Surfstat.australia: an online text in introductory Statistics

SUMMARISING AND PRESENTING DATA

NORMAL DISTRIBUTIONS

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.

Quincunx

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, etc.

The quincunx can also be used to demonstrate the effects of changing the process, e.g. moving the position of the funnel produces distributions like

i.e. with increased spread/variability

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