# Harmonic mean using R with examples

In this tutorial, you will learn about what is harmonic mean, when harmonic mean is used and how to calculate harmonic mean in R.

## What is harmonic mean?

Harmonic mean is a measure of central tendency. It is useful in finding the averages involving time, rate, price and ratio.

Let $x_i, i=1,2, \cdots , n$ be $n$ positive observations then the harmonic mean of $X$ is denoted by $HM$ and is defined as

$$HM = \dfrac{n}{\sum_{i=1}^{n} \dfrac{1}{x_i}}$$

Thus, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of the observations.

## When to use harmonic mean?

Harmonic mean is often used in averaging the data about rates and ratios. Harmonic mean is commonly used in finance to calculate average price to earning ratio, average speed.

## How to calculate Harmonic mean in R?

As such there is no specific function available in base package. But you can define your own function for harmonic mean or use harmonic.mean() function from psych package.

### Harmonic mean using user-defined function

Let us define our own function to calculate harmonic mean as follows:

Harmonic.Mean <- function(x, na.rm = TRUE) {
if (na.rm == FALSE & any(is.na(x))) {
avg <- NA
} else{
avg <- 1 / mean(1 / x, na.rm = TRUE)
}
return(avg)
}

### Using harmonic.mean() function from psych package

One can use the harmonic.mean() function from psych package. First you need to install the psych package using install.packages("psych"). Once the package is installed, then load the package using library(psych).

The syntax of the harmonic.mean() function is

harmonic.mean(x,na.rm=TRUE,zero=TRUE)

where,

• x : a vector, matrix or data.frame,
• na.rm : na.rm=TRUE remove NA values before processing,
• zero : if TRUE, then if there are any zeros, return 0, else, return the harmonic mean of the non-zero elements

For more information about psych package check this link psych package.

Note that, the harmonic mean is not useful if there are elements that are zero.

## Numerical example on harmonic mean using R

### Example 1: Harmonic Mean Using R

A bus driver travels total 30 miles. He travel first 10 miles with speed of 30 miles per hour, second 10 miles with the speed of 35 miles per hour and last 10 miles with the speed of 45 miles per hour. What is the average speed?

#### Harmonic mean using formula

The harmonic mean is the reciprocal of the mean of reciprocal of the observations. That is

$$HM = \dfrac{1}{\text{mean(1/x)}}$$

# create a data vector
x <- c(30, 35, 45)
# calculate harmonic mean
HM <- 1/mean(1/x)
HM
[1] 35.66038

The average speed is 35.6603774 miles per hour.

#### Harmonic mean using user-defined function

Earlier we have defined the function for harmonic mean as Harmonic.Mean(x). Let us calculate the harmonic mean using user-defined function Harmonic.Mean(x).

# create a data vector
x <- c(30, 35, 45)
Harmonic.Mean(x)
[1] 35.66038

The average speed is 35.6603774 miles per hour.

#### Harmonic mean using psych package

One can use harmonic.mean() function from psych package to calculate harmonic mean.

# load the package
library(psych)
Warning: package 'psych' was built under R version 4.0.5
# use the function to compute harmonic mean
harmonic.mean(x)
[1] 35.66038

The average speed is 35.6603774 miles per hour.

### Example 2: Harmonic Mean using R

Compute the harmonic mean of data 10,25,36,23,NA,17.

#### Harmonic mean using formula

The harmonic mean is the reciprocal of the mean of reciprocal of the observations. That is

$$HM = \dfrac{1}{\text{mean(1/x)}}$$

# create a data vector
x <- c(10,25,36,23,NA,17)
# calculate harmonic mean
HM <- 1/mean(1/x,na.rm=TRUE)
HM
[1] 18.51306

The harmonic mean of given data is 18.5130628.

Note that the default value of na.rm is FALSE in mean() function. Therefore, to remove NA values before processing use the argument na.rm=TRUE.

#### Harmonic mean using user-defined function

Use the user-defined function Harmonic.Mean(x) with additional argument na.rm=TRUE.

# create a data vector
x <- c(10,25,36,23,NA,17)
Harmonic.Mean(x)
[1] 18.51306

The harmonic mean of given data is 18.5130628.

Note that the default value of na.rm is TRUE in our user-defined function Harmonic.Mean() function. Therefore, no need to specify the argument na.rm=TRUE.

#### Harmonic mean using psych package

Use harmonic.mean() function from psych package to calculate harmonic mean.

# load the package
library(psych)
# use the function to compute harmonic mean
harmonic.mean(x)
[1] 18.51306

The harmonic mean of given data is 18.5130628.

Note that the default value of na.rm is TRUE in harmonic.mean() function from psych package. Therefore, the NA values are removed before processing.

## Endnote

In this tutorial, you learned about what is harmonic mean and how to compute harmonic mean for the data using R.

To learn more about descriptive statistics using R, please refer to the following tutorials:

Hopefully you enjoyed this tutorial on harmonic mean using R.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.