Distributions and Capability Joint File4
Distributions and Capability Joint File4
Distributions and Capability Joint File4
Student's t distribution
Chi-square distribution
F distribution
Discrete vs Continuous Distribution
Uniformly distributed
z is the z-score,
X is the value of the element,
μ is the population mean,
σ is the standard deviation.
Discrete Probability Distributions
Binomial probability distribution
Poisson probability distribution
Binomial Distribution
A binomial experiment (also known as a
Bernoulli trial) has the following properties:
• The experiment consists of n repeated
trials.
• Each trial can result in just two possible
outcomes. We call one of these outcomes a
success and the other, a failure.
• The probability of success, denoted by P,
is the same on every trial.
• The trials are independent; that is, the
outcome on one trial does not affect the
outcome on other trials.
Binomial Distribution
x: The number of successes that result from the
binomial experiment.
n: The number of trials in the binomial experiment.
P: The probability of success on an individual trial.
Q: The probability of failure on an individual trial.
(This is equal to 1 - P.)
n!: The factorial of n (also known as n factorial).
b(x; n, P): Binomial probability - the probability that
an n-trial binomial experiment results in exactly x
successes, when the probability of success on an
individual trial is P.
nCr: The number of combinations of n things, taken
r at a time.
Binomial Distribution
The binomial probability refers to the
probability that a binomial experiment
results in exactly x successes.
Suppose a binomial experiment consists of n
trials and results in x successes. If the
probability of success on an individual trial is
P, then the binomial probability is:
b(x; n, P) = nCx * Px * (1 - P)n - x
or
b(x; n, P) = { n! / [ x! (n - x)! ] } * Px * (1 - P)n - x
Binomial Distribution Properties
The mean of the distribution (μx) is
n*P
Poisson Distribution
A Poisson experiment has the following
properties:
The experiment results in outcomes that
can be classified as successes or failures.
The average number of successes (μ) that
occurs in a specified region is known.
Outcomes are random. Occurrence of one
outcome does not influence the chance of
another outcome of interest.
The outcomes of interest are rare relative
to the possible outcomes.
Poisson Distribution
e: A constant equal to approximately
2.71828. (Actually, e is the base of the
natural logarithm system)
μ: The mean number of successes that
occur in a specified region.
x: The actual number of successes that
occur in a specified region.
P(x; μ): The Poisson probability that exactly
x successes occur in a Poisson experiment,
when the mean number of successes is μ.
Poisson Distribution
Poisson Formula. Suppose we conduct a
Poisson experiment, in which the average
number of successes within a given region
is μ. Then, the Poisson probability is:
Process Capability
Conditions to be met:
Cp
Cpk
Process Capability
LSL
USL
LCL
UCL
Cp
Cpk
Process Capability
Why to do Process Capability study?
Process Capability
USL−LSL 6σ 8σ 10σ 12σ
Process Capability vs Rejections Cp 1.00 1.33 1.66 2.00
Rejects 0.27 % 64 ppm 0.6 ppm 2 ppb
Cp vs Cpk
Pp and Ppk
Ppk is an index similar to Cpk but considers
more sources of variation in the process
over a longer period of time.
Cpk and Ppk
Cp and Cpk is based off the sample of data
you have collected.
Cpk is Short-Term
Ppk is Long-Term