I can't see anything wrong with your hair settings, so the problem is likely to be elsewhere.
Check each modifier above the hair particle modifier and make sure their viewport settings are the same as their render settings. This means they should all be visible in the viewport and if there are any settings that are render specific (e.g. subdivision count) that they match viewport. In particular ones that add or remove geometry can break the hair cache and cause odd results.
If you want to alter subdivision between display and render, you need to either move it below the hair modifier or split it in two and have one before the hair with display and render settings equal, and one after with only the additional render subdivisions.
After playing with the blend far a while I can't find anything obvious, I slowly turned off / reset to default all the settings in the hair particle system and eventually managed to make the system render more or less the same as the viewport, however resetting just the last setting changed (brownian in the physics tab) did not fix the original group. From this it looks like the problem is some strange interaction of settings in the group.
My recommendation would be to recreate the hair systems from scratch, doing quick renders after each change until you get something satisfactory. Hopefully someone else can come along and give a better answer though.