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Bessel's Statistical Formula

Let x^__1 and s_1^2 be the observed mean and variance of a sample of N_1 drawn from a normal universe with unknown mean mu_((1)) and let x^__2 and s_2^2 be the observed mean and variance of a sample of N_2 drawn from a normal universe with unknown mean mu_((2)). Assume the two universes have a common variance sigma^2, and define

w^_=x^^_1-x^__2
(1)
omega=mu_((1))-mu_((2))
(2)
N=N_1+N_2.
(3)

Then

t=(w^_-omega)/(sigma_w/sqrt(N))=(w^_-omega)/(sqrt((sum_(i=1)^(n)(w_i-w^_)^2)/(N(N-1))))
(4)

is distributed as Student's t-distribution f_n(t) with n=N-2.

See also

Student's t-Distribution

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References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 186, 1951.

Referenced on Wolfram|Alpha

Bessel's Statistical Formula

Cite this as:

Weisstein, Eric W. "Bessel's Statistical Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BesselsStatisticalFormula.html

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