The harmonic mean is a type of average commonly used when dealing with rates or ratios, especially when the values are defined in terms of a unit per quantity, such as speed (distance per time) or efficiency (output per unit input). Mathematically, the harmonic mean of a set of n non-zero numbers is defined as the reciprocal of the arithmetic mean of their reciprocals. In formula terms, it is expressed as:
H = n / (1/x₁ + 1/x₂ + ... + 1/xₙ).
This calculation emphasizes smaller values in the dataset more than larger values, making it particularly useful when one or more small values would otherwise be masked by an arithmetic mean.
A classic example of the harmonic mean in action is in calculating average speed over equal distances. If a car travels a certain distance at 60 km/h and returns over the same distance at 40 km/h, the average speed is not 50 km/h (the arithmetic mean) but 48 km/h, calculated using the harmonic mean. The harmonic mean ensures more accurate representations in such scenarios because it accounts for the time spent at each speed rather than just the numerical values of the speeds themselves. Due to its sensitivity to small values, it is also applied in finance, such as averaging price-to-earnings (P/E) ratios.
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The harmonic mean is a type of average commonly used when dealing with rates or ratios, especially when the values are defined in terms of a unit per quantity, such as speed (distance per time) or efficiency (output per unit input). Mathematically, the harmonic mean of a set of n non-zero numbers is defined as the reciprocal of the arithmetic mean of their reciprocals. In formula terms, it is expressed as:
H = n / (1/x₁ + 1/x₂ + ... + 1/xₙ).
This calculation emphasizes smaller values in the dataset more than larger values, making it particularly useful when one or more small values would otherwise be masked by an arithmetic mean.
A classic example of the harmonic mean in action is in calculating average speed over equal distances. If a car travels a certain distance at 60 km/h and returns over the same distance at 40 km/h, the average speed is not 50 km/h (the arithmetic mean) but 48 km/h, calculated using the harmonic mean. The harmonic mean ensures more accurate representations in such scenarios because it accounts for the time spent at each speed rather than just the numerical values of the speeds themselves. Due to its sensitivity to small values, it is also applied in finance, such as averaging price-to-earnings (P/E) ratios.
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