6. Pythagorean Means

Suppose we have data points \(x_1, x_2, \dots, x_n\). The aritmetic mean, \(M_A\), which is the mean we are most familiar with is defined as follows,

\[ M_A = \frac{x_1 + x_2 + \dots + x_n}{n} \]

There are two other pythagorean means: the harmonic mean and the geometric mean. We will call these \(M_H\) and \(M_G\) respectively. They are defined as follows,

\[\begin{split} \begin{align} M_G &= \sqrt[n]{x_1 \cdot x_2 \dots x_n}, \\ M_H &= \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}} \end{align} \end{split}\]

7. When should the geometric mean be used?

The geometric mean should be used when your data can be naturally multiplied together. For example when working out the average return from compounding interest.

Let us suppose we invest £100 and get 10% return for 50 years (unrealistic I know). Calculate the aritmetic and geometric mean for this data.

import numpy as np
import matplotlib.pyplot as plt

n = 50
x = np.array([100 * 1.10 ** i for i in range(n)])
am = x.sum() / n
gm = x.prod() ** (1 / n)
hm = n / (1 / x).sum()

fig, ax = plt.subplots()
ax.plot(x)

mean_types = ["Arithmetic Mean", "Geometric Mean", "Harmonic Mean"]
colors = ["tab:orange", "tab:green", "tab:purple"]
for mean, mean_type, color in zip([am, gm, hm], mean_types, colors):
    ax.hlines(mean, 0, n, label=mean_type, linestyle="--", color=color)
    ax.legend()

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[1], line 1
----> 1 import numpy as np
      2 import matplotlib.pyplot as plt
      4 n = 50

ModuleNotFoundError: No module named 'numpy'