The X square statistic is a useful measure of the discrepancy between the actual distribution of a set of data points and the theoretical distribution of a random variable of which the data points supposedly are values. Thus the X square statistic is frequently used in goodness of fit tests

Argentiero, P. D.

Morris, R. A.

Tolson, R. H.

English

NASA Technical Reports Server

April 1968	 a	 Brief 68-10136 
NASA TECH BRIEF 
NASA Tech Briefs are issued to summarize specific innovations derived from the U.S. space program, 
to encourage their commercial application. Copies are available to the public at 15 cents each 
from the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
2 
The X Statistic and Goodness of Fit Test 
The test for goodness of lit consists of testing the 
hypothesis that a given set of N observations con-
stitutes values of a random variable with a specified 
frequency function f(x). The X2 statistic has been 
found to be a useful measure of the discrepancy 
between the actual distribution of a set of data points 
and the theoretical distribution of a random variable 
of which the data points supposedly are values. Thus 
the X2 statistic is frequently used in goodness of lit 
tests. Unfortunately the probability density function 
of X2 is quite involved. In fact it has been only 
recently that explicit information, either in terms of a 
generating function or an analytical representation, on 
the distribution of X2 has been provided. This dif-
ficulty has been mitigated somewhat by the fact that 
for fairly large samples the distribution of the X2 
statistic is closely approximated by the distribution of 
the well known chi-square variable. Hence in a good- 
ness of lit test, although it is the X2 statistic which is 
calculated, it is from the appropriate table of the chi-
square distribution that a critical value for the test 
is chosen. The subject of the chi-square approximation 
to the X2 statistic and the circumstances under which 
it properly can be used has had much discussion in 
the statistical literature. Authors treating this ques-
tion frequently have been vague and have not been in 
general agreementThus when confronted with margin-
ally small samples to be tested for goodness of fit,
the investigator discovers in the literature imprecise 
and contradictory advice as to how to proceed. This 
unsatisfactory situation is corrected in a report that 
provides tables of critical values of the X2 statistic 
which are applicable to those cases in which the use 
of the chi-square approximation is questionable. The 
X2 statistic and its distribution, its applicability to 
goodness of fit tests, and its approximation by the chi-
square variable are discussed in some detail. An 
example is also provided to demonstrate that the 
proper application of the tables can add considerable 
precision to a goodness of lit test when small samples 
are involved. 
Note: 
Copies of the report may be obtained from: 
Technology Utilization Officer 
Goddard Space Flight Center 
Greenbelt, Maryland 20771 
Reference: B68-10136 
Patent status: 
No patent action is contemplated by NASA. 
Source: P. D. Argentiero 
of Goddard Space Flight Center, 
R. A. Morris 
of University of Maryland, 
and R. H. Tolson 
of Langley Research Center 
(GSC- 10547) 
Category 02 
This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States
Government assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately owned rights.
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The X square statistic and goodness of fit test

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