Surfstat.australia: an online text in introductory Statistics
SUMMARISING AND PRESENTING DATA
As the number of observations increases, a histogram can be
approximated by a continuous function f(x). A common function is the
normal curve (bell-shaped curve).
Example - Distribution of blood pressure
Distribution of blood pressure can be approximated as a normal
distribution with mean 85 mm. and standard deviation 20 mm. A histogram
of 1,000 observations and the normal curve is shown below.
The normal distribution has about 68% of the observations lying within
one standard deviation of the mean, 95% within two standard deviations
and 99.7% within 3 standard deviations. This allows for a simple description
of where most values are to be found. The standard normal distribution
has a mean of 0 and a variance of 1 and is shown below.
This device shown in the diagram below was invented by the scientist F.
Galton (1822-1911), who also worked on statistical methods for
correlation and regression. It is designed to study the shape of
distributions when objects are acted on by many factors.
Beads are put into the hopper and then dropped one by one through
the funnel. They bounce through an arrangement of pins and depending
on which pins they hit, they fall into one of many slots at the bottom.
If the process is not manipulated in any way the resultant
distribution of beads looks like
This is the Normal Distribution whose probability density
function can be worked out mathematically.
The quincunx demonstrates what happens in a stable process affected
by many factors.
eg. output of some production process is affected by raw materials,
production process (environmental, machine and operator factors),
packing, transport, etc.
eg. growth of plants or animals is affected by genetics, nutrition,
The quincunx can also be used to demonstrate the effects of changing
the process, e.g. moving the position of the funnel produces
i.e. with increased spread/variability