Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean

The average of the collection of values present in any set is referred to as the mean. It’s used in finance to compute growth rates and risk factors, in biology to calculate cell division rates, and in math to solve linear transformations.

Arithmetic mean, Geometric mean, and Harmonic mean are the letters AM, GM, and HM, respectively.

• The Arithmetic Mean (AM) is the mean or average of a set of numbers that is calculated by summing all of the terms in the set and dividing the sum by the total number of terms.

• Geometric or GM The mean value or core term in a group of integers in geometric progression is called the mean. The nth root of the product of all the terms in a geometric sequence with ‘n’ terms is computed as the geometric mean of the sequence.

• The Harmonic Mean (HM) is one of the methods for calculating the average. The harmonic mean is calculated by multiplying the number of values in the sequence by the sum of the terms’ reciprocals.

Relationship Between AM GM HM

Relationship between AM GM HM helps you comprehend the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM). The product of the arithmetic and harmonic means equals the square of the geometric mean.

AM × HM = GM².

The arithmetic mean is greater than the geometric mean, and the geometric mean is greater than the harmonic mean among the three means. The statement that the value of AM is greater than the value of GM and HM explains the relationship between AM, GM, and HM. The arithmetic mean is greater than the geometric mean, and the geometric mean is greater than the harmonic mean for the same set of data points. The following expression can be used to represent the relationship between AM, GM, and HM.

AM > GM > HM

Formula for Relation between AM GM HM

The product of arithmetic mean and harmonic mean equals the square of the geometric mean is the formula describing the relationship between AM, GM, and HM. This can be expressed as this expression.

AM × HM = GM²

Let us try to deduce this formula in order to better comprehend it.

AM × HM = [(a+b)/2] × [2ab/(a+b)] = ab

ab = √(ab)² = GM²

Conclusion

The product of the arithmetic and harmonic means equals the square of the geometric mean.

The product of the arithmetic and harmonic means equals the square of the geometric mean.

AM × HM = GM².

The arithmetic mean is greater than the geometric mean, and the geometric mean is greater than the harmonic mean among the three means.

AM > GM > HM