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Statistics Calculators ▶ Harmonic Mean Calculator

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An online harmonic means calculator allows you to calculate harmonic mean from the dataset, by dividing the sum of reciprocals of the dataset. Remember that this calculator allows you to perform H.M calculations for both positive and negative integer’s dataset.

Keep reading to completely know about its definition, formula, how to calculate it manually & different other useful data related to harmonic mean!

Read on!

It is one of the three most important central tendency types, along with the arithmetic & geometric mean. The harmonic mean represents the central tendency by dividing the total integers with the sum of the integers. It is the reciprocal of the arithmetic mean.

It shows the lowest value among all the means. It is sometimes called subcontrary mean.

This harmonic mean calculator uses the following formula for the calculations:

$$ H = \frac {n}{\frac {1}{x_1} + \frac {1}{x_1} + . . . + \frac {1}{x_1}} = \frac {n}{\sum_{i=1}^n \frac {1} {x_i}}$$

Where,

\(n\) is the total number of values and \(x (x_1, x_2 ,x_3,………,x_n)\) are the numbers in the data set.

If the set of weights \(\omega_1, \omega_2, \omega_3, . . . \omega_n\) is associated with data sets of \(x_1, x_2, x_3… x_n\), then the weighted harmonic mean of data set will be equal to:

image

The harmonic mean always gives the shortest value from all of the other means. Its relation with the other means (Arithmetic & Geometric) is as follow:

\(A.M > G.M > H.M\)

If you have only two integers, you can also compute the harmonic mean from the ratio of squared geometric mean and arithmetic mean.

\(H.M = \frac {G.M^2}{A.M}\)

Get this free online geometric mean calculator to determine the geometric mean for any date set of numbers or percentages.

To find the harmonic mean between positive or negative numbers becomes very easy with this online harmonic mean calculator. Just follow the given steps for the accurate results:

Swipe on!

**Inputs:**

- First of all, select how numbers are separated from the drop-down menu. It is either separated by comma, space or user defined. (Enter the separation technique if you select the “user define”)
- Very next, enter the numbers for which you want to do calculations.
- Lastly, hit the calculate button.

**Outputs:**

Once you fill all the fields of the calculator, it will show:

- Harmonic mean of given data points
- Numbers in Ascending order.
- Numbers in Descending order.
- Even numbers in data.
- Odd numbers in data.
- Total sum of the numbers.
- Maximum value in numbers.
- Minimum value in numbers.
- Total numbers.

It have many applications in different fields of science so that the experts of calculator-online made this online harmonic mean calculator for you to calculate the harmonic mean accurately for a given set of numbers. It is widely used in:

- In Geometry, the radius of the incircle is equal to the one-third of the harmonic mean of altitude of the triangle.
- In Finance, it is used to calculate the ratio between the price and earnings of the several products.
- In physics, it is helpful to determine the average speed, resistance and capacitance of a capacitor.

The formula used for the calculation is as follow:

$$ H = \frac {n}{\frac {1}{x_1} + \frac {1}{x_1} + . . . + \frac {1}{x_1}} = \frac {n}{\sum_{i=1}^n \frac {1} {x_i}}$$

Let’s have an example to better understand the concept:

**For example:**

Find the harmonic mean between \(12, 23, 34, 45,\) and \(56\)?

**Solution:**

Here,

\(n = 5\)

\(x_1= 12\)

\(x_2 = 23\)

\(x_3= 34\)

\(x_4 = 45\)

\(x_5 = 56\)

So,

\(H.M = \frac {5}{\frac {1}{12}+\frac{1}{23}+\frac {1}{34}+\frac {1}{45}+\frac {1}{56}}\)

\(H.M = \frac {5}{(0.083)+(0.043)+(0.029)+(0.022)+(0.017)}\)

\(H.M = \frac {5}{0.194}\)

\(H.M = 25.47\)

For the calculation between n numbers, divide the reciprocals of the numbers with total numbers for which you want to calculate the harmonic mean. It is the reciprocal of arithmetic mean.

To calculate the harmonic mean between n numbers in excel, use the HARMEAN function in excel. The syntax of the function is”

\(=HARMEAN\) \((number1, [number2]…)\)

There are two harmonics in the waves. They are:

1. Even harmonics.

2. Odd harmonics.

The merits and demerits of harmonic mean is discussed below:

**Merits:**

• Harmonic mean is capable of further algebraic treatments.

• It is rigidly defined.

• It cannot ignore any value.

• It gives a straight curve than the arithmetic and geometric.

**Demerits:**

• It cannot understand by a person who has moderate knowledge.

• Its calculation is complex, as involve the reciprocal of the numbers

• It is affected by the values of extreme items.

• If any one of the item is zero, it can’t be calculated.

The harmonic mean is very helpful in many conditions like to determine the price to earnings ratio, averaging things, capacitance and resistance of capacitors & resistors respectively and many others. Simply, use this online harmonic mean calculator that helps you to give speedy calculations between the n numbers. Typically, students and professionals use this online tool to find out the solution of their harmonic mean problems in Statistics.

From the authorized source of Wikipedia: Harmonic mean, Relationship with other means, and all related approaches.

From the source of sciencedirect: Ultimate Guide on H.M (statistical data)

The information from ck12: The entire overview of H.M Statistical Concepts