The "ca. 1806" birthdate appears to be a guess based on her marriage date.

1.  Anon.  1966.  Marriage Index:  Ohio, 1789-1850.  Liahona Research, Orem, UT (Broderbund CD-400):

Frazier, Henry

Stroup, Barbara Ann

10 Jul 1828

Clermont Co.

2.  1830 Census Index and Digital Images (online at Ancestry.com; Image #3 of 14; extracted by Diana Gale Matthiesen):¤•

1830

OH

Clermont Co.

Goshen Twp.

p. 297

Ln. 23

Henry Frazier

100 001 - 100 010

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	4 or under	1825-1830	= Son A (b. 1828-30)
1 male	30-39	1790-1800	= Henry
1 female	4 or under	1825-1830	= Daughter A
1 female	20-29	1800-1810	= Barbara

Listed three lines below Barbara's sister, Susanna (STROUP) MORRIS, who is three lines below Barbara's father, Jacob STROUP (Sr.), and three lines above Barbara's brother, Peter STROUP, who is one line above Barbara's brother, Jacob STROUP (Jr.), who is one line above Barbara's sister, Elizabeth (STROUP) DOUGHMAN. 

3.  1840 Census Index and Digital Images (online at Ancestry.com; Image #9-10 of 22; extracted by Diana Gale Matthiesen):¤•

1840

OH

Clermont Co.

Goshen Twp.

pp. 260B-261A

Ln. 27

Henry Frazier

001 000 100 - 010 000 100

0100000

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	10-14	1825-1830	= Son A (b. 1828-30)
1 male	40-49	1790-1800	= Henry
1 female	5-9	1830-1835	= Daughter A (b. 1830)
1 female	40-49	1790-1800	= Barbara
1	person engaged in agriculture

4.  Census Index:  U.S. Selected Counties, 1850 (Broderbund CD-317):  can't find, and I checked all FRAZIER/etc. households in Clermont Co. to see if Barbara was living as a widow with one of her children.

5.  AIS (Accelerated Index System) 1860 Census Index (online at Ancestry.com):  not found — the Henry FRAZER in Clermont Co. is age 41, wife Manerva.

6.  Family Quest Archives:  Ohio 1870 Census Index.  Heritage Quest ACD-0014:  not found.

7.  1880 United States Census and National Index (LDS Family History Resource File: CD-ROM Library):  not found.

8.  LDS.  Family Search: Internet Genealogy Service:  AF - Ancestral File (online at FamilySearch.org).

9.  WorldConnect / Ancestry World Trees (online at RootsWeb.com/Ancestry.com).