## DEFINITION of 'Harmonic Mean'

The harmonic mean is an average. It is calculated by dividing the number of observations by the reciprocal of each number in the series. Thus, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.

The harmonic mean of 1,4, and 4 is:

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## BREAKING DOWN 'Harmonic Mean'

The weighted harmonic mean is used in finance to average multiples like the price-earnings ratio, because it gives equal weight to each data point. Using a weighted arithmetic mean to average these ratios would give greater weight to high data points than low data points, because price-earnings 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:

As an example, take two firms. One has a market capitalization of \$100 billion and earnings of \$4 billion (P/E of 25) and one with 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 weighted arithmetic mean: P/E = 0.1x25 + 0.9x 250 = 227.5

Using the weighted harmonic mean: P/E = (0.1 + 0.9) / (0.1/25 + 0.9/250) ≈ 131.6

As can be seen, the weighted arithmetic mean significantly overestimates the mean price-earnings ratio.

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