Geometric Mean: Definition, Formula, Calculation

The geometric mean has from time to time been used to calculate financial indices (the averaging is over the components of the index). For example, in the past the FT 30 index used a geometric mean.8 It is also used in the CPI calculation9 and recently introduced “RPIJ” measure of inflation in the United Kingdom and in the European Union. For various reasons, the geometric mean is an important tool for calculating portfolio performance. One of the most significant of those reasons is that it takes into account the effects of compounding. I think some of the comments are related to a false-equivalency of NA values in the data and zeros.

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1. The arithmetic mean is the calculated average of the middle value of a data series.
2. Among these, the mean of the data set provides the overall idea of the data.
3. Its closing prices and the resulting figure for the moving average are shown below.
4. The error bars appear visually symmetrical, which means they are numerically far from symmetrical.

The Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values. In mathematics and statistics, measures of central tendencies describe the summary of whole data set values. The most important measures of central tendencies are mean, median, mode, and range. Among these, the mean of the data set provides the overall idea of the data. The mean defines the average of numbers in the data set. The different types of mean are Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM).

Here is a vectorized, zero- and NA-tolerant function for calculating geometric mean in R. The verbose mean calculation involving length(x) is necessary for how to calculate geometric mean the cases where x contains non-positive values. By default, geomean operates along the first dimension of X whose size does not equal 1. In this case, this dimension is the first dimension of X. By comparing the result with the actual data shown on the table, the investor will find a 1% return is misleading. In computer implementations, naïvely multiplying many numbers together can cause arithmetic overflow or underflow.

The geometric mean is best used to calculate the average of a series of data where each item has some relationship to the others. That’s because the formula takes into account serial correlation. Or, the arithmetic mean could be used to determine a moving average for a stock price. A moving average is helpful for traders and investors because, when calculated and plotted over time, it smooths out a long series of price movements to present a big picture of a price trend. Market participants can also chart long-term points of support and resistance with a moving average.

The Geometric Mean Calculator is an online tool for calculating the geometric mean of numbers.

Aspect ratios

From your data table, click the Analyze button and select Descriptive Statistics. The median would be the value found where half the items in the measured sample appear above it and half appear below it. In an ellipse, the semi-minor axis is the geometric mean of the maximum and minimum distances of the ellipse from a focus; it is also the geometric mean of the semi-major axis and the semi-latus rectum. The semi-major axis of an ellipse is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix. Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result. Thus, the geometric mean provides a summary of the samples whose exponent best matches the exponents of the samples (in the least squares sense).

The Formula for Arithmetic Mean

However, the arithmetic mean is not an appropriate method for calculating an average where the data exhibit serial correlation, or have some relationship to each other. We used an arithmetic mean for a moving average because the closing prices have no correlation. One closing price may be higher or lower than the next, but there’s no intrinsic relationship. In finance and investing, one might use the arithmetic mean to get an idea of the average earnings estimate for a series of estimates issued by a number of analysts covering a stock. Simply add up the various estimates and divide by the number of estimates.

1. Market participants can also chart long-term points of support and resistance with a moving average.
2. To calculate the geometric mean of two numbers, you would multiply the numbers together and take the square root of the result.
3. For the graph in this example, choose the data table option “Enter or import data into a new table”, and “Enter replicate values, stacked into columns”.
4. The geometric mean is a value that represents the middle of a set of numbers, a measure of central tendency.
5. Programs exist where the data is entered correctly and the proper rule is invoked when the calculation is made.
6. Annual investment returns over the years have an impact on each other.

Geometric mean of a continuous function

It is best used in calculations involving items that, while the same type, have no relationship with each other. The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount. Thus, the geometric mean is also defined as the nth root of the product of n numbers. In the arithmetic mean, data values are added and then divided by the total number of values.

Adam received his master’s in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.

When the return or growth amount is compounded, the investor needs to use the geometric mean to calculate the final value of the investment. This sort of relationship is useful when comparing portfolio returns, bond yields, and total returns on equities. They affect the return for each succeeding period measured. To calculate a 14-day moving average for a stock, simply add up its closing price for the past 14 days and then divide that sum by 14. Its closing prices and the resulting figure for the moving average are shown below.

GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The mean of a data set, for example, provides an overview of the data. To create a graph in Prism displaying the geometric mean of a dataset along with its 95% confidence interval, start by selecting the Column table type on the Welcome dialog. For the graph in this example, choose the data table option “Enter or import data into a new table”, and “Enter replicate values, stacked into columns”.

The investor will compare both investment options by analyzing the interest rate or the final equity value with the same initial equity. Note that it is not possible to compute a logarithm of zero or any negative number. Therefore, it is only possible to compute a geometrical mean when every value is positive and none are negative or zero. Annual investment returns over the years have an impact on each other. If you lose a substantial amount of money in a particular year, you have that much less capital with which to invest and generate returns in the following years.