I compiled a list of Bauder vital statistic records on file in NY State. Official state records were not kept until the 1880's and only a few locations sent in records from the early 1880's. I have some (most, but not all) death, marriage, and birth vital statistic dates from 1881 - (1901: birth and marriage) (1900: death). About one half of the birth records say simply male or female, the other half has names.
I tried to attach the file, but this system does not seem to accept either pdf's or word documents. Contact me with your e-mail if you would like a copy. Please specify either pdf format or word format - both are four pages in length.
Sample:
marriage cert. Edson B. Bauder on 10-26-1882 in Corfu NYS VR # 6230

death cert. Elizabeth Bauder on 9-28-1886 in Gilbert Mills NYS VR # 35632

birth cert. (male) Bauder on 4/28/1888 in Canastota NYS VR # 15209

Having the Vital Record #'s makes it easy for you to order a copy of the original document from the NYS health dept.
Even for some of the birth records that specify only male or female, the original documents would have mother and father's names.