Harmonic Mean definition, formula and applications

The harmonic mean is the inverse of the arithmetic mean of the reciprocals of the observations of a set. It is a type of average generally used for numbers that represent a rate or ratio such as the precision and the recall in information retrieval and can be described as the reciprocal of the arithmetic mean of the reciprocals of the data.

Formula of harmonic mean

The formula for harmonic mean is,

where, H= Harmonic mean

n= number of observations

Mathematical example

Aassume a vehicle travels 10 miles at 60 mph (this would take 10 minutes) and then travels another 10 miles at 30 mph (this would take 20 minutes). So, 20 miles were traveled in 30 minutes which means the vehicle’s average speed should be 40 mph. Now, let’s compare this average speed to the arithmetic mean of the two speeds

The arithmetic mean results in a noticeably higher speed. 20 miles at 45 mph would take 26.67 minutes; not a match. Now, let’s try the harmonic mean of the two speeds

This is happen because arithmetic average can not deal with outlier and in this case harmonic average properly work. And it is considered to be the true average compared to the arithmetic average and this can easily be proven by calculating average speed in above.

When to use Harmonic mean

The harmonic average is suitable when the values are pertaining to the rate of change per unit time, such as speed, number of hems produced per day, contracts completed per year, etc. In general, it is suitable for time, speed, rates, prices, etc.

There are some merits as following,
• Harmonic mean is based on all observations of a set.
• It is a good mean for a highly variable series.
• It gives more weightage to the small values and leas weightage to the large value.
• It is better than the weighted mean since, in this, values are automatically weighted.