Geometric Mean Definition Formula Calculation Example

Geometric Mean Definition

The method is suitable for determining the average value appreciation of a particular investment or the overall portfolio—for a given time frame. It accurately evaluates average values pertaining to a continuous series of interdependent values.

Key Takeaways

The geometric mean (GM) refers to a central tendency measure that evaluates the average of a series by multiplying all the numbers and then finding the nth root of the product. It is the power mean used for a continuous data series with a close-end distribution of the interdependent values or items. The GM method is applied to compute the proportional growth of stock indices. Further, it ascertains the average returns of investments or portfolios that provide compounding benefits.

Geometric Mean Explained

You are free to use this image on you website, templates, etc., Please provide us with an attribution linkHow to Provide Attribution?Article Link to be HyperlinkedFor eg:Source: Geometric Mean (wallstreetmojo.com)

The geometric mean is used with time-series data to ascertain the compoundingCompoundingCompounding is a method of investing in which the income generated by an investment is reinvested, and the new principal amount is increased by the amount of income reinvested. Depending on the time period of deposit, interest is added to the principal amount.read more average. Time series data is a culmination of observations collected through repeated measurements over time. When time-series data is represented on a graph, one of the axes would always be time.

This metric is often referred to as power mean. It is mostly used to compare the growth averages of different investment products or portfolios. It has an exponential relationship with the arithmetic meanArithmetic MeanArithmetic mean denotes the average of all the observations of a data series. It is the aggregate of all the values in a data set divided by the total count of the observations.read more of logarithms.

This method normalizes ranges, reducing the impact of dominant values on the weightage. Thus, enormous values no longer influence skewed distribution patterns. This method provides better results when variables are widely skewed.

SkewnessSkewnessSkewness is the deviation or degree of asymmetry shown by a bell curve or the normal distribution within a given data set. If the curve shifts to the right, it is considered positive skewness, while a curve shifted to the left represents negative skewness.read more refers to symmetry. If skewness is 0, the data is perfectly symmetrical. If the normal distributionNormal DistributionNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more is uneven with a skewness greater than zero or positive skewnessPositive SkewnessA positively skewed distribution is one in which the mean, median, and mode are all positive rather than negative or zero. The data distribution is more concentrated on one side of the scale, with a long tail on the right.read more, then its right tail will be more prolonged than the left.

Properties

The following properties make geometric mean stand out among other central tendencyCentral TendencyCentral Tendency is a statistical measure that displays the centre point of the entire Data Distribution & you can find it using 3 different measures, i.e., Mean, Median, & Mode.read more measures:

Even if each value in the data series is replaced with the GM, their product will still be the same.In a particular data set, the geometric mean value is always lower than the value of its arithmetic mean.When the corresponding observations of GM of two data series are multiplied, the value acquired is equal to the multiplicative result of their GM values.

Geometric Mean Formula

The geometric mean calculation helps investors ascertain the compounding average for a given data series. It is evaluated using the following formulas:

Or,

In the above equations, n is the total number of values in a given data series. Also, x1, x2, andx3 are the provided data series’s first, second, and third values. In addition, xn is the nth value of the provided data series.

Geometric Mean Calculation

Given below are the basic steps of Geometric Mean calculation:

Make sure that the provided data series has interdependent and progressive values.Count the number of values or items in the series to determine “n.”Now, multiply all the values in the given sequence.Then, find the nth root of the product acquired in the previous step to obtain GM.

Example

Rhonda is a soccer player. In a five-month selection process for the interstate competition, she scored 3,7, 8, 11, and 17 goals respectively. Determine the geometric mean to analyze her overall performance.

Solution:

Note: In the above sheet, we have applied the formula n√ (x1x2 x3…xn) or (x1x2 x3…xn) ^(1/n).

Hence, on average, Rhonda scores 7.93 or 8 goals.

Application

Geometric Mean is a reliable method for determining the average rate of returnAverage Rate Of ReturnAverage Rate of Return (ARR) is the expected rate of return on an investment or asset divided by the initial investment cost or average investment during the project’s life. The formula for calculating the average rate of return is: Average annual net earnings after taxes/Initial investment * 100%read more of an investment or the investment portfolio—for a given year. It facilitates the comparison of the two or more investment opportunities or assets and helps select the better option.

It is used to determine the average compounded growth rateGrowth RateThe Growth rate formula is used to calculate the annual growth of the company for a particular period. It is computed by subtracting the prior value from the current value and dividing the result by the prior value.read more of stocks and other securities. Further, it is applied on stock indicesStock IndicesThe stock index, which is also known as the stock market index, is a tool used to determine the performance of shares/securities in the market and to calculate the return on the stock of their investment investors use it to have knowledge about the performance of investments and access the total value they possess.read more that hold stocks of equivalent weightage—varying in market capitalizationMarket CapitalizationMarket capitalization is the market value of a company’s outstanding shares. It is computed as the product of the total number of outstanding shares and the price of each share.read more.

In geometry, the geometric mean is applied to exponential values. GM is also applied in biology to research bacterial growth, viral mutation rates, and cell division. Economists employ this method for understanding population growth and average voter turnouts.

Advantages and Disadvantages

The geometric mean is beneficial for identifying the average of rigid sequences. However, unlike the arithmetic mean, a higher level of weightage is vested upon small values. Moreover, sampling fluctuations have little impact on GM results.

The method sure has limitations, the foremost being its complexity. As a result, it is less popular. Users require thorough mathematical knowledge of ratios, roots, and logarithms—application is difficult for a layman.

The method is not a suitable measure for data series with zero or negative values. Similarly, the method cannot be applied to data series with open-end distributions.