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The geometric mean, also known as the geometric average, is a specialised average defined as the n-th root of the product of n numbers of the same sign. In an arithmetic mean, we sum the numbers and then divide by their number, whereas in a geometric mean, we calculate the product of the numbers and then take the n-th root. The geometric mean is the answer when there are multiple elements that contribute to a product and you wish to determine the “average” of the factors.
It can be used in a variety of circumstances where a growth rate is needed, such as calculating compound interest rates, financial returns or risk and losses, area and volume averages, and indexes like the US Consumer Price Index (inflation index), among others. If you’re working on such a project, a geometric mean calculator like ours will come in handy.
The formula for calculating the geometric mean is:
where n is the number of numbers and X1…Xn are the numbers from the first to the n-th. An alternative way to write the formula is (X1 x X2 … x Xn)^1/n . This formula is used in our calculator.
Rectangles and squares are a geometric technique to explain the formula. The perimeter of a rectangle with sides of 4 and 16 is equal to the total of all four sides: 4 + 4 + 16 + 16 = 40. The arithmetic mean of 4 and 16 is 10, therefore a square with 10 sides has the same perimeter as one with 4 and 16 sides.
The area of our 4 x 16 rectangle, on the other hand, is the product of 4 and 16 and equals 64. The geometric mean provides an answer to the dilemma of what side a square should have in order to have a 64-square-foot area. The correct answer is 8, which is the geometric mean of the numbers 4 and 16.
The area calculation correlates to the geometric mean in the image above, while the perimeter calculation corresponds to the arithmetic mean.
Obviously, assuming you don’t want to use a calculator. Let’s imagine we have a set of numbers: 1 5 10 13 30 and wish to find the arithmetic mean of them. We may get an arithmetic mean of 11.80 by adding the integers (1 + 5 + 10 + 13 + 30) and then dividing by 5. Instead, we use their product to calculate the geometric mean: Calculate the 5-th root of 19,500 = 7.21 by multiplying 1 x 5 x 10 x 13 x 30. This is the same as multiplying 19,500 by a fifth power.
The geometric mean can also be calculated using logarithms, as it is the average of logarithmic numbers transformed back to base 10. Assume you want to find the geomean of 2 and 8. Here, log with base 2 is convenient, therefore 2 = 21 and 8 = 23. The geometric mean is 22 = 4 since the arithmetic mean of the exponents (1 and 3) is 2. This is also something that our geometric average calculator can confirm.
As can be seen, the geometric mean is much more resistant to outliers and extreme values. Replacing 30 with 100, for example, would result in an arithmetic mean of 25.80 but a geometric mean of only 9.17, which is highly desirable in some instances. However, before you decide to use the geometric mean to answer your issue, you should think about whether it is the proper statistic to utilise.
The median allowance of ten might be a better estimate in this scenario.
When it comes to calculating the success of investment portfolios, the arithmetic mean isn’t ideal, especially when compounding, or the reinvestment of dividends and earnings, is involved.It’s also not commonly used to compute present and future cash flows, which analysts use to make predictions. This will very certainly result in erroneous data.
To avoid computing the root of a negative product, which would result in imaginary numbers, we can only calculate the geometric mean of positive numbers, or, more accurately, the numbers must be of the same sign. However, this does not rule out the possibility of working with negative numbers. Let’s imagine we have the following relative changes over the course of three years: 8% growth, 10% decline, and 11% growth.
At the end of the day, the overall growth rate is 7.89 percent, but how do we compute the average annual growth rate? 10% would normally be -10%, which has a different sign and prevents us from doing the calculation, but we can use a little trick and express the numbers as proportions, so 8% growth becomes 1 + 8% x 1 = 1.08, 10% decline becomes 1 – 10% x 1 = 0.9, and % growth becomes 1 + 11 % x 1 = 1.11. The geometric mean is 1.0256, which translates to 2.56 % annual increase.
This is handled automatically by our geometric mean calculator, so no manual transformations are required. You can also specify percentages, such as “2 % 10 % -10 % 8 %,” and the programme will handle it (it simply strips the percent ).
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