Here are some more records of the Mailes/Males family I found. There are still problems with the son named William since his name is William A. Males in 1880, and the mother Margaret J. is from IL instead of MO, but since the family was in Maricopa Co, AZ in 1920 I think it's your family.

1900 United States Federal Census
Mathewson, Canadian, Oklahoma
Household Members: Name Age
M J Malis (or Males) 47 w f Apr 1853 md 28y 11ch 7liv IL IRE TN farmer
Sarah Malis 25 dau w f Apr 1875 AR AR IL
William Malis 22 son w m May 1878 AR AR IL farm laborer
Sorvana Malis 20 dau w f Feb 1880 AR AR IL
Armmie Malis 14 dau w f Jun 1885 TX AR IL
John Malis 9 son w m Mar 1891 TX AR IL
Gus Chapman 21 son-in-law w m 1875? md US unk unk farm laborer
Nettie Chapman 18 dau w f Apr 1882 och oliv AR AR IL
Major Faut 31 son-in-law w m Mar 1869 md 7y TX AL AL farm laborer
Ella Faut 24 dau w f Oct 1875 5ch 4liv AR AR IL
Chester Faut 6 grson w m Mar 1894 TX TX AR
Vesta Faut 3 grdau w f Jun 1896 TX TX AR
Jessie Faut 1 grdau w f Jan 1899 TX TX TX


1880 United States Federal Census
Brawley, Scott, Arkansas
Household Members: Name Age
Archibald Males 28 w m farmer AR IN AR
Margarette Males 26 w f wife keeping house IL IL IL
Ella Males 6 w f dau at home AR AR IL
Sarah Males 5 w f dau at home AR AR IL
Calidonia Males 3 w f dau at home AR AR IL
William A. Males 2 w m son at home AR AR IL
Arizonia Males 4M w f dau at home AR AR IL

Arkansas Marriages, 1851-1900
Name: ARCHIBALD MALES
Spouse: MARGRET J. BEAN
Marriage Date: 12 Aug 1872
County: Washington
State: AR