STUDENTS t DISTRIBUTION

The Shape of the Student's t distribution is determined by the degrees of freedom. As shown in the animation above, its shape changes as the degrees of freedom increases. For more information on how this distribution is used in hypothesis testing, see t-test for independent samples and t-test for dependent samples in Basic Statistics and Tables. See also, Student's t Distribution. As indicated by the chart below, the areas given at the top of this table are the right tail areas for the t-value inside the table. To determine the 0.05 critical value from the t-distribution with 6 degrees of freedom, look in the 0.05 column at the 6 row: t_(.05,6) = 1.943180.

CHI-SQUARE DISTRIBUTION

Like the Student's t-Distribution, the Chi-square distribution's shape is determined by its degrees of freedom. The animation above shows the shape of the Chi-square distribution as the degrees of freedom increase (1, 2, 5, 10, 25 and 50). For examples of tests of hypothesis that use the Chi-square distribution, see Statistics in crosstabulation tables in Basic Statistics and Tables as well as Nonlinear Estimation . See also, Chi-square Distribution. As shown in the illustration below, the values inside this table are critical values of the Chi-square distribution with the corresponding degrees of freedom. To determine the value from a Chi-square distribution (with a specific degree of freedom) which has a given area above it, go to the given area column and the desired degree of freedom row. For example, the .25 critical value for a Chi-square with 4 degrees of freedom is 5.38527. This means that the area to the right of 5.38527 in a Chi-square distribution with 4 degrees of freedom is .25.

 

Right tail areas for the Chi-square Distribution 

STANDARD NORMAL DISTRIBUTION

The Standard Normal distribution is used in various hypothesis tests including tests on single means, the difference between two means, and tests on proportions. The Standard Normal distribution has a mean of 0 and a standard deviation of 1. The animation above shows various (left) tail areas for this distribution. For more information on the Normal Distribution as it is used in statistical testing, see Elementary Concepts. See also, Normal Distribution.


As shown in the illustration below, the values inside the given table represent the areas under the standard normal curve for values between 0 and the relative z-score. For example, to determine the area under the curve between 0 and 2.36, look in the intersecting cell for the row labeled 2.30 and the column labeled 0.06. The area under the curve is .4909. To determine the area between 0 and a negative value, look in the intersecting cell of the row and column which sums to the absolute value of the number in question. For example, the area under the curve between -1.3 and 0 is equal to the area under the curve between 1.3 and 0, so look at the cell on the 1.3 row and the 0.00 column (the area is 0.4032). 

Area between 0 and z