the harmonic mean is a numerical average calculated by dividing the number of observations or entries in the series by the reciprocal of each number in the series thus the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals 1
for example to calculate the harmonic mean of 1 4 and 4 you would divide the number of observations by the reciprocal of each number as follows 3 1 1 1 4 1 4 3 1 5 2 frac 3 left frac 1 1 frac 1 4 frac 1 4 right frac 3 1 5 2 11 41 41 3 1 53 2the harmonic mean has uses in finance and technical analysis of markets among others 2understanding the harmonic meanthe harmonic mean helps to find multiplicative or divisor relationships between fractions without worrying about common denominators harmonic means are often used in averaging things like rates e g the average travel speed given a duration of several trips the weighted harmonic mean is used in finance to average multiples like the price to earnings p e ratio because it gives equal weight to each data point 3 using a weighted arithmetic mean to average these ratios would give greater weight to high data points than low data points because p e ratios aren t price normalized while the earnings are equalized the harmonic mean is the weighted harmonic mean where the weights are equal to 1 the weighted harmonic mean of x1 x2 x3 with the corresponding weights w1 w2 w3 is given as i 1 n w i i 1 n w i x i displaystyle frac sum n i 1 w i sum n i 1 frac w i x i i 1n xi wi i 1n wi the reciprocal of a number n is simply 1 n harmonic mean vs arithmetic mean and geometric meanother ways to calculate averages include the simple arithmetic mean and the geometric mean taken together these three types of mean harmonic arithmetic and geometric are known as the pythagorean means the distinctions between the three types of pythagorean mean makes them suitable for different uses an arithmetic average is the sum of a series of numbers divided by the count of that series of numbers if you were asked to find the class arithmetic average of test scores you would simply add up all the test scores of the students and then divide that sum by the number of students for example if five students took an exam and their scores were 60 70 80 90 and 100 the arithmetic class average would be 80 the geometric mean is the average of a set of products the calculation of which is commonly used to determine the performance results of an investment or portfolio it is technically defined as the nth root product of n numbers the geometric mean must be used when working with percentages which are derived from values while the standard arithmetic mean works with the values themselves the harmonic mean is best used for fractions such as rates or multiples example of the harmonic meanas an example take two firms one has a market capitalization of 100 billion and earnings of 4 billion p e of 25 and the other has a market capitalization of 1 billion and earnings of 4 million p e of 250 in an index made of the two stocks with 10 invested in the first and 90 invested in the second the p e ratio of the index is using the wam p e 0 1 25 0 9 250 227 5 using the whm p e 0 1 0 9 0 1 25 0 9 250 131 6 where wam weighted arithmetic mean p e price to earnings ratio whm weighted harmonic mean begin aligned text using the wam p e 0 1 times25 0 9 times250 227 5 text using the whm p e frac 0 1 0 9 frac 0 1 25 frac 0 9 250 approx 131 6 textbf where text wam text weighted arithmetic mean text p e text price to earnings ratio text whm text weighted harmonic mean end aligned using the wam p e 0 1 25 0 9 250 227 5using the whm p e 250 1 2500 9 0 1 0 9 131 6where wam weighted arithmetic meanp e price to earnings ratiowhm weighted harmonic mean as can be seen the weighted arithmetic mean significantly overestimates the mean price to earnings ratio advantages and disadvantages of the harmonic meanthe harmonic mean is effective because it incorporates all the entries in the series and remains impossible to compute if any item is disallowed using the harmonic mean also allows a more significant weighting to be given to smaller values in the series and it can also be calculated for a series that includes negative values in comparison with the arithmetic mean and geometric mean the harmonic mean generates a straighter curve however there are also a few downsides to using the harmonic mean first and foremost because it requires using the reciprocals of the numbers in the series the calculation of harmonic mean can be complex and time consuming in addition because of the impossibility of finding the reciprocal of zero it is not feasible to calculate the harmonic mean if the series contains a zero value finally any extreme values on the high or low end of the series have an intense impact on the results of the harmonic mean 