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.
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
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.
A distribution used to describe the probability
of a response when the output is continuous (variable)
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
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
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
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
All values can be obtained from the Normal Distribution
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:
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
The shape of the t distribution is similar to
the normal distribution, and converges on the
normal distribution as the number of degrees of
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
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.