PCM-2 Manual.pdf - Voss Associates

Chi-squared Distribution
0.995
1.5
···························r················7:··:::-···-l.;.;,···;,;,;··:.:.:··.:.:··.:.:···:··:.:.:··.:··:··:.:.···~··~··fr·=·:-::··~··~··~ ..=.::.=.:::: ..~..:..:..~.="' ••:.J====:.: ..:..:.:.:.: ..~..~..~.'j.j
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0.990
0.900
............,
. ~ ,
1 ~ ~ ~ .
0.500
~--l ;
0.750
0.250
'" u
0.5
5 10 15
Degrees of Freedom
20 25 30
Figure 5, Chi-squared Distribution
Table 3 is entered at the row that corresponds to the number of degrees of freedom (n - 1).
Associated with the closest x 2 /v found in that row is a probability listed at the top of the column.
That value is the probability that x2/V would be less than or equal to the number found in the table.
Very large or very small probabilities indicate poor fit. When considered in terms of observed results
divided by expected results, intuitively, an ideal experiment would produce a ratio of 1. A 50 %
probability is equivalent to saying that in a random sampling x 2 /v is just as likely to be above as
below the number indicated, and therefore indicative of a good fit of real data to ideal.
NOTE: The fact that the degrees of freedom is one less than the number of data points can cause
some consternation to students of statistics. To clarify the relationship between "n" and "v", take the
case of n=2, the smallest value of n from which an average can be computed. Knowing the average
and the value of only one of the two data points, it is possible to derive the second data point. Also,
the two data points are equidistant from their average. In like fashion, if the average of n data points
is known, then the value of anyone of those data points can be derived from the values of the
remaining (n - 1) data points. Any value could be assigned to any of the (n - 1) points, but the n tl1e
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