLs  B3 s LCLX  X  A3 s

Islamic University, Gaza - Palestine Distribution of Data

• Normal distributions • Skewed distribution

٧ Islamic University, Gaza - Palestine Control Charts for Variables

Trial Control Limits

• The control limits should be treated as trial control limits. • If this process is in control for the m samples collected, then the system was in control in the past. • If all points plot inside the control limits and no systematic behavior is identified, then the process was in control in the past, and the trial control limits are suitable for controlling current or future production.

Islamic University, Gaza - Palestine Control Charts for Variables

Trial control limits and the out-of-control process

• If points plot out of control, then the control limits must be revised. • Before revising, identify out of control points and look for assignable causes. – If assignable causes can be found, then discard the point(s) and recalculate the control limits. – If no assignable causes can be found then 1) either discard the point(s) as if an assignable cause had been found or 2) retain the point(s) considering the trial control limits as appropriate for current control.

٨ Islamic University, Gaza - Palestine Control Charts for Variables

Control Limits, Specification Limits, and Natural Tolerance Limits • Control limits are functions of the natural variability of the process • Natural tolerance limits represent the natural variability of the process (usually set at 3-sigma from the mean) • Specification limits are determined by developers/designers.

Islamic University, Gaza - Palestine Control Charts for Variables

Control Limits, Specification Limits, and Natural Tolerance Limits

• There is no mathematical relationship between control limits and specification limits. • Do not plot specification limits on the charts – Causes confusion between control and capability – If individual observations are plotted, then specification limits may be plotted on the chart.

٩ Islamic University, Gaza - Palestine Rational subgroup from homogeneous lot

- Rational subgroup from homogeneous lot : same machine, same operator - Decisions on size of sample empirical judgment + relates to costs • choose n = 4 or 5  use R-chart • when n  10  use s-chart - frequency of taking subgroups often enough to detect process changes - Guideline of sample sizes/frequency using • Say, 4000 parts/day  75 samples • if n = 4  19 subgroups • or n = 5 15 subgroups

Islamic University, Gaza - Palestine Control Charts for Variables

Rational Subgroups

• X bar chart monitors the between sample variability • R chart monitors the within sample variability.

١٠ Islamic University, Gaza - Palestine Control Charts for Variables

Guidelines for the Design of the Control Chart • Specify sample size, control limit width, and frequency of sampling • if the main purpose of the x-bar chart is to detect moderate to large process shifts, then small sample sizes are sufficient (n = 4, 5, or 6)

• if the main purpose of the x-bar chart is to detect small process shifts, larger sample sizes are needed (as much as 15 to 25)…which is often impractical…alternative types of control charts are available for this situation…

Islamic University, Gaza - Palestine Sample Std. Deviation Chart (x - s control chart )

• Both R and s measure dispersion of data

• R chart - simple, only use XH (highest) and XL(lowest) • s chart - more calculation - use ALL UCL  X  A s xi’s  more accurate, need x 3 LCL  X  A s calculate sub-group sample standard x 3 deviation UCLs  B4s • When n < 10 R chart  LCLs  B3s s chart so s • n  10 - s chart better , R not o   C4 C4 accurate any more

١١ Islamic University, Gaza - Palestine Control Charts for Variables

Guidelines for the Design of the Control Chart

• If increasing the sample size is not an option, then sensitizing procedures (such as warning limits) can be used to detect small shifts…but this can result in increased false alarms.

• R chart is insensitive to shifts in process standard deviation.(the range method becomes less effective as the sample size increases) may want to use S or S2 chart.

• The OC curve can be helpful in determining an appropriate sample size.

Islamic University, Gaza - Palestine Control Charts for Variables

Guidelines for the Design of the Control Chart Allocating Sampling Effort

• Choose a larger sample size and sample less frequently? or, Choose a smaller sample size and sample more frequently?

• The method to use will depend on the situation. In general, small frequent samples are more desirable.

١٢ Islamic University, Gaza - Palestine Statistical Control

• Statistical Process Control: Statistical evaluation of the output of a process during production

• Quality of Conformance: A product or service conforms to specifications

Islamic University, Gaza - Palestine Control Chart

• Control Chart – Purpose: to monitor process output to see if it is random – A time ordered plot representative sample statistics obtained from an on going process (e.g. sample means) – Upper and lower control limits define the range of acceptable variation

١٣ Islamic University, Gaza - Palestine Statistical Control

Abnormal variation Out of due to assignable sources control UCL

Mean Normal variation due to chance LCL Abnormal variation due to assignable sources

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Sample number

Islamic University, Gaza - Palestine Statistical Process Control

• The essence of statistical process control is to assure that the output of a process is random so that future output will be random.

١٤ Islamic University, Gaza - Palestine Statistical Process Control

• The Control Process – Define – Measure – Compare – Evaluate – Correct – Monitor results

Islamic University, Gaza - Palestine Statistical Control

• Variations and Control – Random variation: Natural variations in the output of a process, created by countless minor factors – Assignable variation: A variation whose source can be identified

١٥ Islamic University, Gaza - Palestine

Sampling Distribution

Sampling distribution

Process distribution

Mean

Islamic University, Gaza - Palestine

Normal Distribution

Standard deviation

    Mean 95.44%

99.74%

١٦ Islamic University, Gaza - Palestine

Control Limits

Sampling distribution

Process distribution

Mean

Lower Upper control control limit limit

Islamic University, Gaza - Palestine SPC Errors

• Type I error – Concluding a process is not in control when it is actually in Control. (Good Items rejected)

• Type II error – Concluding a process is in control when it is not.

١٧ Islamic University, Gaza - Palestine SPC Errors: Type I and Type II Error

In control Out of control

In control No Error Type I error (producers risk)

Out of control Type II Error No error (consumers risk)

Islamic University, Gaza - Palestine

Type I Error

/2 /2

Mean

Probability LCL UCL of Type I error

١٨ Islamic University, Gaza - Palestine

Observations from Sample Distribution

UCL

LCL

1 2 3 4 Sample number

Islamic University, Gaza - Palestine Mean and Range Charts

(process mean is shifting upward) Sampling Distribution

UCL

x-Chart Detects shift

LCL

UCL Does not R-chart detect shift LCL

١٩ Islamic University, Gaza - Palestine

Mean and Range Charts

Sampling Distribution (process variability is increasing)

UCL

Does not x-Chart reveal increase LCL

UCL

R-chart Reveals increase LCL

Islamic University, Gaza - Palestine Run Tests

Run test – a test for randomness • Any sort of pattern in the data would suggest a nonrandom process • All points are within the control limits - the process may not be random

٢٠ Islamic University, Gaza - Palestine Nonrandom Patterns in Control charts

• Trend • Cycles • Bias • Mean shift • Too much dispersion

Islamic University, Gaza - Palestine Counting Runs

Counting Above/Below Median Runs (7 runs)

B A A B A B B B A A B

Counting Up/Down Runs (8 runs)

U U D U D U D U U D

٢١ Islamic University, Gaza - Palestine NonRandom Variation

• Managers should have response plans to investigate cause • May be false alarm (Type I error) • May be assignable variation

Islamic University, Gaza - Palestine Micro Card Company

Micro Card Company (MCC) produces collectible sports cards of college and professional athletes. MCCs card-front design uses a picture of the athlete, however, the process used to center an athlete’s picture does not function perfectly. Five cards are randomly selected from each 1000 cards produced and measured to determine the degree of off-centeredness of each card’s picture. The measurement taken represents percentage of total margin (.25”) that is on the left edge of a card. Data from 30 consecutive samples is included with your materials, and summarized on the following slides.

٢٢ Islamic University, Gaza - Palestine Micro Card Company Sample X-bar R Sample X-bar R Sample X-bar R 1 55.6 22 11 51.2 15 21 50.0 11 2 61.0 23 12 49.4 14 22 47.0 14 3 45.2 20 13 44.0 32 23 50.6 15 4 46.2 11 14 51.6 14 24 48.8 16 5 46.8 18 15 53.2 12 25 44.6 22

6 49.8 23 16 52.4 23 26 46.8 16 7 46.8 18 17 50.6 8 27 49.2 8 8 44.2 20 18 56.0 18 28 45.6 19 9 50.8 32 19 50.2 19 29 57.6 40 10 48.4 16 20 44.0 23 30 51.4 17

Islamic University, Gaza - Palestine Micro Card Company Summary Calculation

n = 5 A3 = 1.427

X = 49.63 B3 = NA

S = 7.42 B4 = 2.089

R = 18.63 D3 = NA

d2 = 2.326 D4 = 2.115

A2 = 0.577  = R/d2 = 8.01

٢٣ Islamic University, Gaza - Palestine

Micro Card Company: X-bar and R Control Chart Limits

X based on R R UCL 60.38 39.40 U2SW 56.80 32.48 L U1SL 53.22 25.55 CL 49.63 18.63 L1SL 46.05 11.71 L2SWL 42.47 4.79 LCL 38.89 ------

Islamic University, Gaza - Palestine Micro Card Company

X Bar ChartLimitsfor Sp Basedorts C ona rRds CenteringValues

1 60 3.0SL=60.38

2.0SL=56.80 n a

e 1.0SL=53.22 M

e 50 l X=49.63 p m

a -1.0SL=46.05 S -2.0SL=42.47 40 -3.0SL=38.89

0 10 20 30 Sample Number Samples of 5 from each 1000 Cards Printed

٢٤ Islamic University, Gaza - Palestine Micro Card Company

R Chart for Sports Card Centering

40 3.0SL=39.40

2.0SL=32.48

e 30 g

n 1.0SL=25.55 a R

e 20 l R=18.63 p m a -1.0SL=11.71 S 10 -2.0SL=4.791 0 -3.0SL=0.000

0 10 20 30 Sample Number Samples of 5 Cards from each 1000 Produced

Islamic University, Gaza - Palestine

Micro Card Company: X-bar & R Chart Interpretation

• Application of all eight PATs to the X-bar chart indicated a violation of PAT 1 (one point plotting above the UCL) at sample 2. Apparently, a successful process adjustment was made, as suggested by examination of the remainder of the chart.

• Application of PATs one through four to the R chart indicated a violation of PAT 1 at sample 29. Measures would be investigated to reduce process variation at that point. The violation was a “close call” and was out of character with the remainder of the data.

• We are close to being able to apply PDCA to the process for the purpose of achieving lasting process improvements.

٢٥ Islamic University, Gaza - Palestine

Coordinates for the X bar Control Chart: “S”

CL= X UCL= X + A3S LCL= X- A3S

U2SWL= X+ 2A3S/3 L2SWL= X- 2A3S/3 U1SL= X+ A3S/3 L1SL= X- A3S/3 where A3 depends only on n

Islamic University, Gaza - Palestine Coordinates on an S Control Chart

• CL= S

• UCL= B4S

• LCL= B3S

• U2SWL= S+ 2(B4-1)S/3

• L2SWL= S- 2(B4-1)S/3

• U1SL= S+ (B4-1)S/3

• L1SL= S- (B4-1)S/3

• where B3 and B4 depend only on n

٢٦ Islamic University, Gaza - Palestine Micro Card Company Sample X-bar S Sample X-bar S Sample X-bar S 1 55.6 9.63 11 51.2 6.83 21 50.0 5.15 2 61.0 8.63 12 49.4 5.46 22 47.0 5.15 3 45.2 7.40 13 44.0 14.35 23 50.6 5.55 4 46.2 4.09 14 51.6 5.18 24 48.8 6.50 5 46.8 7.22 15 53.2 5.36 25 44.6 8.96

6 49.8 8.76 16 52.4 9.48 26 46.8 6.50 7 46.8 6.72 17 50.6 3.44 27 49.2 3.19 8 44.2 8.53 18 56.0 7.00 28 45.6 7.96 9 50.8 11.95 19 50.2 7.60 29 57.6 14.38 10 48.4 6.19 20 44.0 8.46 30 51.4 6.80

Islamic University, Gaza - Palestine Micro Card Company X-bar and S Chart Limits

X based on S S UCL 60.22 15.49 U2SWL 56.69 12.80 U1SL 53.16 10.11 CL 49.63 7.42 L1SL 46.11 4.72 L2SWL 42.58 2.03 LCL 39.05 ------

٢٧ Islamic University, Gaza - Palestine Micro Card Company

X Bar Chart Limitsfor Sp Basedorts C onar dSs CenteringValues

1 60 3.0SL=60.22

2.0SL=56.69 n a

e 1.0SL=53.16 M e

l 50 X=49.63 p m

a -1.0SL=46.11 S -2.0SL=42.58 40 -3.0SL=39.05

0 10 20 30 Sample Number Samples of 5 from each 1000 Cards Printed

Islamic University, Gaza - Palestine Micro Card Company

S Chart for Sports Card Centering Values

15 3.0SL=15.49 2.0SL=12.80 v e

d 10 1.0SL=10.11 t S e l S=7.416 p m a 5 -1.0SL=4.724 S

-2.0SL=2.032 0 -3.0SL=0.000

0 10 20 30 Sample Number 5 Cards Sampled from each 1000 Cards Produced

٢٨ Islamic University, Gaza - Palestine

Micro Card Company X-bar & S Chart Interpretation

• Application of all eight PATs to the X-bar chart indicates a violation of PAT 1 (one pt. above the UCL) at sample 2. Judging from the remainder of the chart, the process was successfully adjusted. • Application of the first four PATs to the S chart indicates no violations. • In summary, the process appears to have been temporarily “out-of-control” w.r.t. its mean at sample 2. The process was successfully adjusted and may now be subjected to PDCA for permanent improvement purposes.

Islamic University, Gaza - Palestine

Another Example of X-bar & R chart

٢٩ Islamic University, Gaza - Palestine Example: Control Charts for Variable Data Slip Ring Diameter (cm) Sample 1 2 3 4 5 X R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15

Islamic University, Gaza - Palestine Calculation

From Table above: • Sigma X-bar = 50.09 • Sigma R = 1.15 • m = 10 Thus; • X-Double bar = 50.09/10 = 5.009 cm • R-bar = 1.15/10 = 0.115 cm

Note: The control limits are only preliminary with 10 samples. It is desirable to have at least 25 samples.

٣٠ Islamic University, Gaza - Palestine Trial control limit

• UCLx-bar = X-D bar + A2 R-bar = 5.009 + (0.577)(0.115) = 5.075 cm

• LCLx-bar = X-D bar - A2 R-bar = 5.009 - (0.577)(0.115) = 4.943 cm

• UCLR = D4R-bar = (2.114)(0.115) = 0.243 cm

• LCLR = D3R-bar = (0)(0.115) = 0 cm For A , D , D : see Table B, Appendix 2 3 4 n = 5

Islamic University, Gaza - Palestine 3-Sigma Control Chart Factors

Sample size X-chart R-chart

n A2 D3 D4 2 1.88 0 3.27 3 1.02 0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48 0 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86

٣١ Islamic University, Gaza - Palestine X-bar Chart

5.10 UCL 5.08 5.06 5.04 r a

b 5.02

X 5.00 CL 4.98

4.96 LCL 4.94 0 1 2 3 4 5 6 7 8 9 10 11 Subgroup

Islamic University, Gaza - Palestine R Chart

0.25 UCL

0.20

e 0.15 g CL n a

R 0.10

0.05 LCL 0.00 0 1 2 3 4 5 6 7 8 9 10 11 Subgroup

٣٢ Islamic University, Gaza - Palestine 6.70 Run Chart 6.65 6.60 r a 6.55 b - X

, 6.50 n a

e 6.45 M 6.40 6.35 6.30 0 5 10 15 20 25 Subgroup number 0.35 0.3 0.25 R

, 0.2 e g n

a 0.15 R 0.1 0.05 0 0 5 10 15 20 25 Subgroup number

Islamic University, Gaza - Palestine

Common Questions for Investigating an out-of-Control Process

• Are there differences in the measurement accuracy of instruments / methods used? • Are there differences in the methods used by different personnel? • Is the process affected by the environment, e.g. temperature/humidity? • Has there been a significant change in the environment? • Is the process affected by predictable conditions such as tool wear? • Were any untrained personnel involved in the process at the time? • Has there been a change in the source for input to the process such as a new supplier or information? • Is the process affected by employee fatigue?

٣٣ Islamic University, Gaza - Palestine

Common Questions for Investigating an Out-of-Control Process

• Has there been a change in policies or procedures such as maintenance procedures? • Is the process frequently adjusted? • Did the samples come from different parts of the process? Shifts? Individuals? • Are employees afraid to report “bad news”?

Islamic University, Gaza - Palestine Control Charts for Variables

• Control limits are established at standard deviations from the centerline for the process using the following formulas:

UCLX  X  3 x

LCLX  X  3 x

٣٤