Harmonic Mean calculator

Harmonic mean is a measure of central tendency in statistics, just like the arithmetic mean and geometric mean. For a given set of values, the harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals of the values. It accounts for all the values in the set. To find the harmonic mean, we first take the reciprocals of each value and add them up. Finally, we divide the number of values by the sum of these reciprocals.

The harmonic mean is primarily used when there is a necessity to give greater weight to the smaller values. A key characteristic of the harmonic mean is that, it assigns greater weights to smaller values and lesser weights to the larger values of the data set. This is done to balance the values evenly. The harmonic mean is used to calculate average rates and average ratios because it equalizes the weight of each value. Some common examples are that of speed and time, and work and time.

Of the three types of means, the harmonic mean is the smallest, and the arithmetic mean is the largest.

For example, consider the values 10, 11 and 12.

The arithmetic mean (AM) = $$ {(10+11+12) \over3 } = 11 $$

The geometric mean (GM) = $$ 3 \sqrt {(10×11×12)} = 3 \sqrt{1320} = 10.96 $$

The harmonic mean (HM) = $$ {3 \over { {1\over10}+{1\over 11} + {1\over 12}}} = {3 \over 0.274} = 10.94. $$

This clearly illustrates that $$ AM > GM > HM.$$

Given below are some important properties of the harmonic mean.

Data set – The values for which the harmonic mean is to be calculated.

Sum of reciprocals – The total obtained after adding up all the reciprocals of the values in the data set.

Harmonic mean – The value obtained by dividing the total number of values in the set by the sum of the reciprocals.

Weighted harmonic mean – When the values in the data set have some weights assigned to them, we calculate the weighted harmonic mean. Here, each value is assigned a fixed weight, depending on how much they influence the data. To find the weighted harmonic mean, we divide each weight by the corresponding data value, and add them up. Finally, we divide the sum of all the weights with the sum obtained by dividing the weights and data values. The usual harmonic mean is a special case of the weighted harmonic mean where each value has the same weight.

The formula for harmonic mean is: $$ harmonic \,mean \,= \,{number \,of \,values \over sum \,of \,the \,reciprocals \,of \,all \,the \,values} $$

This can also be written as $$ HM = {n \over {\sum_{i=1}^n}{1\over x+i}} \\ HM = {n \over { {1\over x_1}+{1\over x_2}+ ... +{1 \over x+3} }} $$

In this formula, each x_i indicates one value of the data set, 1/x_i denotes the reciprocal of each value and n is the total number of values.

The formula for weighted harmonic mean is: $$ weighted \,harmonic \,mean \,= {sum \,of \,the \,weights \over sum \,of \,the \,reciprocals \,of \,the \,weights \,divided \,by \,the\, corresponding \,values} $$

This can also be written as $$ HM = {W \over {\sum_{i=1}^n}{W_i \over X_i}} $$

where $$ W = \sum_{i=1}^n w_i = w_1 + w_2 + ⋯ + w_n, $$ and each w_i denotes the weight corresponding to x_i.

$$ HM = {{w_1 + w_2 + w_3 + ... + w_n} \over {{w_1 \over x_1}+{w_2 \over x_2}+{w_3 \over x_3}+ ... +{w_n \over x_n}}} $$

The harmonic mean cannot be calculated if any of the values is zero or negative.

Consider the three means below.

$$ AM = { {x_1 + x_2 + x_3 + ... + x_n} \over n } $$

$$ GM = n \sqrt {x_1x_2x_3...x_n} $$

$$ HM = {n \over {{1\over x_1}+{2\over x_2}+{3\over x_3}+...+{n\over x_n}}} $$

Then, the relationship between the three of them is $$ GM^2 = AM × HM, or GM = \sqrt{AM * HM} $$

The harmonic mean accounts for every single value in the data set.

If the values of the data set are the same value c, the harmonic mean will also be c.

For any set of data values, the arithmetic mean is the largest and harmonic mean is the smallest. That is, AM > GM > HM.

The harmonic mean cannot be calculated if any of the values is zero or negative. It works for only positive real numbers.

It assigns the highest weight to the smaller value in the data set, and the lowest weight to the largest value. In other words, it assigns unequal weights to each value.

It is not significantly affected if any of the values of the data set suffer fluctuations.

The harmonic mean is significantly affected by the extreme values of the data set.

The harmonic mean has quite a few real-world applications.

It is used in finance to calculate the price-earnings ratio. In this case, the weighted harmonic mean is used where the price is in the numerator.

It is used by market analysts to find patterns like the Fibonacci sequence.

It is used in physics to find relationships between the distances and speeds.

It also has a quite a few applications in geometry. One example is, in any triangle, the harmonic mean of the three altitudes is equal to one-third of the radius of the incircle.

Given below is an example making use of the harmonic mean formula.

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