When distances were carefully measured from earth to globular clusters above and below the milky way plane (where our view of them is not obscured by interstellar dust and gas), their distribution was found to be

They were distributed in a Spherical way in a space of about 35 Kpc¹ (inside the Halo) around the Milky Way plane.

Explanation:

Globular clusters in the Milky Way belong to two groups, which are in the thick disk and in a halo that is around the Milky Way plane.

The stars that belong to the halo have a low metallicity² compared to the members from globular clusters in the thick disk that rise a distintion of different populations based on the age of the stars, chemical abundances and kinematics, giving a better understanding of the formation and evolution of the Milky Way. In the other hand, knowing the distances to the globular clusters allows to schematize the structure of the Galaxy.

Stars that are members of the halo are considered as old (Population II³), because they are metal-poor. The reason of this is that when they were formed, the interstellar media has less enrichment of heavy elements (elements heavier than Helium).

According with the theories of formation and evolution of the Galaxy, the stars from the globular clusters in the halo were formed first, which can be confirmed when it is compared to the chemical composition of its members against the chemical composition of stars that are in the thin disk, thick disk or bulge of the Galaxy.

¹Kiloparsec (Kpc): Astronomical unit to measure distances.

²Metallicity: the abundance of heavier elements against the presence of Helium or Hydrogen.

³Population II: stars that are metal-poor and have differential velocities.