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.
What is mean?
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 below.
How to find the mean with our 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).
By default, the mean calculator returns all three means. You can also choose the specific type of mean you want to determine:
- Choose the mean type — all three at once, or just the arithmetic, geometric, or harmonic mean.
- Select the unit type — plain number, currency (USD, RUB, EUR, GBP and 20+ more), US imperial, or metric units.
- 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.
- Read the result 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:
A = (x₁ + … + xₙ) / n
Arithmetic mean definition: the sum of values divided by the number of values (n) — basically known as the average.
Geometric mean formula:
G = ⁿ√(x₁ × … × xₙ) = (x₁ × … × xₙ)^(1/n)
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:
H = n / (1/x₁ + … + 1/xₙ)
Harmonic mean definition: the number of values, n, divided by the sum of reciprocals of the values (recall that the reciprocal of x equals 1/x).
How to calculate mean…
How to calculate the arithmetic mean:
- Add all the numbers together, and denote their sum by s. For the numbers 1, 2, 4, 17: s = 1 + 2 + 4 + 17 = 24.
- Divide the sum, s, by the number of values, n: A = 24 / 4 = 6.
How to calculate the geometric mean:
- Multiply the values and denote their product by p. For 2, 4, 8: p = 2 × 4 × 8 = 64.
- Take the n-th root of the product. Here there are three numbers, so we take the cube root: G = ∛64 = 4.
How to calculate the harmonic mean:
- Determine the reciprocal of each value, e.g. for 6, 50, 75: 1/6, 1/50, 1/75.
- Add the reciprocals, and denote the sum by s: s = 1/6 + 1/50 + 1/75 = 1/5.
- Divide the number of values, n, by the sum of the reciprocals, s: H = 3 / (1/5) = 3 × 5 = 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 notice that the three means are always ordered in the same way. This is the geometric–arithmetic means inequality: 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. The harmonic mean of a list of values is the reciprocal of the arithmetic mean of the reciprocals of those values:
H(x₁, …, xₙ) = A(1/x₁, …, 1/xₙ)⁻¹
Geometric and arithmetic mean. The logarithm of the geometric mean of a list of values equals the arithmetic mean of the logarithms of those values:
G(x₁, …, xₙ) = e^A(ln x₁, …, ln xₙ)
Weighted means
In some situations, not all values should count equally. A weighted mean multiplies each value by a weight that reflects its importance, sums the products, and divides by the sum of the weights. A familiar example is a grade point average, where courses with more credit hours carry more weight. When all weights are equal, the weighted mean reduces to the ordinary arithmetic mean.
Arithmetic, geometric, and harmonic mean applications
- The arithmetic mean is the right choice for quantities that add up — test scores, temperatures, or total costs — where every value should contribute equally to the total.
- The geometric mean is used for quantities that multiply, such as growth rates, investment returns, and ratios. It answers "what constant factor, applied each period, gives the same overall change?"
- The harmonic mean is used for rates and ratios defined per unit, such as average speed over equal distances or the average of price-to-earnings ratios. It gives more weight to smaller values.
Mean in statistics: A deeper understanding
In statistics, the mean is a measure of central tendency that summarizes an entire dataset with a single representative value. The population mean is written μ, while the mean of a sample is often written x̄ or M — the calculation is identical, only the notation differs. For perfectly symmetric (normal) distributions, the mean sits exactly at the center. In skewed distributions it is pulled toward the long tail, which is why analysts often report the median alongside the mean for data such as income or house prices.
FAQs
What is the mean and how do I calculate it?
The mean is a central value of a dataset. For the arithmetic mean, add all the numbers and divide by how many there are. For example, the mean of 2, 5, 6, 8, 9 is (2 + 5 + 6 + 8 + 9) / 5 = 6.
What is the difference between arithmetic, geometric, and harmonic mean?
The arithmetic mean sums values and divides by the count; the geometric mean multiplies values and takes the n-th root; the harmonic mean divides the count by the sum of the reciprocals. For any set of positive numbers, harmonic ≤ geometric ≤ arithmetic.
Why can't I compute a geometric or harmonic mean with negative numbers?
The geometric mean involves an n-th root of a product and the harmonic mean involves reciprocals, both of which are only well defined for strictly positive values. If your dataset contains a zero or a negative number, only the arithmetic mean is shown.
Can I calculate the mean of currency values?
Yes. Select the Currency option and choose from 25 world currencies including USD, RUB, EUR, GBP, JPY, INR, BRL, CAD, and AUD. All results are displayed with the appropriate currency symbol.
Does this calculator support Imperial and Metric units?
Yes. It supports US Imperial units (inches, feet, yards, miles, ounces, pounds, gallons, etc.) and Metric (SI) units (mm, cm, m, km, mg, g, kg, mL, L). The formulas work the same regardless of unit.
How many numbers can I enter?
You can enter up to 50 numbers. New input fields appear automatically as you fill out the form, and a live preview shows all three means as you type.