The 50' (read as 50 minutes) means 50 60ths of a degree, so that part makes sense. And not sure about the rounding, but 38N, 97W just "looks" more central look

The 50' (read as 50 minutes) means 50 60ths of a degree, so that part makes sense. And not sure about the rounding, but 38N, 97W just "looks" more central looking at the map, so I'm guessing they just played around with the numbers for a few minutes to get it to look nice. I've definitely done this, picking arbitrary points to place a marker on a map for, say, a big city, county or national park.