arXiv:2610.33859v1 Announce Type: new Abstract: Kerman's approximation to the median of the Beta$(\alpha,\beta)$ distribution is the ratio $g(\alpha)/(g(\alpha) + g(\beta))$ with $g(x) = x - 1/3$, using the first two terms of the asymptotic expansion of the median of the standard gamma distribution. We extend this elegant ratio form to the one-parameter family with $g(x) = x - 1/3 + c_1/x$, in which $c_1 = 0$ gives Kerman's. For large shape parameters along any ray $\beta = r\alpha$ with $r > 1$, the resulting approximation is eventually a lower bound on the median if $c_1 \le 8/405$ and an upper bound if $c_1 \ge 1/45$. We recommend $c_1 = 1/36$ for its simplicity: it makes $g(x) = (x - 1/6)^2/x$ a perfect square over $x$, and it is within 1% of the minimax value of $c_1$ for $\alpha, \beta \ge 1$ under several error measures, while the theoretically distinguished values $8/405$ and $1/45$ are less accurate. The approximation is then $((\alpha - 1/6)^2/\alpha)/((\alpha - 1/6)^2/\alpha + (\beta - 1/6)^2/\beta)$. Its relative error lies between -0.171% and +0.212% for all $1 \le \alpha \le \beta$, compared with up to -3.8% for Kerman's. It is more accurate than Kerman's approximation wherever the latter is defined, with relative error never more than 13/32 of Kerman's, the ratio approached asymptotically. The same $g(\alpha)$ is a useful approximation to the gamma distribution median, overestimating it by at most 0.212% for shape parameter $\alpha \ge 1$.