Statistical Quality Control

STATISTICAL QUALITY CONTROL

HISTOGRAM OR FREQUENCY DISTRIBUTION • Plot frequency (or relative frequency) versus the values of the variable • Shape • Location or central tendency • Scatter or spread

Histogram for discrete data

Numerical Summary of Data

The Box Plot

The Binomial Distribution Basis is in Bernoulli trials

The Poisson Distribution

IMPORTANT CONTINUOUS DISTRIBUTIONS The Normal Distribution

The Exponential Distribution

The Gamma Distribution

• When r is an integer, the gamma distribution is the result of summing r independently and identically exponential random variables each with parameter λ

• The gamma distribution has many applications in reliability engineering; see Example 2-121, text page 71

QUALITY

Definitions and Meaning of Quality The Eight Dimensions of Quality 2. Performance 3. Reliability 4. Durability 5. Serviceability 6. Aesthetics 7. Features 8. Perceived Quality 9. Conformance to Standards

STATISTICAL PROCESS CONTROL

Basic SPC Tools

Check Sheet

Pareto Chart

Cause-and-Effect Diagram

Defect Concentration Diagram

Scatter Diagram

CHANCE AND ASSIGNABLE CAUSES OF QUALITY VARIATION A process is operating with only chance causes of variation present is said to be in statistical control. A process that is operating in the presence of assignable causes is said to be out of control

STATICAL BASIS OF CONTROL CHART A control chart contains – A center line – An upper control limit – A lower control limit

Shewhart Control Chart Model

Reasons for Popularity of Control Charts • • • • •

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.

CHOICE OF CONTROL LIMIT • •

•

3-Sigma Control Limits – Probability of type I error is 0.0027 Probability Limits – Type I error probability is chosen directly – For example, 0.001 gives 3.09-sigma control limits Warning Limits – Typically selected as 2-sigma limits

More Basic Principles • Charts may be used to estimate process parameters, which are used to determine capability

• Two general types of control charts – Variables • Continuous scale of measurement • Quality characteristic described by central tendency and a measure of variability • Counts – Attributes • Conforming/nonconforming

CONTROL CHARTS FOR ATTRIBUTE • P-chart Control chart for Fraction nonconforming • C-chart Control chart for Defects • U-chart Control chart for the average number of nonconformities

CONTROL CHARTS FOR ATTRIBUTE • Fraction nonconforming is the ratio of the number of nonconforming items in a population to the total number of items in that population. • Control charts for fraction nonconforming are based on the binomial distribution

Control charts for fraction nonconforming Recall: A quality characteristic follows a binomial distribution if: 1. All trials are independent. 2. Each outcome is either a “success” or “failure”. 3. The probability of success on any trial is given as p. The probability of a failure is 1-p. 4. The probability of a success is constant.

Control charts for fraction nonconforming • The binomial distribution with parameters n ≠ 0 and 0 < p < 1, is given by n  x n −x  p( x ) = p ( 1 − p ) x   

• The mean and variance of the binomial distribution are

µ = np

σ = np(1 − p) 2

Control charts for fraction nonconforming Development of the Fraction Nonconforming Control Chart Assume • n = number of units of product selected at random. • D = number of nonconforming units from the sample • p= probability of selecting a nonconforming unit from the sample.

D ˆ = p n

Control charts for fraction nonconforming Development of the Fraction Nonconforming Control Chart n x P(D = x ) =  p (1 − p) n − x x where

is a random variable with mean and variance

µ=p

p(1 − p) σ = n 2

Control charts for fraction nonconforming Standard Given • If a standard value of p is given, then the control limits for the fraction nonconforming are p(1 − p ) UCL = p + 3 n CL = p p(1 − p ) LCL = p − 3 n

Control charts for fraction nonconforming No Standard Given • If no standard value of p is given, then the control limits for the fraction nonconforming are UCL =p +3

p (1 −p ) n

CL =p LCL =p −3 m

∑Di

p (1 −p ) n m

∑ pˆ i

p = i =1 = i =1 mn m

Control charts for fraction nonconforming • The np control chart • The actual number of nonconforming can also be charted. Let n = sample size, p = proportion of nonconforming. The control limits are:

UCL = np + 3 np(1 − p) CL = np LCL = np − 3 np(1 − p) •

(if a standard, p, is not given, use

)

Operating-characteristic function (OC) and Average RUN length (ARL) The oc function • The number of nonconforming units, D, follows a binomial distribution. Let p be a standard value for the fraction nonconforming. The probability of committing a Type II error is D ˆ = p n

β = P(pˆ < UCL | p) − P(pˆ ≤ LCL | p) = P(D < nUCL | p) − P(D ≤ nLCL | p)

OC

B

1.0

0.5

0.0 0.0

0.1

0.2

0.3

p

0.4

0.5

0.6

Average RUN length (ARL) ARL •

•

The average run lengths for fraction nonconforming control charts can be found as before: ARL =

1 p (out-of-control)

•

The in-control ARL is

•

The out-of-control ARL is

1 ARL 0 = α 1 ARL1 = 1 −β

Control charts for defects • There are many instances where an item will contain nonconformities but the item itself is not classified as nonconforming.

• Poisson Distribution −c

e c p( x ) = x!

x

C-chart • Standard Given: UCL = c + 3 c CL = c LCL = c − 3 c

No Standard Given

UCL = c + 3 c CL = c LCL = c − 3 c

Control chart for the average number of nonconformities U-chart • If we find c total nonconformities in a sample of n inspection units, then the average number of nonconformities per inspection unit is u = c / n. • The control limits for the average number of nonconformities is u UCL = u + 3 n CL = u u LCL = u −3 n

Demerit Systems Demerit Schemes 3. 4. 5. 6.

Class A Defects - very serious Class B Defects - serious Class C Defects - Moderately serious Class D Defects - Minor

•

Let ciA, ciB, ciC, and ciD represent the number of units in each of the four classes.

• Demerit Schemes The following weights are fairly popular in practice: – Class A-100, Class B - 50, Class C – 10, Class D - 1 di = 100ciA + 50ciB + 10ciC + ciD di - the number of demerits in an inspection unit

CONTROL CHARTS FOR VARIABLES

CONTROL CHARTS FOR VARIABLES • Variable - a single quality characteristic that can be measured on a numerical scale. • When working with variables, we should monitor both the mean value of the characteristic and the variability associated with the characteristic.

Control Charts for X and R • X bar chart monitors the between sample variability • R chart monitors the within sample variability.

x1 + x2 +  + xn x= n

σx =σ / n Ri = range of the values in the ith sample Ri = xmax - xmin R = average range for all m samples

Control Charts for X and R • Control Charts for

x and R

UCL = x + A 2 R Center Line = x LCL = x −A 2 R A2 is found in Appendix VI for various values of n.

Control Charts for X and R Control Limits for the R chart

UCL = D4 R Center Line = R LCL = D3 R • D3 and D4 are found in Appendix VI for various values of n.

Estimating Process Capability • The x-bar and R charts give information about the capability of the process relative to its specification limits. • Assumes a stable process. • Assume the process is normally distributed, and x is normally distributed, the fraction nonconforming can be found by solving:

P(x < LSL) + P(x > USL)

Process-Capability Ratios (Cp) • If Cp > 1, then a low # of nonconforming items will be produced. • If Cp = 1, (assume norm. dist) then we are producing about 0.27% nonconforming. • If Cp < 1, then a large number of nonconforming items are being produced.

USL − LSL Cp = 6σ

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

• There is no mathematical relationship between control limits and specification limits.

Interpretation of X and R charts • Patterns of the plotted points will provide useful diagnostic information on the process, and this information can be used to make process modifications that reduce variability. – – – – –

Cyclic Patterns Mixture Shift in process level Trend Stratification

Control chart for X and S If a standard σ is given the control limits for the S chart are: UCL = B6 σ CL = c 4 σ LCL = B5σ

• B5, B6, and c4 are found in the Appendix for various values of n.

Control chart for X and S No Standard Given

• If σ is unknown, we can use an average m 1 sample standard deviation, S = ∑ S m i =1

UCL =B 4 S CL =S LCL =B3 S

i

Control chart for X and S x

Chart when Using S

x The upper and lower control limits for the chart are given as UCL = x + A 3 S CL = x LCL = x − A 3 S

where A3 is found in the Appendix