Overlapping Normal Distributions. John M. Linacre

Are men taller than women? Are women more patient than men? We answer such questions by comparing putatively normal distributions. A typical analysis computes a t-test of differences between sample means. When samples are large, any difference between means will be declared "statistically significant". Before drawing substantive conclusions from a "significant" result, however, it is wise to discover how much the distributions overlap, contradicting any finding of clear "difference".

Proportions of distributions for overlapping normal distributions


Proportions of distributions for overlapping normal distributions


Consider two normal distributions, N1 and N2, with the same sample sizes but different means (M1 and M2) and standard deviations (SD1 and SD2). Number the samples so that N1 has the smaller standard deviation, SD1. To discover the expected percentage by which each sample distribution overlaps the other, consult the nomogram. For the x-coordinate, compute a standardized absolute distance between the means, |M2- M1|/SD1. For the y-coordinate, compute the ratio of the standard deviations, SD2/SD1. Interpolate by eye between contours to estimate the percent, p%, of each distribution that is in common with the other. The unique amount of each distribution is then 100-p%.

If M1=2.4, SD1=1.6, and M2=3.2, SD2=2.0, then |M2-M1|/SD1=0.5, SD2/SD1=1.25, and, by reference to the nomogram, about 80% of each distribution overlaps the other.

These results were computed using numerical integration of the areas under the normal curves.


Overlapping normal distributions. Linacre JM. … Rasch Measurement Transactions, 1996, 10:1 p.487



Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

To be emailed about new material on www.rasch.org
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Rasch.org

www.rasch.org welcomes your comments:

Your email address (if you want us to reply):

 

ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
May 17 - June 21, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden http://www.hkr.se/samc2024
June 21 - July 19, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 5 - Aug. 6, 2024, Fri.-Fri. 2024 Inaugural Conference of the Society for the Study of Measurement (Berkeley, CA), Call for Proposals
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com

 

The URL of this page is www.rasch.org/rmt/rmt101r.htm

Website: www.rasch.org/rmt/contents.htm