Part 1: Introduction

Part 2: Probability Density Function Implementation

Part 3: Cumulative Distribution Function Implementation 1

Part 4: Cumulative Distribution Function Implementation 2

Part 5: Cumulative Distribution Function Implementation 3

**Part 6: References and Notes**

# References and Notes

As stated in Part 1 the implementations presented in this series of posts are my own, however I conducted no original research in their development.In this part I list the sources I used while preparing this series of posts as well as notes and additional sources of interest.

### Main Sources

Wikipedia article on the Normal Distribution - the Wikipedia article was my first stop when looking for information for these series of posts. It contains the basic information and formulas for the PDF and CDF functions, but more important the section about numerical approximations contains a reference to algorithm 26.2.17 from Abramowitz & Stegun (1964) used in Part 4**Milton Abramowitz and Irene Stegun (1964) - Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables**- referred in these posts as "Abramowitz & Stegun (1964)"; this book contains a wide array of formulas and approximation in several fields. The Wikipedia article has more information about this work.

The approximations used in Part 3 and Part 4 are taken from this source.

**W. J. Cody (1969) - Rational Chebyshev Approximations for the Error Function, Mathematics of Computation, Vol. 23, No. 107 (Jul., 1969), pp. 631-637**- Referred in these posts as "W. J. Cody (1969)", I came accross this paper from one of the pioneers in numerical analysis while looking for Hart (1968) (see Addtional References below) and decided to use his approximation for the "ultimate" implementation in Part 5.

**Improvents in Excel 2007 and 2010**are discussed in the following sources:

- Description of improvements in the statistical functions in Excel 2003 and in Excel 2004 for Mac
- Functions and Solver Improvements in Excel 2010 - Microsoft Office 2010 Engineering - Site Home - TechNet Blogs
- Function Improvements in Microsoft Office Excel 2010

The white paper in the last reference in particular is very interesting as it contains a primer on common approximation methods.

### Additional References

**Cecil Hastings, Jr. (1955) - Approximations for Digital Computers**- this is the source for the algoritms presented in Abramowitz & Stegun (1964), thought the actual approximations in this book are for the erf function.

Part 1 of this book describes the methods used to develop the approximations.

**John F. Hart, et al (1968) - Computer Approximations**- this oft sited book contains a wide array of approximations for several functions including highly accurate ones for the erf function.