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Statistics and Probability: Cumulative Distribution Function

Updated on November 18, 2016
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? | Source

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.

Prediction and Analysis

Statistics and Probability help us to predict and to plan in several fields of study and many industries. They are useful in Aerospace, Engineering, Health & Medical, Information Technology, and many other endeavors.

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. Using this online version, click on a tab at the top of the page for Reference Tables. I think you will be pleased at the extensive material you find in this book.
  • 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.
  • Statistics on The Library Spot - Statistics from a range of various government bureaus.
  • Robert Niles - Statistics for journalists.
  • For additional resources, visit you nearest college or university mathematics or engineering department library and review the software programs on public-access computers.

Understanding Distributions

Definitions of the Distribution


Explanations and Examples of the Distribution






Cumulative Distribution Function of the Standard Normal Distribution

Reference: 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 Internet page above applies the symmetry of the normal distribution, so the relevant expression for the variable of interest given is really:





Texts and Manuals

An abundance of statistics texts and manuals, as well as hand held statistical and graphing computers are available on the market to enable students and researchers to use the CDF and other complex functions easily and with great speed.

© 2007 Patty Inglish


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    • Patty Inglish, MS profile image

      Patty Inglish 10 years ago from USA. Member of Asgardia, the first space nation, since October 2016

      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 profile image

      Maddie Ruud 10 years ago from Oakland, CA

      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 profile image

      Patty Inglish 10 years ago from USA. Member of Asgardia, the first space nation, since October 2016

      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 profile image

      Earth Angel 10 years ago

      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!!