3.4:
Harmonic Mean
The harmonic mean is one of the three Pythagorean means. It is calculated by taking the reciprocal of the arithmetic mean of reciprocals.
The harmonic mean is used to calculate the average of ratios or rates, such as speed of a vehicle, or in business to find out the price-to-earnings ratio of a company.
Consider a car moving from point A to B with a speed of 30 miles per hour. Then to point C with 70 miles per hour and finally return to point A with 80 miles per hour.
The arithmetic mean of the speed in the above journey is 60 miles per hour, which is skewed towards the larger values. In contrast, the harmonic mean avoids this bias in data by giving more weight to the smaller values.
Begin by taking the reciprocal of the given values. Then calculate their arithmetic mean. Finally, take the reciprocal of this arithmetic mean to get the harmonic mean.
Note that the harmonic mean should not be used if any of the data values is zero.
3.4:
Harmonic Mean
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.