Arithmetic mean or geometric mean ?

A quick sympy/numpy toy exercice.

In this short post I want to show few things about arithmetic mean and geometric mean. We’ll use 2 variables that span the [0–1] segments. See this post as a toy snippet for sympy, numpy and matplotlib.

Analytical solution with sympy

Let’s first use sympy to check in which case one is greater that the other

import sympy as sp
x, y = sp.symbols("x, y")
geom = sp.sqrt(x * y)
arit = (x + y)/2
display(geom) # jupyter display function
display(arit)

Now let’s create a new equation as the difference between them:

poly = arit - geom
display(poly>=0)

Now lets plot the xy plane and highlight a blue zone where the artihmetic mean is greater than the geometrical mean.

sp.plot_implicit(poly>=0, x_var=(x, -0.1, 1.1), y_var=(y, -0.1, 1.1))

So sympy tells use that the geometric mean is always smaller than the aritmetic mean.

Hand analytical solution