The chi^2 distribution can be used to compare the goodness of fit of observed frequencies to the frequency that is expected under a hypothetical distribution. A chi^2 distribution with v degrees of freedom is produced from summing the squares of v independent, normally distributed random variables.
Compare the tests Frequency, Poker, Runs and Serial in the menu Analysis \ Analyse Randomness.
A number of reference tables exist for evaluating the probability density of the chi^2 distribution. The table below provides a few selected quantiles for the chi^2 distribution for v degrees of freedom:
v alpha = 0.01 alpha = 0.05 alpha = 0.1
(P[X>x]<0.01) (P[X>x]<0.05) (P[X>x]<0.1)
x= x= x=
1 6.63 3.84 2.71
2 9.21 5.99 4.61
3 11.34 7.82 6.25
4 13.28 9.49 7.78
5 15.09 11.07 9.23
6 16.81 12.59 10.64
7 18.48 14.07 12.01
8 20.09 15.51 13.36
9 21.67 16.92 14.68
10 23.21 18.31 15.98
15 30.58 25.00 22.30
20 37.57 31.41 28.41
30 50.89 43.77 40.25
255 310.40 293.20 284.30
The entry (v = 10, alpha = 0.05, x = 18.31) means that assuming that X has a chi^2 distribution with 10 degrees of freedom the probability of obtaining a value for X of 18.31 or more is only 5%.
Reference: Christian Schiestl, Pseudozufallszahlen in der Kryptographie, in Klagenfurt, 1999.