3.4: 谐波平均值

The arithmetic mean is usually skewed towards the larger values in the data set. Therefore, to avoid this inherent bias towards smaller values, the harmonic mean is used.

Take the example of the speed of a car, which is the measure of the rate of distance traveled. If the vehicle traverses the same distance back-and-forth, its average speed equals the total distance traveled divided by the total time taken. However, if the car moves with varying speeds, then the arithmetic mean is more skewed towards the larger value. Therefore, the arithmetic mean of the reciprocal speed is first calculated. Then, this quantity’s reciprocal is determined, also referred to as the harmonic mean of the original quantity.

Physical quantities with zero values should not be considered for calculating the harmonic mean because division by zero is undefined.

It can be shown that the harmonic mean of a data set with distinct positive values is always smaller than its geometric mean, which in turn is smaller than its arithmetic mean.