Tables of used for tests of significance are typically based on one tail only (the tail to the right) of the sampling distribution of . These are NOT critical values for directional (or one-tailed) tests). Although only one tail of the sampling distribution of is used, the table values are those required for testing the significance of a difference regardless of direction (that is, for a two-tailed test).

For one degree of freedom:

0.05 of the area of the distribution falls to the right of .

0.01 of the area of the distribution falls to the right of .

These are NOT critical values for a one-tailed test.

For 1 degree of freedom the square root of is a normal distribution and can be used with reference to the standard normal curve in applying two-tailed tests. In effect, because the distribution is the square of the standard normal distribution, both tails of the normal curve are incorporated in the right tail of the curve. For the standard normal distribution:

Significance at the 0.05 level for a two-tailed test is z = 1.96. Note that 1.96^2 = 3.84.

Significance at the 0.01 level for a two-tailed test is z = 2.58. Note that 2.58^2 = 6.64 (within rounding error).

In tests of goodness of fit and in most tests of independence you won't usually be concerned with the direction of the difference observed. The value of (with one degree of freedom) required for significance at the:

0.05 level for a one-tailed test is 2.71 (this is 1.64^2 and 1.64 is the value of z required for significance at the 0.05 level for a one tailed test).

0.01 level for a one-tailed test is 5.41 (this is 2.33^2 and 2.33 is the value of z required for significance at the 0.01 level for a one tailed test).