Harmonic Mean Calculator

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!

What is Harmonic Mean?

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.

What is the Harmonic Mean Formula?

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

\[ H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} \]

Where:

• n = Total number of values
• x = Data set numbers \(x_1, x_2, x_3, \dots, x_n\)

What is Weighted Harmonic Mean?

If each value in the data set has an associated weight \(\omega_1, \omega_2, \omega_3, \dots, \omega_n\), then the Weighted Harmonic Mean (WHM) is calculated as:

\[ H_w = \frac{\sum_{i=1}^{n} \omega_i}{\sum_{i=1}^{n} \frac{\omega_i}{x_i}} \]

Where:

• \(\omega_i\) = weight associated with \(x_i\)
• \(x_i\) = value of the ith observation
• \(n\) = total number of values

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Relation to Other Means:

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:

Relationship between Means:

\[ A.M \ge G.M \ge H.M \]

Where:

• A.M = Arithmetic Mean
• G.M = Geometric Mean
• H.M = Harmonic Mean

Harmonic Mean Shortcut for Two Numbers

If you have only two numbers, you can compute the harmonic mean using the ratio of the squared geometric mean to the arithmetic mean:

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

This formula is especially useful when working with just two values \(x_1\) and \(x_2\), where:

• \(A.M = \frac{x_1 + x_2}{2}\)
• \(G.M = \sqrt{x_1 x_2}\)

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

How to Find Harmonic Mean With Harmonic mean Calculator:

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.

Applications of Harmonic Mean:

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.

How to Find Harmonic Mean (Step-By-Step)

Formula:

\[ H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} \]

Example:

Find the harmonic mean of the numbers: 12, 23, 34, 45, 56.

Solution:

Here, \(n = 5\), \(x_1 = 12\), \(x_2 = 23\), \(x_3 = 34\), \(x_4 = 45\), \(x_5 = 56\)

Step 1: Sum of reciprocals

\[ \frac{1}{12} + \frac{1}{23} + \frac{1}{34} + \frac{1}{45} + \frac{1}{56} \approx 0.083 + 0.043 + 0.029 + 0.022 + 0.018 = 0.195 \]

Step 2: Apply the formula

\[ H = \frac{5}{0.195} \approx 25.64 \]

Final Answer: Harmonic Mean ≈ 25.64

Frequently Ask Questions (FAQ’s):

How do you calculate harmonics?

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.

How do you calculate harmonic mean in Excel?

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

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

Where:

• number1 = First number or range of numbers
• number2, ... = Optional additional numbers or ranges

This function automatically computes the harmonic mean of the given numbers, similar to the manual formula:

\[ H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} \]

How many harmonics are there?

There are two harmonics in the waves. They are:

1. Even harmonics.

2. Odd harmonics.

What are the merits and demerits of harmonic mean?

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.

Wrapping it up:

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.

References:

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