According to the United States Census Bureau, the town has a total area of 3.5 km² (1.4 mi²). 3.5 km² (1.4 mi²) of it is land and none of it is covered by water.
There are 1,369 households out of which 27.9% have children under the age of 18 living with them, 52.8% are married couples living together, 9.5% have a female householder with no husband present, and 35.0% are non-families. 29.9% of all households are made up of individuals and 8.7% have someone living alone who is 65 years of age or older. The average household size is 2.24 and the average family size is 2.79.
In the town the population is spread out with 22.2% under the age of 18, 7.7% from 18 to 24, 29.7% from 25 to 44, 28.0% from 45 to 64, and 12.4% who are 65 years of age or older. The median age is 39 years. For every 100 females there are 92.9 males. For every 100 females age 18 and over, there are 91.5 males.
The median income for a household in the town is $41,581, and the median income for a family is $53,854. Males have a median income of $41,538 versus $24,662 for females. The per capita income for the town is $24,537. 2.5% of the population and 0.0% of families are below the poverty line. Out of the total people living in poverty, 0.0% are under the age of 18 and 5.1% are 65 or older.
On July 18, 1862, Newburgh was the first town north of the Mason-Dixon Line to be captured by the Confederate forces during the War Between the States. Brig. General Adam R. Johnson, with a guerrilla band, crossed the Ohio River and confiscated supplies and ammunition without a shot being fired.
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