Replacing column values in pandas
As Emre already mentioned, you may use the groupby function. After that, you should apply reset_index to move the MultiIndex to the columns:
import pandas as pd

df = pd.DataFrame( [ ['Hospital 1', 'District 19', 5],
                     ['Hospital 1', 'District 19', 10],
                     ['Hospital 1', 'District 19', 6],
                     ['Hospital 2', 'District 10', 50],
                     ['Hospital 2', 'District 10', 51]], columns = ['Hospital_ID', 'District_ID', 'Employee'] )

df = df.groupby( ['Hospital_ID', 'District_ID'] ).mean()

which gives you:
  Hospital_ID  District_ID  Employee
0  Hospital 1  District 19       7.0
1  Hospital 2  District 10      50.5