Harmonic mean

Harmonic Mean is a type of average that is often used in situations where the average rate is desired. It is the reciprocal of the arithmetic mean of the reciprocals of a set of observations.

Definition

The harmonic mean (H) of a set of observations is defined as:

H = n / (1/x1 + 1/x2 + ... + 1/xn)

where:

• n is the number of observations
• xi are the observations

Properties

The harmonic mean has several important properties:

1. It is always less than or equal to the arithmetic mean.
2. It is equal to the arithmetic mean when all observations are equal.
3. It is undefined when any observation is zero.

Applications

The harmonic mean is often used in situations where the average rate is desired. For example, it is used in the calculation of average speed when the speed varies over different segments of a journey. It is also used in the calculation of average price when the price varies over different quantities of a commodity.

See also

• Arithmetic mean
• Geometric mean
• Mean
• Average speed
• Average price

• 
  Mathematical Means
• 
  Harmonic Mean Graphical Computation
• 
  Crossed Ladders
• 
  Harmonic Mean for Beta Distribution for Alpha and Beta Ranging from 0 to 5 - J. Rodal
• 
  (Mean - Harmonic Mean) for Beta Distribution versus Alpha and Beta from 0 to 2 - J. Rodal
• 
  Harmonic Means for Beta Distribution Purple=H(X), Yellow=H(1-X), Smaller Values Alpha and Beta in Front - J. Rodal
• 
  Harmonic Means for Beta Distribution Purple=H(X), Yellow=H(1-X), Larger Values Alpha and Beta in Front - J. Rodal