keisoku

Noncentral t Percent Point

Enter a probability p, tail mode, degrees of freedom nu and noncentrality delta to find the noncentral t quantile t by bisection, plus CDF, upper tail, density, mean and variance.

Input

Find the percent point (quantile) of the noncentral t distribution. Enter the probability p, the tail mode, the degrees of freedom nu and the noncentrality delta.

Tail mode

Finds the t that satisfies P(T ≤ t) equals p.

Enter a value greater than 0 and less than 1.

Enter a positive value.

A real number for the shift. A value of 0 gives the central t distribution.

Result

Percent point for Lower tail, p equals 0.95, nu equals 10, delta equals 2

4.35747638

Degrees of freedom nu

10

Noncentrality delta

2

Probability p

0.95

Lower tail P(T ≤ t)

0.95

Upper tail P(T ≥ t)

0.05

Density f(t)

0.05751984

Mean

2.16744462

Variance

1.55218384

Probability density function

Cumulative distribution function

How it works

  • The noncentral t distribution is defined from a standard normal variable Z and a chi-square variable V with nu degrees of freedom as T equals (Z plus delta) divided by sqrt(V over nu), where delta is the noncentrality and nu the degrees of freedom.
  • The cumulative distribution function is evaluated as a Poisson-weighted series, a mixture of central distributions. Poisson weights with mean lambda equals delta squared over 2 are combined with regularized incomplete beta functions.
  • The percent point is found by bisection on the CDF, which is monotonically increasing in t. The lower mode solves P(T ≤ t) equals p and the upper mode solves P(T ≥ t) equals p.
  • The mean exists when nu greater than 1 and equals delta multiplied by sqrt(nu over 2) multiplied by gamma((nu minus 1) over 2) divided by gamma(nu over 2). The variance exists when nu greater than 2; otherwise these are undefined.
  • The regularized incomplete beta function uses a continued fraction (Lentz method), the log gamma function uses the Lanczos approximation, and the standard normal CDF uses a rational approximation of the error function. When delta equals 0 the result matches the central t distribution.

Reviews

Tell us what you think of this calculator.

Write a review

  1. Home

  2. Noncentral t Percent Point