Statistical Process Control

(multi-billion dollar application of the humble central limit theorem!)

Process

• series of interconnected steps in producing a product or service–producing a transmission in Ford plant– transporting components from Brazil and

China to assembly plant in Arizona– reviewing applicants to BYU– serving patients in emergency room–ordering food at Taco Bell

make wafer

cut to specs

bond die to carrier

bond wire to package

encapsulate in plastic

trim wire leads test device

prepare for shipment

ship device

e.g. Microchip Production Process

Statistical Dogma for Processes• all processes have natural variation– raw material, human performance,

equipment performance, measurement • all processes occasionally susceptible to

unnatural variation–bad batch of raw material–broken machine–poorly trained operator

Process Control• monitor process variables (inputs, outputs,

etc.) over time to decide if variability consistent with natural variation– if consistent, continue process– if inconsistent, stop process, find cause of

unnatural variation, fix problem, resume process

• decision often based simply on central limit theorem

Definitions

Control Chart: tool for monitoring variables of a process, alerting us when unnatural variation seems to have occurred

Process in Statistical Control: process whose output exhibits only natural variation over time– We will characterize each process as “in control”

or “out of control”

Which control plot below, in your opinion, shows only natural variation (ie, “process is in control”?

iClicker

Advantages of Control Charts

• investigate unnatural variation • avoid reacting to natural variation

both are important!

x chart

• most common kind of control chart• used when output variable is quantitative• theory based on sampling distribution of

and central limit theoremx

Summaryprobability distribution of (or “sampling

distribution of ”) for SRS of size n from population with mean μ and st. dev.

mean of dist. of = μ

standard deviation of dist. of =

approximately normal (if population not normal and n large) or exactly normal (if population normal )

Center

Spread

Shape

x

x

x / n

x

iClicker

What is the approximate probability that will be between if the process is in control?

a. 95%b. 99%c. 99.7%d. impossible

to say

x - 3 3 n and n

iClickerThe target wt. is μ = 1.0875 lb. The st. dev. is σ = 0.015 lb. For monitoring, n = 8 bottles are sampled every 10 min. What are the control limits for the filling process?

a. 1.0875±3(0.015) b.

1.0875±3(0.015/2.83)c. 1.0875±3(0.015/8)d. 1.0875±8(0.015/3)e.

1.0875±8(0.015/10)

(insert sections 14 of Flash lesson 21)

Vocabulary

control chartcontrol limitsnatural variationout-of-control signalsprocessprocess controlstatistical controltarget valueunnatural variationx chart