What is a nomogram?
A nomogram is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a function. The field of nomography was invented in 1884 by the French engineer Philbert Maurice d’Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision. Each variable is marked along a scale, and a line drawn through known scale values (or a straightedge placed across them) will cross the value of the unknown variable on its scale.
Nomograms serve a dual purpose: they allow nitty-gritty fast computation—answers in the form of unambiguous numbers—and at the same time provide tremendous insight through the relationship of the various scales, their labeling, limits, and gradations. The better nomograms are self-documenting. They provide a visual model of a system and manifest a wonderful ability to imply interrelationships and cross-variable sensitivities. As the mathematician and computer scientist Richard Hamming remarked, "The purpose of computing is insight, not numbers."
The three of us have significant backgrounds in computer software—we welcome new technologies and are certainly not Luddites. We all have graduate degrees in physics and have professional backgrounds. One of us has written a respected book on software development, another a book on advanced algorithms for mental calculation, and another a number of scientific journal publications. We maintain websites with technical information on nomograms (see below), and we have published journal articles on nomography (one example is found here, but also see our Bayes' Theorem article below).
We recognize well-designed nomograms as tools that can be very useful, enlightening, and easy-to-use in many applications. In that regard, we have prepared a light-hearted comparison of mathematical solution by nomogram vs. computer that you may enjoy reading here.
Applications of Nomograms
For more information on nomograms
Our team has an unparalleled experience in the design of nomograms. For more information on nomography, read this introductory essay and visit our sites on the Internet:
SCAN - An overview of nomography and some new designs.
Dead Reckonings - A blog offering several essays on the history of nomography.
PyNomo - Sophisticated software for designing your own nomograms, with examples.
Nomograms for Hollow Helical Springs
The nomograms for Maximum Shear Stress and Spring Deflection that are presented in the paper, Nomograms for the Design of Light Weight Hollow Helical Springs, by William J. Bagaria, Jr., Ron Doerfler and Leif Roschier (to be published) are available for download here.
Our Current Offerings
NOMOGRAM POSTERS FOR BAYES' THEOREM CALCULATIONS
Unique circular nomograms were created by our team for calculating statistical outcomes from Bayes' Theorem. This approach is used in a variety of fields, such as medicine, to calculate how the probability of a conjecture changes after a test is performed.
Intuition often fails when interpreting statistical results from Bayes' Theorem when the pretest probabilities are low. A second version of the nomogram shifts the scale values to make it easy to calculate these special cases.
Posters of the nomograms are shown here. More information is provided below them.
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An 18"x24" color poster printed with superior resolution on heavy (>65lb) stock.
The top two nomograms provide individual results for the positive and negative test results for the Common Case scenario (optimized for common pretest probabilities). These are merged into single Common Case nomograms in the posters shown above.
The large nomogram at the bottom is optimized for the Rare Case scenario (for very low pretest probabilities). This is often the situation of most value because this is where Bayes' Theorem produces results that can be quite counter-intuitive.
Smaller nomograms and their accompanying text descriptions demonstrate the operation of the nomogram.
Click on the image to see a larger view.
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Example use of the nomograms by medical practitioners
A doctor knows the prevalence of a particular disease (the pretest probability) is 30%. Treatments have risks, and should be administered only if the patient has a certain probability of having the disease. A diagnostic test having a certain discriminatory power (a likelihood ratio) of 2.8 is run and comes back positive. What is the probability now of the patient having the disease?
The calculation is found by placing a straightedge (such as a ruler or simply the edge of a sheet of paper) on the tick mark for 30% on the pretest probability scale along the bottom half of the circle, and crossing the likelihood ratio scale value of 2.8 on the horizontal diameter. This will then cross the posttest probability scale along the top of the circle at just under 55%, the new probability of the patient having the disease. Sometimes diagnostic tests are characterized by sensitivity and specificity rather than likelihood ratio; a straightedge connecting these values on the inner elliptical scales crosses the horizontal likelihood ratio at its value. The calculation of the posttest probability then proceeds as before. Including negative test results, a complete set of 10 variables of Bayes' Theorem calculations are embedded in this nomogram for easy solution.
The Rare Case nomogram’s scales show very graphically that to have higher multipliers (likelihood ratios) of the pretest probability, you need to have increased specificity rather than sensitivity. That is, once you get to a certain point it doesn’t help much to make the test more sensitive—you must find a test that is more specific. Being able to visualize that difference on the Rare Case nomogram is an example of the better “feel” you have for what’s going on with a graphical calculator—more than you ever could by punching numbers into an app.
DOWNLOAD our article on Bayes' Theorem and these nomograms
Our article on Bayes' Theorem, with instructions on using the nomograms, has been published in the UMAP Journal:
Marasco, J., Doerfler, R., and Roschier, L. 2011. Doc, What Are My Chances? The UMAP Journal 32 (4): 279—298.
A PDF of this article can be downloaded from the link below. Please read and respect the copyright notice below from COMAP, the publisher of the UMAP Journal.
COMAP Copyright Notice
Copyright ©2010 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP. Request permissions from COMAP, Inc., 175 Middlesex Turnpike, Suite 3B, Bedford, MA 01730, USA, 1–800–772–6627 = (800) 77–COMAP or (617) 862–7878; (617) 863–1202 (fax); or firstname.lastname@example.org.
©2012 Joe Marasco, Ron Doerfler and Leif Roschier. All Rights Reserved.