|
A Bimodal distribution has two peaks. It
occurs when the output from two process
streams are mixed, thus combining two distributions.
Each individual process stream often conforms
to a normal distribution.
An example could be when the
output from two similar machines, each set
to a slightly different size, is mixed.
| Bivariate
Probability Distribution |
|
The Bivariate distribution
shows the joint probability distribution
of two random variables. The bivariate distribution
forms a three dimensional surface.
Suppose that the position of the center
of a hole can vary in both the x and y directions.
The distribution of the center position
in both the x and y directions conforming
to a normal distribution.
A distribution of the hole center positions
would conform to a bivariate normal distribution.
If the standard deviations in x and y were
the same then the surface formed would be
a bivariate normal distribution. The shape
of the surface would depend on the standard
deviation in the x and y directions, and
whether x any y were correlated.
A statistical distribution with the pdf:
The symbol 'G'
is the 'Gamma Function'.
The chi-square distribution
is used for the chi-square goodness of fit
test.
Continuous distributions are used when
the characteristic being investigated is
a continuous variable, for example length,
weight etc.
The 'normal distribution', also known as
the 'gaussian' distribution' is the most
common. It governs the behavior of many
processes, and most types of parametric
statistical analysis depend on the characteristic
conforming to a normal distribution. Most
of the other common types of distribution
are closely related to the normal distribution.
An extremely important statistical concept
is the 'central limit theorem'. In simple
terms, this theorem states that even if
individual items taken from the process
do not conform to a normal distribution
the means of samples taken from the process
will conform, as long as the sample size
is sufficient large.
Some of the distributions are associated
with other types of study. For example,
the Exponential Distribution and Weibull
distribution are used in reliability studies.
| Cumulative
Distribution Function (cdf) |
|
Cumulative Distribution Function.
The mathematical function that gives the
cumulative proportion of a distribution:
For a normal distribution the cdf is:
This cannot be easily solved,
and so normal distribution tables, or computers
are used.
The exponential distribution is represented
by:
The exponential distribution
is closely related to the Poisson distribution,
and is also used for reliability engineering.
It models situations where the failure rate
is constant. This occurs in the 'useful
life' phase, when the item concerned will
operate until it is affected by some external
factor.
An example would be tire punctures.
Most punctures are caused by objects on
the road and the likelihood of a puncture
is constant as long as the tread is not
badly worn, at which time the tire is hopefully
discarded.
A distribution that is used to test the
hypothesis concerning sample variances.
The F Distribution is formed from the ratios
of two chi-squared variables. If X1
and X2 are independent chi-square
variables then:
The pdf is:
Very few people can calculate f(x) using
mental arithmetic.
An alternative name for the Normal Distribution,
see Normal Distribution
Kurtosis is the degree of 'peakedness'
of a distribution. It is calculated from
the formula:
The '3' is included in the
formula to give the normal distribution
a kurtosis of zero (some published versions
do not include it).
A distribution with a kurtosis
greater than zero is more peaky than a normal
distribution and is 'Leptokurtic'. A distribution
that is flatter than a normal distribution
is 'Platykurtic'.
The lognormal distribution has the pdf:
The lognormal distribution is often used
for modeling material properties.
The normal distribution (also known as
the Gaussian distribution) is a continuous
probability distribution. Most processes
that are 'in control' conform to a normal
distribution, or to some variant of it.
Normality is also a condition of many statistical
tests.
The normal distribution has a 'bell shape':
The proportion of the process output that
falls within a range of values is represented
by the area of the curve. Values are:
| Between -1 and +1 Standard Deviation |
68.3% (about two thirds) |
| Between -2 and +2 Standard Deviation |
95.5% (about 95%) |
| Between -3 and +3 Standard Deviation |
99.7% |
All values can be obtained from the Normal
Distribution Tables.
A graphical method for testing normality
and identifying points that do not conform
to the normal distribution. The values are
plotted on a special graph paper called
'Normal Probability Paper' that has the
'y' axis stretched so that points that conform
to a normal distribution will form a straight
line:
| Probability
Distribution Function (pdf) |
|
Probability Distribution Function. The
equation that describes the distribution.
The pdf gives the height of the distribution:
See t Distribution
The t distribution has the probability
distribution function:
The shape of the t distribution is similar
to the normal distribution, and converges
on the normal distribution as the number
of degrees of freedom increases:
A distribution that has a single local
maximum (cf bimodal distribution).
The Weibull distribution is a very flexible
distribution. It is useful in reliability
engineering because the parameters can be
tailored to suit the product characteristics,
particularly in the infant mortality and
wearout phases of products:
The parameters may be based on a theoretical
model, but are often just those that best
fit the data.
A graphical technique to determine if a
data set comes from a 2 parameter Weibull
distribution. The Weibull plot has non-linear
scales that are arranged so that if the
data set does conform to a normal distribution
the points will form a fairly straight line.
The Normal Probability Plot is a similar
concept.
|
