Harmonic Mean Calculator

This tool calculates the harmonic mean, a type of average that is particularly useful when dealing with rates or ratios and in handling datasets with extreme values.
CalculationInputs﻿﻿x1
:2.00​
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﻿﻿x2
:0.00​
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﻿﻿x3
:0.00​
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Add more values
Output﻿﻿Harmonic mean, H
:2.00​
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H=n∑i=1n1xiH = \dfrac{n}{\sum_{i=1}^{n}\frac{1}{x_i}}H=∑i=1n​xi​1​n​
Where:
﻿﻿H
 is the harmonic mean
﻿﻿n
 is the number of terms in the dataset
﻿﻿xi​
 for ﻿i=1
 to ﻿n
 is a number in the dataset
💡Why are the values unitless?
ExplanationThe harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of a dataset. It is especially relevant when dealing with rates or ratios and in handling datasets with extreme values. 
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H=n∑i=1n1xi=n1x1+1x2+...+1xnH = \dfrac{n}{\sum_{i=1}^{n}\frac{1}{x_i}} = \dfrac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}H=∑i=1n​xi​1​n​=x1​1​+x2​1​+...+xn​1​n​
Characteristics of the harmonic mean:
It is always less than or equal to the arithmetic and geometric mean for a given set of positive values
It gives more weight to smaller numbers in the dataset, providing a more representative average in skewed distributions
Interpreting the harmonic mean:
A lower harmonic mean indicates that the dataset contains some smaller numbers that significantly influence the average
In scenarios involving rates or ratios, such as speed or density, the harmonic mean provides more accurate representation 
When the harmonic mean is compared with arithmetic and geometric means, it can provide insights into the skewness and distribution characteristics of the data.
Use cases of the harmonic mean:
The harmonic mean of speeds at which you travel equal distances will give you the true average speed over the entire distance
For analyzing investment portfolios, the harmonic mean can provide a more accurate measure of central tendency when evaluating the performance of diverse assets, particularly when dealing with rates of return, by minimizing the distortion effect of outliers.
Related ResourcesArticle: "Mean, Median, and Mode: Understanding the Differences" by Robert Niles
Book: "The Art of Statistics: Learning from Data" by David Spiegelhalter
Book: "Statistics in Plain English" by Timothy C. Urdan
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