Any time you have several factors contributing to a product, and you want to calculate the “average” of the factors, the answer is the geometric mean. The geometric mean is a common alternative when the numbers being averaged are to be multiplied together, as in the case of exponents such as interest rates. It can give the average rate for multiple rates that are multiplied together, such as rates of growth.

  1. Due to its qualities in correctly reflecting investment growth rates the geometric mean is used in the calculation of key financial indicators such as CAGR.
  2. Geometric mean is found by taking the multiple of all the number and then taking the n th root of the number.
  3. While most values tend to be low, the arithmetic mean is often pulled upward (or rightward) by high values or outliers in a positively skewed dataset.
  4. Geometric means will always be slightly smaller than the arithmetic mean, which is a simple average.
  5. The geometric mean is an important tool for calculating portfolio performance for many reasons, but one of the most significant is it takes into account the effects of compounding.

The different types of mean are Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM). In this article, let us discuss the definition, formula, properties, applications, the relation between AM, GM, and HM with solved examples in detail. The geometric mean is used in finance to calculate average growth rates.

Geometric mean triangles and other applications in geometry

Let’s have a look at geometric mean triangles and proof of this theorem. We’ll show that in two ways – using the similarity of the triangles and the Pythagorean theorem. Because it is determined as a simple average, the arithmetic mean is always higher than the geometric mean.

Geometric means with zeros in the dataset

You add 100 to each value to factor in the original amount, and divide each value by 100. In the second formula, the geometric mean is the product of all values raised to the power of the reciprocal of n. In the first formula, the geometric mean is the nth root of the product of all values.

Geometric mean for negative numbers

Geometric mean of the two segments of a hypotenuse equals the altitude of a right triangle from its right angle. As a result, investors consider the geometric mean to be a more accurate indicator of returns than the arithmetic mean. The additive means is known as the arithmetic mean where values are summed and then divided by the total number of values as a calculation.

Example Question Using Geometric Mean Formula

The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc. Using the geometric mean allows analysts to calculate the return on an investment that gets paid interest on interest. This is one reason portfolio managers advise clients to reinvest dividends and earnings. Geometric means will always be slightly smaller than the arithmetic mean, which is a simple average. To calculate compounding interest using the geometric mean of an investment’s return, an investor needs to first calculate the interest in year one, which is $10,000 multiplied by 10%, or $1,000. In year two, the new principal amount is $11,000, and 10% of $11,000 is $1,100.

Formula and Calculation of the Geometric Mean

Examples of this phenomenon include the interest rates that may be attached to any financial investments, or the statistical rates of human population growth. Even though the geometric mean is a less common measure of central tendency, it’s more accurate than the arithmetic mean for percentage change and positively skewed data. The geometric mean is often reported for financial indices and population growth rates. The geometric mean provides a way of finding the average of a group of values by using multiplication instead of addition. It is ideal for numbers that are usually used with multiplication, such as rates and percentages. The procedure for finding the geometric mean is to multiply all of the numbers together, then take the nth root of the product, where n is the total number of values.

The geometric mean is also used in the geometric mean theorem, which allows one to find the lengths of parts of a triangle. Taking the altitude of a hypotenuse divides it into two line segments; the length of the altitude is equal to the geometric mean of those two segments. To review, the geometric mean is the nth root when you multiply n numbers together.

The geometric mean is calculated by raising the product of a series of numbers to the inverse of the total length of the series. The geometric mean is most useful when numbers geometric mean formula in the series are not independent of each other or if numbers tend to make large fluctuations. The arithmetic mean is used when the growth is determined by addition.

Let us learn the with a few solved examples. In any case, the geometric mean is equal to zero for any data set where one or more values is equal to zero. The geometric mean can be an unreliable measure of central tendency for a dataset where one or more values are extremely close to zero in comparison to the other members of the dataset. The geometric mean, sometimes referred to as compounded annual growth rate or time-weighted rate of return, is the average rate of return of a set of values calculated using the products of the terms.

For example, for the product of two numbers, we would take the square root. The geometric mean theorem gives a new relationship between sides of a right triangle. When taking the altitude of the hypotenuse, the hypotenuse is naturally divided into two line segments, one on either side of where the altitude intersects. The theorem states that the length of the altitude is equal to the geometric mean of these two segments. Multiply all of your values together to get the geometric mean, then take a root of it.

The arithmetic mean is defined as the ratio of the sum of given values to the total number of values. Whereas in geometric mean, we multiply the “n” number of values and then take the nth root of the product. Using the arithmetic average of 0.4% growth per year we expect to see an end capital of $1020.16, with the geometric average of -2.62% we see exactly $875.83. In other cases, zeros mean non-responses and in some cases they can just be deleted before calculation.

Among these, the mean of the data set provides the overall idea of the data. The different types of mean are Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). If you are dealing with such tasks, a geometric mean calculator like ours should be most helpful. It is the mathematical average of a set of two or more numbers. An arithmetic mean adds up all the numbers in a set and then divides the sum by the total number of data points. A geometrical mean, on the other hand, refers to the average values calculated using the products of the terms.

The geometric mean of n number of data values is the nth root of the product of all the data values. This is a kind of average used like other means (like arithmetic mean). To calculate the geometric mean of two numbers, you would multiply the numbers together and take the square root of the result. Both the geometric mean and arithmetic mean are used to determine the average.