Harmonic mean

The harmonic mean is a type of average that is used to find an average based on a set of numbers. It is calculated by taking the reciprocal of each number, then making the sum of these reciprocals, and finally taking the reciprocal of the sum.

The harmonic mean is often used for data sets that contain at least one number that is much greater than the others. It gives more weight to the smaller values in the set and less weight to the larger values; thus, it can more accurately represent the average of the set. It is typically used for data sets with continuous data, such as distances, areas, or speeds.

Example of Harmonic mean

To illustrate the use of the harmonic mean, consider a set of three numbers: 2, 4 and 8. To calculate the harmonic mean of this set, we first need to take the reciprocal of each number:

• 1/2 = 0.5
• 1/4 = 0.25
• 1/8 = 0.125

Then, add the reciprocals together: 0.5 + 0.25 + 0.125 = 0.875. Finally, take the reciprocal of the sum: 1/0.875 = 1.14. Therefore, the harmonic mean of the set is 1.14.

Formula of Harmonic mean

The formula of harmonic mean is:

$H={\frac {n}{\sum _{i=1}^{n}{\frac {1}{x_{i}}}}}$

where H is the harmonic mean, n is the number of numbers in the set, and x_i is each of the numbers in the set.

When to use Harmonic mean

The harmonic mean is typically used when:

• The data set contains at least one number that is much greater than the others.
• The data set contains continuous data, such as distances, areas, or speeds.
• It is desired to give more weight to the smaller values in the set and less weight to the larger values.

The harmonic mean is a type of average used to find an average based on a set of numbers. It is particularly useful when the data set contains at least one number that is much greater than the others, and when it is desired to give more weight to the smaller values in the set and less weight to the larger values.

Types of Harmonic mean

• Arithmetic-Harmonic Mean: The Arithmetic-Harmonic Mean (AHM) is a method that combines the arithmetic mean and the harmonic mean. It is calculated by taking the arithmetic mean of the set of numbers, then taking the harmonic mean of the same set. Mathematically, the AHM is expressed as:

${\frac {1}{AHM}}={\frac {1}{2}}\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}+{\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{x_{i}}}\right)$

• Weighted Harmonic Mean: The Weighted Harmonic Mean (WHM) is a method that incorporates weights into the harmonic mean calculation. It is calculated by taking the harmonic mean of the set of numbers, but multiplying each number by its corresponding weight before taking the reciprocal of each number. Mathematically, the WHM is expressed as:

${\frac {1}{WHM}}={\frac {1}{n}}\sum _{i=1}^{n}w_{i}{\frac {1}{x_{i}}}$

where w_i is the weight of the i-th element in the set.

The harmonic mean is a type of average that is used to find an average based on a set of numbers. It is used for data sets with continuous data, such as distances, areas, or speeds, and gives more weight to the smaller values in the set and less weight to the larger values. There are two types of harmonic mean: the Arithmetic-Harmonic Mean (AHM) and the Weighted Harmonic Mean (WHM). The AHM is calculated by taking the arithmetic mean of the set of numbers, then taking the harmonic mean of the same set. The WHM is calculated by taking the harmonic mean of the set of numbers, but multiplying each number by its corresponding weight before taking the reciprocal of each number.

Steps of Harmonic mean

To calculate the harmonic mean, the following steps should be followed:

• Step 1: Calculate the reciprocal of each number in the set. This can be done by dividing 1 by each number.
• Step 2: Add up all of the reciprocals from Step 1.
• Step 3: Divide 1 by the sum of all of the reciprocals. This is the harmonic mean of the set.

The harmonic mean is an important measure of central tendency that is used to accurately represent the average of a data set. It is especially useful for data sets with one number that is much greater than the others, giving more weight to the smaller values in the set. Following the steps above can help calculate the harmonic mean of any set of numbers.

Advantages of Harmonic mean

• It is better than the arithmetic mean when extremes values are present in the data set. It gives more accurate results when the data set includes extreme values.
• It is used to calculate the average speed, as it takes the reciprocals of the speed values.
• It is useful in calculating the average of rates and ratios.

Disadvantages of Harmonic mean

• The harmonic mean is not defined for negative numbers, as it is impossible to take the reciprocal of a negative number.
• It is also not defined for data sets with a zero in it, as it is not possible to take the reciprocal of zero.
• The harmonic mean is not as widely used as the arithmetic mean, and is usually only used when the data set includes extreme values.

The harmonic mean is a useful tool for calculating an average when the data set includes extreme values. It gives more accurate results than the arithmetic mean by giving more weight to the smaller values in the set and less weight to the larger values. It is useful for calculating the average of rates and ratios, as well as the average speed.

Limitations of Harmonic mean

The harmonic mean has some limitations that should be taken into consideration.

• The harmonic mean cannot be used for data sets that contain negative numbers, as it will yield an imaginary result.
• The harmonic mean can be heavily influenced by outliers in the data set, as it gives more weight to the smaller numbers and less weight to the larger numbers.
• The harmonic mean is not an appropriate measure for data sets that contain zero or infinite values.

Overall, the harmonic mean is best used for data sets that contain a few numbers that are much greater than the others, as it can better represent the average of the set. However, its limitations should be taken into consideration before it is used.

Other approaches related to Harmonic mean

• Arithmetic mean: The arithmetic mean is the most common type of average. It is calculated by adding all of the numbers in the set together and dividing by the total number of numbers in the set. Mathematically, the arithmetic mean is expressed as:

${\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}$

where \bar{x} is the arithmetic mean, n is the number of numbers in the set, and x_i is each of the numbers in the set.

• Geometric mean: The geometric mean is used to calculate an average for a set of numbers with exponential growth. It is calculated by multiplying all of the numbers in the set together and taking the nth root of the product, where n is the number of numbers in the set. Mathematically, the geometric mean is expressed as:

$G={\sqrt[{n}]{x_{1}x_{2}x_{3}...x_{n}}}$

where G is the geometric mean, n is the number of numbers in the set, and x_i is each of the numbers in the set.

The harmonic mean, arithmetic mean, and geometric mean are all useful tools for calculating an average from a set of numbers. Each of these approaches has its own purpose and use in different circumstances. The harmonic mean is best used for data sets with one outlier, while the arithmetic mean is the most common approach and the geometric mean is best used for data sets with exponential growth.

Harmonic mean — recommended articles

Kurtosis — Confidence level — C chart — Central tendency — Coefficient of determination — Multidimensional scaling — Statistical significance — Probability density function — Lorenz curve

References

• Ferger, W. F. (1931). The nature and use of the harmonic mean. Journal of the American Statistical Association, 26(173), 36-40.