Mathematics and Psychology
Kashif Shah
University of Southern Indiana
2012-04-08 

			Mathematics and Psychology 

	The Use of Statistics in Psychology Research in any scientific field requires 
precise observations and record keeping. Psychology is no different when it 
comes to needing a mechanism to ensuring the validity of a research project, 
providing precise data, or analyzing their observations. Psychologists often 
use statistics to organize and analyze their data, and to also uncover any 
hidden patterns that may not be noticeable through pure observation. Through 
the development of statistics it is now possible to contain “messy” data, and 
it has shown us the need for descent research practices in order to obtain 
valid conclusions (Brysbaert & Rastle, 2009, p. 68). 

	Whenever researchers apply statistical tests to assess the results of their 
studies the term “statistical significance” is often used. Statistical significance 
is the “process of judging whether or not a particular statistical outcome is likely 
or unlikely to be due to chance” (Cowles, 2001, p.7). The statistical test 
themselves serve as an evaluation method to make sure that a researcher’s 
findings can be verified and gives the researcher a way to figure if the 
outcome of the experiment was due to an unforeseen situation (Statistical 
significance, 2004). So, as a result, statistical significance helps to 
maintain a more precise study. 

	Statistics not only help with organizing the end results of a study, but they 
also help researcher’s design proper studies that will ultimately lead to a valid 
conclusion. It has been shown that the construction of appropriate questions in a 
“systematic framework” deliver data that, with the tools of statistics, lead to valid 
answers (Cowles, 2001, p. 31). A researcher can develop and run a study from beginning 
to end, however if the study is not constructed carefully the end result will produce 
patterns that become complex and almost impossible to draw any conclusion 
from. So, when the researcher takes advantage of the statistical tools that 
they have at their disposal it is much more advantageous to design a study 
from the start that would rule out any confounding variables. As a result the 
end data would be organized and easier to draw conclusions from (Brysbaert & 
Rastle, 2009, p. 69). 

			Quantitative / Mathematical Psychology

	According to the American Psychological Association (APA), there are more jobs
for quantitative psychologists than there are students of quantitative psychology 
(APA, 2012). Quantitative Psychology is a field of study that encompasses all 
aspects of psychology itself. Research in quantitative psychology is 
important for developing research methods, new techniques for measuring 
psychological phenomenon, data analysis, and modeling of psychological 
processes (APA, 2012). Mathematical Psychology in particular is heavily based 
in mathematics. Mathematical models are inherently necessary for studying and 
making hypothesis for brain mechanism, decision-making, understanding 
perception, language, and behavior, as well as the mechanisms by which 
environmental stimuli are converted into biopsychological representations. 
Mathematical psychologists utilize game theory, probability and statistics, 
as well as general mathematical approaches and physics based models (APA, 
2012). Some of the earliest mathematical models in Psychology were developed 
as ways to map physical stimuli to psychological perception. The early 
psychologist Ernst Weber developed the differential equation dp = k*(dS/S) to 
describe the relationship between change in weight and change in perception 
of weight. Further analysis of this equation produces the equation p = k* ln 
(S/S0) which describes the intensity of weight perception in terms of the 
product of an experimentally determined constant k and the natural log of the 
ratio of stimulus intensity to the stimulus threshold. Later, Stanley Stevens 
developed what is now called Steven’s Power Law, Psi(I) = k*I^alpha, which 
relates the magnitude of physical stimulus to the psychological intensity. 
Stevens derived this equation by fitting a power function to data collected 
from different participants across many different sensory modalities 
(Smelser, 2001). These early examples of mathematical models of perception 
are still discussed in introductory sensation and perception courses but the 
techniques used to model perception have gone far beyond these simplistic 
models to include signal detection theory (Luce, 1990) and neural network 
models. 

			Parallel Distributed Processing 
		
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			Psychology of Mathematics 
			
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			Psychometrics 

	Psychometrics is the study of measuring psychological 
phenomenon such as knowledge, intelligence, personality, education, and 
attitudes. Psychometric measurements are taken by way of questionnaires, 
surveys, and assessments. According to Mitchell (1997), “measure is the 
numerical estimation and expression of the magnitude of one quantity 
relative to another.” In response to a physicist, one psychologist said: 

	"Measurement in psychology and physics are in no sense different. 
	Physicists can measure when they can find the operations by which they may
	meet the necessary criteria; psychologists have but to do the same. They
	need not worry about the mysterious differences between the meaning of 
	measurement in the two sciences." (Reese, 1943, p. 49) 
	
Almost everyone has taken a psychometric test of one sort or the other 
–some common psychometric tests are the SAT, GRE, IQ tests, and the 
Myers-Briggs Personality Type Indicator. Psychometric tests not only require 
means for analyzing test data but also require rigorous procedures for creating 
validated tests. Psychometric tests need to be able to measure the phenomenon 
of interest to the exclusion of other confounding phenomenon, personal biases,
and societal expectations to name just a few problem areas for developers of 
psychometric tests. Psychometricians use advanced mathematical models of their 
variable of interest as well as statistical methods to analyze and compare data from 
tests to behavioral data from psychological experiments. Some mathematical 
methods utilized by Psychometricians include correlations and other 
statistical techniques, multidimensional scaling, data clustering, 
structural equation modeling, and path analysis.

			References

American Psychological Association (2012). “Quantitative Psychology”. 
	http://www.apa.org/research/tools/quantitative/index.aspx 

Brysbaert, M., & Rastle, K. (2009). Historical and conceptual issues in 
	psychology. Harlow, England: Pearson/Prentice Hall. 

Cowles, M. (2001). Statistics in psychology: An historical perspective. 
	Mahwah, NJ: L. Erlbaum Associates. 

Luce, R.D. (1990), "Psychophysical laws: cross-modal matching",Psychological 
	Review 97 (1): 66–77, doi:10.1037/0033-295X.97.1.66 

Michell, J. B (1997). "Quantitative science and the definition of 
	measurement in psychology". British Journal of Psychology 88 (3): 355–383. 
	doi:10.1111/j.2044-8295.1997.tb02641.x. 

Reese, T.W. (1943). "The application of the theory of physical measurement 
	to the measurement of psychological magnitudes, with three experimental 
	examples". Psychological Monographs 55: 1–89. 

Smelser, N.J., & Baltes, P.B. (2001). International encyclopedia of the 
	social & behavioral sciences. pp. 15105–15106. Amsterdam; New York: 
	Elsevier.ISBN 0-08-043076-7. 

Statistical Significance. (2004). In The Concise Corsini Encyclopedia of 
	Psychology and Behavioral Science. Retrieved from 
	http://www.credoreference.com/entry/wileypsych/statistical_significance