Hi there. In this math post, I talk about the harmonic mean. The harmonic mean is an average like the usual arithmetic mean but this harmonic mean is not as well known.
This harmonic mean does have finance applications as shown by this Investopedia page.
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Given a set/list of numbers the harmonic mean is the amount of numbers divided by the sum of each reciprocal of numbers.
In general math notation, given a list of
Note that n is how many numbers in the list of numbers. In addition, the reciprocal of a number j is 1 divided by j.
This harmonic mean is different from the usual arithmetic mean and is different from the geometric mean. In the usual arithmetic mean for averages, we add all the numbers and divide by how many numbers there are. With the geometric mean, we multiply all the numbers together and take the n-th root of this product.
For this example, we have a list of five numbers. These five numbers (n = 5) are 1, 4, 10, 5 and 2. The calculations for the harmonic mean would be as follows:
Here is some supplementary Python code for obtaining the harmonic mean. It is not too difficult with the use of the Numpy library. The code should be self explanatory.
# Harmonic Mean In Python
import numpy as np
numbers_list = np.array([1, 4, 10, 5, 2])
# Take reciprocal (element-wise)
1 / numbers_list
array([1. , 0.25, 0.1 , 0.2 , 0.5 ])
# Harmonic mean: 5 divided by the sum of reciprocals
len(numbers_list) / sum(1 / numbers_list)
2.439024390243903
The Harmonic Mean As A Python Function
# Define Harmonic mean function:
def harmonic_mean(num_list, precision = 2):
return round(len(num_list) / np.sum(1 / np.array(num_list)), precision)
harmonic_mean(numbers_list, precision = 5)
2.43902
The harmonic mean in R requires a little bit less typing compared to Python. Defining a function in R is different compared to Python.
# Create vector of numbers
numbers <- c(1, 4, 10, 5, 2)
# Harmonic mean: 5 divided by sum of reciprocals:
length(numbers) / (sum(1/numbers))
2.439024
Harmonic Mean Function In R
harmonic_mean <- function(num_list, precision = 2) {
return(round(length(num_list) / sum(1/num_list), precision))
}
harmonic_mean(numbers, precision = 3)
2.439
Thank you for reading.