Heronian mean

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The Heronian mean is a type of mean or average named after the ancient Greek engineer and mathematician Hero of Alexandria. It is used in various fields of mathematics and engineering to find an average value that is more representative of the data set than the arithmetic mean in certain contexts.

Definition

The Heronian mean of two non-negative real numbers a and b is given by the formula: \[ H(a, b) = \frac{a + \sqrt{ab} + b}{3} \]

This formula combines the arithmetic mean and the geometric mean of the two numbers, providing a balance between the two.

Properties

• The Heronian mean is always between the arithmetic mean and the geometric mean of the two numbers.
• It is symmetric, meaning that \( H(a, b) = H(b, a) \).
• It is homogeneous, meaning that for any positive real number \( k \), \( H(ka, kb) = kH(a, b) \).

Applications

The Heronian mean is used in various applications where a balance between the arithmetic and geometric means is desired. Some of these applications include:

• Engineering: In certain design calculations where both the sum and the product of quantities are important.
• Statistics: In data analysis to provide a more representative average in skewed distributions.

Related Means

The Heronian mean is part of a family of means that includes:

• Arithmetic mean
• Geometric mean
• Harmonic mean

See Also

• Mean
• Arithmetic mean
• Geometric mean
• Harmonic mean
• Hero of Alexandria