Introduction to Research Design and Statistics

Probability Distributions

Standard Normal Distribution

Students t Distribution

Chi-square Distribution

F Distribution

Some Relationships Among Probability Distributions

z and t

• As degrees of freedom increase towards infinity, the Student's t-distribution converges on the standard normal distribution.

z and chi-square

• z² is distributed as a χ² distribution with one degree of freedom
  
• For two independent standard scores, z₁² + z₂² is distributed as a χ² distribution with two degrees of freedom
• For three independent standard scores, z₁² + z₂² + z₃² is distributed as a χ² distribution with three degrees of freedom, and so on

F and t

• When F has one degree of freedom in the numerator then F(1,df2) = t²(df2)

F and chi-square

• F is defined as the ratio of two chi-square variables
• χ²(df2) = df1*F(df2)
• As df2 increases towards infinity the probabilities for χ² and F converge.

UCLA Department of Education