Why would a woman have less agility and balance than a man? Ever watched the gymnastics at the Olympics?
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Male athletes have greater agility because they have "significantly greater lower body strength, vertical braking force and impulse application, knee and spine flexion, and hip abduction, as well as faster decision time and post COD stride velocity during both agility conditions compared with females."
Now balance in general might not change... but if we're talking in football terms and remaining on your feet after dealing with opposing physical forces in terms of shoulder barges and general defensive duels then there is a significant difference due to the biomechanical make-up of men and women.
Neither of the statements should be in anyway controversial. They're based on biological facts.