t-Tester graphics in accordance to data in the
The t-Tester software uses a numerical solution of the cumulative t- distribution function and from that it derives the t- probability density by numerical differentiation.
Numerical approximation of Student's t- (probability) distribution.
T = reference value for the X-variable following the t-distribution, Po = intermediate probability variable,
Pc(T) = cumulative probability of T, and Pc(T) = P(X < T) ,
the following equations hold:
N (degrees of freedom) even :
Po = sin(z) [1 + cos2(z) / 2 + 3 cos4(z) / 8 + 15 cos6(z) / 48 + 105 cos8(z) / 384 + . . . ]
N (degrees of freedom) uneven (odd) and >1 :
Po = 2Z/pi + (2/pi)cos(z).sin(Z) [1 + 2 cos2(z) / 3 + 8 cos4(z) / 15 + 48 cos6(z) / 105 + . . . ]
N (degrees of freedom) = 1 :
Po = 2z / pi
z = arctan(T / sqrt(N) , and pi = 22 / 7
when T is positive: Pc(T) = Po + (1-Po) / 2
when T is negative: Pc(T) = (1-Po) / 2
The number of terms between the parentheses [ ] to be used is N / 2 when N is even and (N-1) / 2 when N is uneven (odd).
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