Where the 10⁴ comes from

In a Neubauer-improved haemocytometer each large corner square is 1 mm × 1 mm and the chamber is 0.1 mm deep, so one square holds 0.1 mm³ = 0.1 µL = 10⁻⁴ mL. One cell per large square therefore corresponds to 10⁴ cells/mL before correcting for dilution. Mixing equal volumes of cells and trypan blue gives a dilution factor of 2.

Cell counts follow Poisson statistics, so the relative error of a count of N cells is about 1/√N: 100 cells gives roughly ±10%, 400 cells ±5%. Counting more squares improves precision more than recounting the same ones.