The cumulative t-distribution of Student is used
in the T-tester model calculator to test the difference
between the means of data sets.
The confidence interval of the difference of mean values of
data series is shown in green color. This interval is used
by T-tester model calculator in Student's t-test
The t-Tester software model 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).
The flowers are here