The harmonic mean (archaic: subcontrary mean) is a specialised average of a set of numbers. It’s one of the three Pythagorean means that gives the most accurate average of the three. Although they appear to be comparable at first, the harmonic mean is more difficult to solve than the arithmetic. The harmonic mean calculates the reciprocal of the arithmetic mean of reciprocals, whereas the arithmetic mean calculates the reciprocal of the arithmetic mean of reciprocals.
Because it is unaffected by fluctuations, the harmonic mean is commonly employed in scenarios involving quantitative data, such as calculating the average of rates or ratios.
The harmonic mean of a set of non-zero positive numbers is calculated using the formula:
where n is the number of items and X1..X2 are the numbers from 1 to n.
Simply divide the number of items in the set by the sum of their reciprocals to get the answer. In this harmonic mean calculator, we use the formula above.
If you’re asked to discover the harmonic mean of a series of numbers like 3, 12, 20, and 24, the first thing you should do is look for a common denominator. When you sum up the reciprocals, it’ll come in useful. All of the numbers are divisible by 120 in this situation. Next, add the reciprocals to get the sum:
1/3 + 1/12 + 1/20 + 1/24 = 40/120 + 10/120 + 6/120 + 5/120 = 61/120.
Because the set contains four elements, your final computation should be:
4/ (61/120) = 4 x 120/61 = 7.87.
It’s worth noting that if one of the numbers in a set is 0, you won’t be able to find the harmonic mean.
The math is done for you by our harmonic mean calculator!
The reciprocal of the weighted average of the reciprocals of a weights dataset corresponding to a set of numbers is the weighted harmonic mean. It’s calculated using the following formula:
It’s best to use the weighted harmonic mean when averaging multiples with price in the numerator. For example, if you use a weighted arithmetic average to calculate the price-earnings ratio, the results will be skewed upwards because higher data points are given more weight. The weighted harmonic mean, on the other hand, gives equal weight to all data points, resulting in the most accurate average.
Let’s imagine you need to figure out the P/E stock index for three different companies. The first has a market capitalization of $10 billion and $1 billion in earnings; the second has a market capitalization of $27 billion and $3 billion in earnings; and the third has a market capitalization of $100 million and $2 million in earnings. The P/E ratios of the first, second, and third companies are 10, 9, and 50, respectively. The index you must calculate is made up of three equities, with 60 percent invested in the first, 25 percent in the second, and 15 percent in the third. Using the weighted harmonic mean, the P/E ratio is:
 P/E = (0.6 + 0.25 + 0.15)/ 0.6/10 + 0.25/9 + 0.15/50 = 1/ 0.09 = 11.1
Let’s use the weighted arithmetic mean to figure it out:
P/E = 15.75 = 0.6 x 10 + 0.25 x 9 + 0.15 x 50
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Clearly, the weighted arithmetic average overestimates the index, providing you with inaccurate data.
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