The **critical value** of a
statistical test is the value at which, for any per-determined **probability
(***p*), the test would indicate a result
that is less probable than **p**. Such a result is
said to be **statistically ****significant
**at that probability. In experimental biology, the
pre-determined value is typically **p ****= 0.05**,
which indicates a threshold of interest in a result that would
happen by chance only once in twenty trials. Under some
circumstances, the critical value may be set as **p = 0.01**
or **0.001**, that is, the result would occur only once in
a hundred or a thousand trials; **p** may also be
reported as the
range or upper bound of the probability of the observed
result.

The other consideration is the number of**degrees of freedom** (**d** or **df**)
in the data. Typically, if there are** ****C **categories
of data, **df = (C - 1)**. For a total of **N**
observations distributed across **C** categories, the
number of observations in any one category can be anything
between **0 ~ N**. The observations in the second
category can be anything between **0 ~** (**N** - **C**_{1}),
and so on. The number of observations in the last category
however is *not *free to vary, as it is fixed by **N
**minus the sum of all previous category counts.

To evaluate the statistical significance of an experimental result with two categories (**d**** ****= 1**), note that
the critical value of **p**_{0.05} = **3.841**.
If **X**^{2} = 2.0, the
result would *not *be significant, and would
be reported as **p**_{[0.05, df=1]} = 2.0^{ns},
where **ns **means *not significant*. A result of
5.0 would be significant at **p
< 0.05** (or **0.01 < p < 0.05**), a result
of 8.0 significant at **p < 0.01** (or **0.010 < p < 0.001**),
and a result of 20.0 as **p <** or << **0.001**.
The three values might also be reported as 5.0*****, 8.0******,
and 20.0*******, where the
number of stars is a shorthand convention to indicate
significance at 0.05, 0.01, & 0.001, respectively.

The other consideration is the number of

To evaluate the statistical significance of an experimental result with two categories (

Table rearranged from © 2013 by Sinauer; Text material © 2022 by Steven M. Carr