You can not always find an exact birth-date, Look at full date when Census was taken typically listed on top of the page for US Census records. 9 Jul 1870 she is marked as 4, 1 Jun 1880 she is 13 (Her siblings Thomas and Julia ages are switched on the records you linked from 1870 - 1880). Do you have the 1900 Census? Can you get death certificate for her?
This can help narrow things down, on certain Census records people listed a month and year of birth. US Census records are notoriously wrong on ages a lot.
Now with the ages on the census records what is the range, by this time hopefully you have a plus/minus one for the year. Now we are uncertain of exact year, but closer
Unless you find Birth or Death with Exact date of birth, then the year is uncertain. Sometime Obituaries can give you the information. or you may find evidence somewhere where an age is stated like 24 Years, 8 months, 28 Days you can then calculate back to a about mm/dd/yyyy.
Also look at the birth years and/or dates of siblings, this has helped me narrow down the proper half of a year when the birth could have occurred.