Critical values
& degrees of freedom of the Chi-Square (
2) distribution

    The critical value of a statistical test is the numerical value at which an observed result is statistically significant at some per-determined degree of probability. For example, for a 2  test with df = 5, the table shows that if 2 > 11.07, the observed result would be expected to occur by chance less than one time in twenty, that it, is has a probability p < 0.05. If the significance level of the test had been set at p = 0.05, the observed results is said to be statistically significant. Where the observed result exceeds the critical value for p = 0.01, the result is sometimes said to be highly significant. Strictly, the significance level is set a priori (before the experiment is carried out), and any result is either significant or non-significant at that level.

    The simplest kind of statistical test has only two possible outcomes ("either / or") and is also called a single-classification test. In such a test, the number of events in the first category automatically determines the number in the second. Because of this, the experiment is said to have only a single degree of freedom, or  df = 1. For a more complicated experiment with C possible categories of outcome for N events, the number of events that can fall into any of the first C-1 categories can vary freely. The number of events in the last category must make the total add up to N, and is thus constrained. The experimental outcome has lost one degree of freedom,
and in general df = (# of possible outcomes - 1).

All text material 2013 by Steven M. Carr