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# Mean Calculator

By Rita Rain and Anna Szczepanek, PhD candidate

This mean calculator is an arithmetic, geometric, and harmonic mean calculator, all rolled into one. It means no more struggling with the question, "how to calculate mean?" - this tool enables you to find three mean numbers of any dataset. You may demand all of them at once, or choose just one in the final field of this by no means average 😉 calculator. Below, we give the mean formula, the mean math definition, and instruction on how to find the mean by hand. We also describe the inequality between the three means, and explain which situations require the arithmetic, geometric, and harmonic mean.

The other type of averages you may want to calculate are the most common measures of a central tendency - mean, median, and mode.

## How to find the mean with our mean calculator

There are several kinds of mean. This mean calculator incorporates the three most popular means: arithmetic, geometric, and harmonic (also known as the Pythagorean means).

1. By default, the mean calculator returns all three means. You can also choose the specific type of mean you want to determine;
2. Enter each number into a separate field. You can input up to 50 numbers (new rows will appear as you fill out the fields). Remember that harmonic and geometric means can only use positive numbers; and
3. The result will appear at the bottom of the mean calculator.

## Mean math definitions and mean formulas

Let x₁, x₂, ..., xn be a collection of numbers. In the case of the geometric and harmonic means, we assume that they are positive.

• Arithmetic mean formula:

Arithmetic mean definition: the sum of values divided by the number of values, n.

• Geometric mean formula:

The n-th root can be rewritten by raising the product to the power of 1/n, so we have:

Geometric mean definition: the n-th root of the product of n values (the product of the values raised to the power of 1/n).

• Harmonic mean formula:

Harmonic mean definition: the number of values, n, divided by the sum of reciprocals of the values (recall the reciprocal of x equals 1/x).

Well, those math formulas may look overwhelming... Keep calm and keep reading, as in the next section we explain how to calculate mean - step-by-step - and with examples.

## How to calculate mean...

...when you're alone in the woods, without the Internet and calculators?

Make sure you found a water source, some safe shelter, made Chuck chop wood for fire... Take a piece of paper and something to write - charcoal will do. Read on to learn how to find the mean.

• How to calculate arithmetic mean:
1. Add all the numbers together, and denote their sum by `s`. For instance, if the numbers are `1, 2, 4, 17`, then:

`s = 1 + 4 + 2 + 17`

`s = 24`

2. Divide the sum, `s`, by the number of values, `n`:

`A_mean = s / 4 = 24 / 4`

`A_mean = 6`

• How to calculate the geometric mean:
1. Multiply the values and denote their product by `p`. As an example, for `2, 4, 8`, we have:

`p = 4 * 8 * 2`

`p = 64`

2. Take the n-th root of the product, where n is the number of values under consideration. Here, we have three numbers so we take the cube root:

` G_mean = ∛p = ∛64 `

`G_mean = 4`

• How to calculate the harmonic mean:
1. Determine the reciprocal of each value, e.g. for 6, 50, 75:

`1 / x1 = 1 / 6`

`1 / x2 = 1 / 50`

`1 / x3 = 1 / 75`

2. Add the reciprocals, and denote the sum by s:

`s = 1/6 + 1 / 50 + 1 / 75`

`s = 1 / 5`

3. Divide the number of values, n, by the sum of reciprocals, s:

`H_mean = n / s = 3 / (1/5) = 3 * 5`

`H_mean = 15`

## Relationships between the means

• Inequality between the three means:

Experiment a bit with the mean calculator, restricting your input to positive numbers. You'll surely notice that the three means are always different in the same kind of way. You may have already heard about the geometric-arithmetic means inequality: it says that it is not possible to make the geometric mean greater than the arithmetic mean. In addition, the harmonic mean cannot exceed either of the two other means, so we always have:

`Harmonic ≤ Geometric ≤ Arithmetic`

Moreover, the only way to make all three means equal is to have a list of identical numbers. In such a case, all three means are equal to the number appearing in the list.

• Harmonic and arithmetic mean:

That is, the harmonic mean of a list of values is the reciprocal of the arithmetic mean of the reciprocals of those values.

• Geometric and arithmetic mean:

What this means is the logarithm of the geometric mean of a list of values is equal to the arithmetic mean of the logarithms of those values.

## Weighted means

By default, each number in the list contributes equally to the average. However, sometimes we want some values to contribute more than others. In these situations we resort to weighted means, where, apart from a list of numbers x₁, x₂, ..., xn, we also have an associated list of weights w₁, w₂, ..., wn. These weights quantify how much a respective number in the list contributes to the final result.

Each of three means has its weighted version:

• Weighted geometric mean formula (for positive values):

• Weighted harmonic mean formula:

Obviously, if all the weights are equal, the weighted means reduce to their standard versions.

## Arithmetic, geometric, and harmonic mean applications

1. The arithmetic mean of a sample is the most common estimator of the population's mean: if you need a single number as the "typical" value for a set of known numbers, then the arithmetic mean of the numbers does this best, in the sense that it minimizes the sum of squared deviations from the typical value. Learn more with our z-score and z-test calculator.

2. The geometric mean owes its name to its various appearances in geometry, e.g. it is useful in calculating areas, or helping solve triangles (like in the right triangle altitude theorem). Also, geometric mean is very often applied in finance, e.g., in finding the average rate of return.

3. The harmonic mean of two speeds is the proper way of calculating the average speed if we travel a certain distance at some speed, and then return over the same distance at a different speed. In geometry, harmonic mean links the radius of the incircle of a triangle with the height of the triangle, while, in finance, it allows us to determine the price/earnings ratio (P/E) when dealing with an index made of several stocks.

Rita Rain and Anna Szczepanek, PhD candidate

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