Here's my best shot. If you have n numbers chosen at random from a normal distribution, then the middle of the resulting Maximum function should occur where F(x)^n=0.5 where F(x, Md, Sd) is the cumulative distribution function. F(x) = .5*(1/7) = .9057. Looking up the inverse function of F(x) = .9057, x = Md + 1.315 Sd. If I use the one sigma point on the original distribution, I get F(x)^7 = .8413 which is the value of Z on the normal distribution that represents one sigma from the mean. F(x) = 1.971 so Mw(1 sigma) = Mw + Sw = Md + 1.971 Sd. Subbing in Mw = Md + 1.315 Sd I get Sw = .6559 Sd.
I ran this empirically from number of days equals one to seven and the shape of the curve is right, but my means are a hair low and my sigmas are a bit high. I don't know if that helps or not.