Page 443 - Design for Six Sigma for Service (Six SIGMA Operational Methods)

Statistical Basics and Six Sigma Metrics 401

2
The population variance is often denoted by s , and it is often simply called
the variance. The variance of a random variable y is defined as follows:

⎧ ∑ (y − m) p ( ) if y is discrete
2
y
⎪ All
2
s = ⎨ −y
⎪ ∫ +∞ (y − m) f ( )dy if y is continuous
2
y
⎩ −∞
Variance is often denoted by V(y). From the definition of expected value, it
is clear that
2
2
s = Var(y) = E[(y − m) ]
The population standard deviation is often called the standard deviation σ,
2
where s is simply the square root of variance s , that is,
s = s 2

In Six Sigma–related applications, there are several probability distribution
models that are often used as the basis for data-driven decision making. They
are the normal distribution, exponential distribution, binomial distribution
and Poisson distribution. We give an overview for each of these distributions.

Normal Distribution
The normal distribution (or Gaussian distribution) was first proposed by
Gauss (1777–1855). It is often used to model the probability distribution of
continuous variables that have the following properties:
1. There are many random factors that can affect the value of the random
variable.
2. Each of these random factors has relatively small influence on the
random variable; there is no dominate factor.
The normal distribution is the most popular distribution in quality engi-
neering and Six Sigma. It is often used to model the follow random
variables:
1. Quality characteristic of parts from suppliers
2. Students’ test scores or employee performance scores
The probability density function of the normal distribution is as follows:

y− ⎞
fy() = 1 e − 1 2 ⎛ ⎝ s m ⎠ 2 − ∞ < y < +∞
2ps