To do it as a circular problem you'd define Theta(birthday) as (360*Julian(Birthday)/365.25) for degrees, or (2π(Julian(Birthday)/365.25) ) for radians, average the thetas, take remainder after dividing by 360 or π. And that would take care of the Dec31/Jan 1 difficulty.
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I think you want a 2D value for the average: both a vector direction and a length. 100 data points all the same should have a greater length than 100 different ones that barely do more than cancel out.
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Also, the Sardaukar reference I stumbled across is is in appendix 2. "Talmudic Zabur surviving on Salusa Secundus" (where Sardaukar are recruited from).
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