# Non-central t Distribution

Menu location: **Analysis_Distributions_Non-Central t**.

Non-central t (T) represents a family of distributions which are shaped by ν degrees of freedom and a non-centrality parameter (σ).

Non-central t may be expressed in terms of a normal and a chi-square distribution:

- where z is a normal variable with mean d and variance 1 and χ² is a chi-square random variable with degrees of freedom (Owen, 1965).

In the field of meta-analysis some effect size statistics display a non-central t distribution. This function may therefore be useful in hypothesis testing and confidence interval construction for effect sizes (Greenland and Robins, 1985).

__Technical Validation__

StatsDirect evaluates the cumulative probability that a t random variable is less than or equal to a given value of T with n degrees of freedom and non-centrality parameter d (Lenth, 1989; Owen, 1965; Young and Minder, 1974; Thomas, 1979; Chou, 1985; Boys 1989; Geodhart and Jansen, 1992). The inverse of T is found by conventional root finding methods to six decimal places of accuracy.