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Statistics and Probability: Cumulative Distribution Function

Ms. Inglish has 30 years experience in medicine, psychology, STEM instruction, history, and aerospace education for USAF Civil Air Patrol.

What is a Cumulative Distribution Function and how do we calculate, produce, and use the data provided by such a distribution?

What is a Cumulative Distribution Function and how do we calculate, produce, and use the data provided by such a distribution?

Statistics is a Fascinating Field of Study

Statistics are important because the study can help us learn about our universe and the things and beings in it. This is an even more important capability in the face of our flying back to the moon in or before 2018. Trigonometry, calculus, and statistics all help us to fly away from Earth, visit other places, and return.

These divisions of mathematics help us to build bridges as well as to construct bio-robotic arms and legs to replace those loss in battle or lost to injury or genetic events.

Because our universe, though we think it is never ending, has some sort of boundary, numbers behave in a particular number of ways. We have not yet found that number, but we know that equations and distributions are bound by limits concerning results and that the laws of physics for this universe also have boundaries. Quantum physics, not so much, perhaps.

Statistical Resources

What is a Cumulative Distribution Function and how does one calculate, produce, and use the information provided by such a distribution?

This particular distribution function is also known as a CDF, cumulative distribution function, or cumulative frequency function (CFF). Its purpose is to provide the probability that a variable takes on a value that is less-than-or-equal-to a particular number.

A good reference for understanding CDF and CFF is that of Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 4rd ed. New York: Wiley, pp. 6-8, 2000/2010.

Along the right-hand side of this article, you will find various computer programs used for running the CDF. It is difficult, if not impossible to achieve CDF outcomes by hand figuring and hand graphing. A computer program is necessary.

Additional Useful Statistics Resources

Several additional resources can be accessed for use in determining and applying the CDF in real life scenarios. Some of these resources include:

  • Engineers Handbook: This is one of my favorite resources. My father studied mechanical and electrical engineering and always kept an edition of this book on hand. I found it very useful in high school for trigonometry and calculus reference, and was fascinated by all of the tables it included.
  • FedStats: Statistics and additional data provided by various departments of the US Federal Government. Excellent for research and reviews.
  • HYPERSTAT: Simple introduction to statistics.
  • MATLab: A comprehensive set of statistics tools that are used by engineers, financial analysts, and others.
  • Robert Niles: Statistics for journalists.
  • Statistics on The Library Spot: Statistics from a range of various government bureaus.
  • For additional resources, visit you nearest college or university mathematics or engineering department library and review the software programs on public-access computers.

Cumulative Distribution Function of the Standard Normal Distribution

A good reference for this distribution is the Engineering Statistics Handbook. Tests for Probability of Distribution: Cumulative Distribution Function of the Standard Normal Distribution. US Department of Commerce: NIST

NOTE: The table on this federal internet page applies the symmetry of the normal distribution, so the relevant expression for the variable of interest given is really the following:

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This is another useful software program to use for the CDF: Jenness Enterprises Software.

This is another useful software program to use for the CDF: Jenness Enterprises Software.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2007 Patty Inglish MS


Patty Inglish MS (author) from USA and Asgardia, the First Space Nation on January 02, 2008:

The status of multiple formulae or a non-closed or open formula(s) or one or more formula set(s) is a given; along with other mathematic precepts. I listed more than one software program for performing the function and each uses its own 'programming' and formula/ formulae. I did not compare the programs and formula(s), because I leave that up to the user or student needing to answer the question of the request.

As a quirky side note, in research some time ago, I attempted to hand construct a curve from a fractal function and was able to plot only 3 points aver a few hours. Computer use is the most practical application for certain functions.

Thanks for the comment.

Maddie Ruud from Oakland, CA on December 31, 2007:

My father is an econometrician, and he says this:

"You can't. You need a computer program. There is no closed formula."

At least that is my simplification of the stream of math-jargon that came out of his mouth when I read this request aloud to him on my last visit.

Patty Inglish MS (author) from USA and Asgardia, the First Space Nation on October 26, 2007:

Well, I know how to put together my notes and experiences and how to look things up. So, maybe one day...

Thanks for the comment!

Earth Angel on October 26, 2007:

Wow!! You are GREAT Patty!! Have you ever thought of publishing your own book?? I am a huge proponent of self-publishing!! Your Hubs are always full of wonderful and helpful information!! Thank you!! Blessings, Earth Angel!!

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