Mean Calculator

This mean calculator is an arithmetic, geometric, and harmonic mean calculator all rolled into one. No more struggling with the question of how to calculate the mean — this tool enables you to find all three means 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 instructions 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, or harmonic mean.

The other types of averages you may want to calculate are the most common measures of a central tendency included in our mean, median, mode calculator.

The mean, often known as the average, is a statistical measure used to find the central value of a dataset. It is calculated by summing all values in the dataset and dividing by the total number of values.

But did you know that the mean isn’t always just the "normal" average? There are different types of means; arithmetic mean, geometric mean, and harmonic mean, each suited for specific data types.

To explore these variations, keep an eye on the section dedicated to Mean Math Definitions and Mean Formulas in Omni's mean calculator.

There are several kinds of means. 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.
3. The result will appear at the bottom of the mean calculator.

Let x₁, x₂, ..., x_n 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), basically known as average.

• 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).

Those math formulas may seem overwhelming, but keep calm and keep reading. In the next section, we explain how to calculate the mean step-by-step with examples.

If you want to learn more about various types of mean, check out our geometric mean and harmonic mean calculators.

...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 with — 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:

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

• 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:

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:

• How to calculate the harmonic mean:

1. Determine the reciprocal of each value, e.g. for $6$, $50$, $75$:

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

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

• 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:

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.

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, ..., x_n, we also have an associated list of weights w₁, w₂, ..., w_n. These weights quantify how much a respective number in the list contributes to the final result.

Each of the three means has its weighted version:

• Weighted arithmetic mean formula:

It has lots of real-life applications. Most probably, you've used it many times to calculate your college GPA. Our college GPA calculator does precisely that.

• Weighted geometric mean formula (for positive values):

• Weighted harmonic mean formula:

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

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 about it with our z-score calculator!

2. The geometric mean owes its name to its various appearances in geometry. For example, it is useful in calculating areas, or helping to solve triangles (like in the right triangle altitude theorem). Also, the 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, the harmonic mean links the radius of the incircle of a triangle with its height. In finance, it allows us to determine the price/earnings ratio (P/E)** when dealing with an index made of several stocks.

The arithmetic mean, or average value, is a widely used measure of central tendency that summarizes a dataset with a single representative value in statistics. It is very useful to help you to understand the general magnitude of grouped values in both academic and real-world statistical analyses. Here are some facts to note when it comes to the mean in statistics:

Sample Mean vs. Population Mean

Context matters when calculating the mean of a dataset. Are you analyzing an entire population or just a sample?

• Population Mean (μ) — The statistical average of all values in a population.
  Formula:

• Sample Mean (x̄) — The mean of a sample, used when working with a subset of the population.
  Formula:

This distinction is crucial in inferential statistics, where you estimate population parameters from limited data.

Interquartile Range (IQR) vs. Mean

The interquartile range (IQR) complements the mean by describing data spread. While the mean tells you the central value, IQR shows the range in which the middle 50% of the data falls.

Unlike the mean, IQR is not sensitive to outliers, making it valuable when your dataset has extreme values. For example, we can look at the income of people in a population. The mean helps you find the average income, while the IQR helps you to understand how most people’s incomes are distributed—excluding unusually high or low values. Together, they give a fuller picture of the data’s shape and variability.