I like some of the previously posted solutions, but they do have the drawback that you need to keep track of the ranges and means over all your data. This could be cumbersome if you have multiple data sets that will be plotted together. To fix this, I made use of the ax.get_[xyz]lim3d() methods and put the whole thing into a standalone function that can be called just once before you call plt.show(). Here is the new version:
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
import matplotlib.pyplot as plt
import numpy as np
def set_axes_equal(ax):
"""
Make axes of 3D plot have equal scale so that spheres appear as spheres,
cubes as cubes, etc.
Input
ax: a matplotlib axis, e.g., as output from plt.gca().
"""
x_limits = ax.get_xlim3d()
y_limits = ax.get_ylim3d()
z_limits = ax.get_zlim3d()
x_range = abs(x_limits[1] - x_limits[0])
x_middle = np.mean(x_limits)
y_range = abs(y_limits[1] - y_limits[0])
y_middle = np.mean(y_limits)
z_range = abs(z_limits[1] - z_limits[0])
z_middle = np.mean(z_limits)
# The plot bounding box is a sphere in the sense of the infinity
# norm, hence I call half the max range the plot radius.
plot_radius = 0.5*max([x_range, y_range, z_range])
ax.set_xlim3d([x_middle - plot_radius, x_middle + plot_radius])
ax.set_ylim3d([y_middle - plot_radius, y_middle + plot_radius])
ax.set_zlim3d([z_middle - plot_radius, z_middle + plot_radius])
fig = plt.figure()
ax = fig.add_subplot(projection="3d")
# Use this for matplotlib prior to 3.3.0 only.
#ax.set_aspect("equal'")
#
# Use this for matplotlib 3.3.0 and later.
# https://github.com/matplotlib/matplotlib/pull/17515
ax.set_box_aspect([1.0, 1.0, 1.0])
X = np.random.rand(100)*10+5
Y = np.random.rand(100)*5+2.5
Z = np.random.rand(100)*50+25
scat = ax.scatter(X, Y, Z)
set_axes_equal(ax)
plt.show()