Brumbaugh was born in Huntingdon County, Pennsylvania. He grew up in rural Huntingdon County and worked in the combination country store-post office that was operated by his father. He was raised in a small pacifist faith with German roots, the German Baptist Brethren, popularly called Dunkers. Brumbaugh attended Huntingdon Normal School, graduating in 1881. A voracious reader and researcher, he later earned degrees in mechanical engineering, philosophy, and the general sciences before earning a Ph.D. from the University of Pennsylvania in 1894. He returned to Huntingdon Normal School, now renamed Juniata College, in 1895, and continued at the post until 1910. He remained closely connected to the college, returning to the position of the president in 1924.