1

ApplicationsStatistical Quality Control

Dr. Jerrell T. Stracener, SAE Fellow

Leadership in Engineering

EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS

Systems Engineering ProgramDepartment of Engineering Management, Information and Systems

2

Statistical Quality Control is an application of probabilitistic and statistical techniques to quality control

An Application of Probability & Statistics Statistical Quality Control

3

Statistical Quality Control - Elements

Analysisof processcapability

Statisticalprocesscontrol

Processimprovement

Acceptancesampling

4

Statistical Tolerancing - Convention

Normal ProbabilityDistribution

LTL Nominal UTL

0.001350.00135 0.9973

+3-3

5

Statistical Tolerancing - Concept

LTL UTLNominal x

6

For a normal distribution, the natural tolerance limits include 99.73% of the variable, or put another way, only 0.27% of the process output will fall outside the natural tolerance limits. Two points should be remembered:

1. 0.27% outside the natural tolerances sounds small, but this corresponds to 2700 nonconforming parts per million.

2. If the distribution of process output is non normal, then the percentage of output falling outside 3 may differ considerably from 0.27%.

Caution

7

The diameter of a metal shaft used in a disk-drive unitis normally distributed with mean 0.2508 inches andstandard deviation 0.0005 inches. The specificationson the shaft have been established as 0.2500 0.0015 inches. We wish to determine what fraction ofthe shafts produced conform to specifications.

Normal Distribution - Example

8

spec meetingP

91924.00000.091924.0

60.41.400005.0

0.2508-0.24850.0005

0.2508-0.2515x0.2485P0.2515xP

0.2515x0.2485P

0.2508 0.2515 USL

0.2485 LSL

0.2500

f(x)

xnominal

Normal Distribution - Example Solution

9

Thus, we would expect the process yield to be approximately 91.92%; that is, about 91.92% of the shafts produced conform to specifications. Note that almost all of the nonconforming shafts are too large, because the process mean is located very near to the upper specification limit. Suppose we can recenter the manufacturing process, perhaps by adjusting the machine, so that the process mean is exactly equal to the nominal value of 0.2500. Then we have

Normal Distribution - Example Solution

10

0.2515x0.2485P

9973.000135.099865.0

00.33.000005.0

0.2500-0.24850.0005

0.2500-0.2515x0.2485P0.2515xP

0.2500 0.2515 USL

0.2485 LSL

f(x)

xnominal

Normal Distribution - Example Solution

11

12

13

14

15

Using a normal probability distribution as a model for a quality characteristic with the specification limits at three standard deviations on either side of the mean. Now it turns out that in this situation the probability of producing a product within these specifications is 0.9973, which corresponds to 2700 parts per million (ppm) defective. This is referred to as three-sigma quality performance, and it actually sounds pretty good. However, suppose we have a product that consists of an assembly of 100 components or parts and all 100 parts must be non-defective for the product to function satisfactorily.

Normal Distribution - Example

16

The probability that any specific unit of product is non-defective is

0.9973 x 0.9973 x . . . x 0.9973 = (0.9973)100 = 0.7631

That is, about 23.7% of the products produced underthree sigma quality will be defective. This is not anacceptable situation, because many high technology products are made up of thousands of components.An automobile has about 200,000 components andan airplane has several million!

Normal Distribution - Example