Geometric Mean - Explanation, Formula and Properties
Mathematics and statistics use the measures of central tendency to express the summary of all the values in a data collection.
The mean, median, mode, and range are the most essential measurements of central tendency. Among these, the data set's mean provides an overall picture of the data. It's an average of the data set's numbers.
There are different types of mean
Arithmetic Mean (AM)
Geometric Mean (GM)
Harmonic Mean (HM)
The geometric mean is the average value or mean that, by applying the root of the product of the values, displays the central tendency of a set of numbers or data.
It is a metric unit used to calculate the performance of a single investment or an investment portfolio. For example, for a set of 2 numbers such as 24 and 1. The geometric mean for the given set of two numbers is equal to \[\sqrt {(24 + 1)} = \sqrt {25} = 5\]
The geometric mean is also written as G.M.
Fundamentally,
Total 'n' values are multiplied together
The nth root is being taken out of the numbers, where n is the total number of values.
In this article, we will discuss the geometric mean, geometric mean definitions, and formula, the geometric mean formula for grouped data, properties of geometric mean, etc. is.
Formula:
Below is the formula for Geometric mean calculation:
If X1, X2….Xn is the observation, then the Geometric mean is defined as:
GM = \[\sqrt[n]{x_1 \times x_2 \times x_3.....x_n}\]
GM = \[{(x_1 \times x_2 \times x_3.....x_n)^{1/n}}\]
OR
Log GM = \[\frac {1} {n}log{(x_1 \times x_2 \times x_3.....x_n)}\]
= \[\frac {1} {n}{(logx_1 + logx_2 + .....+logx_n)}\]
= \[\frac {\varepsilon log x_i} {n}\]
GM = Anti log \[\frac {\varepsilon log x_i} {n}\]
Where, n = f1 + f2 + …….+fn
It can also be written as:
GM = \[\sqrt[n]{\prod^n_{i=n}x_i}\]
GM can be written as follows for any grouped data:
GM = Antilog \[\frac {\varepsilon flog x_i} {n}\]
The geometric mean, to put it another way, is the nth root of the product of n values.
Multiply all of your values together to get the geometric mean, then take a root of it. The number of values in your dataset determines the root. Take the square root if you have two values. Take the cube root with three values. Take the 4th root using four values, and so on.
Positive values must be present in all of your data. The geometric mean is usually always less than the arithmetic mean for any given dataset. When your dataset contains identical integers, an exception arises (e.g., all 5s). The geometric mean equals the arithmetic mean in this example.
Geometric Mean Formula for Grouped Data
If we have a set of n positive values with some repeated values such as x1,x2,x3…..xn, and the values are repeating s1,s2,s3…….sk times, then the geometric mean formula for grouped data is defined as.
\[G.M. of X = \overline{X} = \sqrt[n]{x_{1}^{s_{1}}, x_{2}^{s_{2}}, x_{3}^{s_{3}} . . . . x_{k}s_{k}}\]
Geometric Mean Formula for Ungrouped Data
The geometric mean formula for ungrouped data for value X consisting n observation such as x1,x2, x3...xn is represented by G, M of X and is derived as
\[G.M. of X = \overline{X} = \sqrt[n]{x_{1}, x_{2}, x_{3} . . . . x_{n}}\]
Properties
The following are the properties of Geometric mean:
The geometric mean for a given data is always less than the arithmetic means for a given data set.
The ratio of the associated observation of the geometric mean in two series is equivalent to the ratio of their geometric means.
The product of the associated observation of the geometric mean in two series is equivalent to the product of their geometric means.
If the geometric mean replaces each observation in the given data set, then the product of observations does nor change.
Algebraic Properties
The geometric mean is less than the arithmetic mean for any set of positive numbers but when all of a series' values are equal, however, G.M. equals the arithmetic mean.
If any value in a series is 0, the geometric mean is infinity, which is unsuitable.
If the number of negative values is odd, it cannot be calculated. This is due to the fact that the product of the values will turn negative, and we will be unable to determine the root of a negative product.
The product of the values equals the geometric mean raised to the nth power.
The geometric mean of any set of numbers with the same N and product is the same.
The product of the geometric mean's each side ratio will be equal to both sides.
Even when each number in a series is replaced by its geometric mean, the series' products remain the same.
The sum of the deviations of the original values' logarithms above and below the G.M.'s logarithm is equal.
Solved Examples:
Find the geometric mean of the numbers 2, 3, and 6?
i) Take the cubed root of the numbers multiplied together (because there are three numbers) = \[\frac {(2 \times 3 \times 6 \times)1}{3}\] = 3.30
Find the geometric mean of 4,8.3,9 and 17?
i) Multiply the numbers together
ii) Take the 5th root (because there are 5 numbers) = \[{(4 \times 8 \times 3 \times 9 \times 17)}^{1/5}\] = 6.81
Geometric mean of \[\frac {1}{2}\] , \[\frac {1}{4}\] , \[\frac {1}{5}\] , \[\frac {9}{72}\] and \[\frac {7}{4}\] is ?
i) Multiply the numbers together
ii) 5th root is to be taken: (\[\frac {1}{2} \times \frac {1}{4} \times \frac {1}{5} \frac {9}{72} \frac {7}{4}^(\frac {1}{5})\] = 0.35.
Over ten years, the average monthly wage in a particular municipality increased from $2,500 to $5,000. What is the average annual increase using the geometric mean?
i) Geometric mean.
\[{(2500 \times 5000)}^{1/2}\] = 3535.53390593.
ii) Divide by 10 (to get the ten-year average increase).
\[\frac {3535.53390593}{10}\] = 353.53.
As per GM, the average increase is 353.53.
Find the geometric mean for the following data.
Solution:
Geometric mean of X = Antilog \[\frac {\sum f log x}{\sum f}\]
= Antilog \[\frac {(119.1074)}{48}\]
= Antilog (2.4814)
= 11.958
Calculate the geometric mean of 10,5,15,8,12.
Solution:
Given, x₁ = 10 ,x₂= 5,x₃ = 15,x₄ = 8,x₅= 12
N= 5
Using the geometric mean formula,
\[\sqrt[n]{x_{1}, x_{2}, x_{3}. . . . x_{n}}\]
\[G.M. of X = \overline{X} = \sqrt[5]{10 \times 5 \times 15 \times 8 \times 12}\]
\[\sqrt[5]{72000}\]
= \[(72000)^{\frac{1}{5}}\]
= 9.36.
Find the geometric mean for the following data.
Solution: Here n = 5
Geometric Mean = \[\frac {\sum log x_i}{n}\]
= Antilog \[\frac {8.925}{5}\]
= Antilog 1.785
Quiz Time
1. The geometric mean is calculated by
(a + b)2
(a - b)/2
± √ab
± ab
2. The three geometric means between 2 and 32 are
6,10,14
10,12,14
6,8,10
4,8,16
3. If a, G and b are in geometric progression the ‘G’ is considered as
Arithmetic mean
Geometric mean
Standard deviation
None of the above
FAQs on Geometric Mean Formula
1. What is the Difference Between Arithmetic Mean and Geometric Mean.
The difference between the arithmetic mean and Geometric mean is given below in the tabulated form.
Geometric Mean | Arithmetic Mean | |
Meaning | It is known as the multiplicative name. | It is known as additive mean. |
Values | The geometric mean is always less than the arithmetic mean because of the compounding effect. | The arithmetic mean is always greater than the arithmetic mean because it is computed as a simple average. |
Data set | The geometric mean formula applied only on the positive set of numbers. | The arithmetic mean formula can be applied on both the positive set of numbers and the negative sets of numbers. |
Uses | The geometric mean is widely used by biologists, economists, and financial analysts. It is most accurate for the dataset that manifests correlation. | The arithmetic mean is used to represent average temperature as well as determine the average speed of a car. |
Effect of outliers | The effect of the outliers on Geometric is moderate. For example, in the given data set 11,13,17 and 1000. In this case, 1000 is the outlier and the average is 39.5. | The effect of the outliers on arithmetic is severe. For example, in the given data set 11,13,17 and 1000. In this case, 1000 is the outlier and the average is 260.25. |
2. What are the Advantages and Disadvantages of the Geometric Mean?
Some Advantages of the Geometric Mean are:
The geometric mean is rigidly defined as their values are always fixed.
It is based on all the elements of the series.
The geometric mean is not much affected by sampling fluctuations.
Geometric means are accurate for algebraic calculations and other mathematical operations.
It gives more weightage to small items.
Disadvantages of Geometric Mean
It is not commonly used as it is difficult to understand.
It is not easy to calculate a geometric mean in case the value of the variable in the series is negative or zero.
The value of geometric mean cannot be obtained in the case of open-end distribution.
3. What do the applications of Geometric mean?
The geometric mean has various advantages and is utilized in a variety of fields. Applications are as follows:
The geometric mean is employed in many of the value line indexes used by financial departments. It is also used in stock indexes.
It is used to compute the annual return on the portfolio.
It's used in finance to calculate average growth rates, commonly known as compounded annual growth rates.
It's also employed in research on cell division and bacterial development, among other things.
4. What are the benefits and limitations of Geometric mean?
Here are the key benefits of geometric mean:
It is unwavering.
The calculation uses all of the sequence's terms.
It can be mathematically analyzed.
It doesn't matter how the sample is fluctuating, the geometric mean doesn't get affected.
It emphasizes minor details.
Disadvantages:
In case, if any one of the observations is negative, then the geometric mean value might result in an imaginary figure despite the quantity of the other observations.
For a non-mathematics person, it becomes quite difficult to compute.
5. What exactly is Arithmetic mean and how is it different from Geometric 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. The calculation is relatively easy when compared to the Geometric mean.
Arithmetic mean is also different from Geometric mean in terms of dependent variables. Arithmetic mean can be applied in conditions where the variables are not dependent on one another whereas Geometric mean can be used in situations where variables are dependent on one another.
6. Is the geometric mean always in the positive number?
The geometric mean is only applicable to positive numbers, not negative ones. It is frequently used to represent a collection of numbers whose values are intended to be multiplied together or are exponential, such as a collection of growth figures. Eg, the population of the world or the interest rates on a financial investment over time.
Negative values, like 0, make it impossible to calculate Geometric Mean. There are, however, several workarounds for this issue, all of which need the negative numbers to be translated or changed into a meaningful positive comparable value.
7. Why is the geometric mean preferable to the arithmetic mean?
The geometric mean takes into consideration the compounding that occurs from period to period, hence it differs from the arithmetic average, or arithmetic mean, in how it is calculated.
As a result, investors consider the geometric mean to be a more accurate indicator of returns than the arithmetic mean.
Because it is determined as a simple average, the arithmetic mean is always higher than the geometric mean. It can only be applied to a positive group of numbers.
It's possible to compute it using both positive and negative values. When the dataset is logarithmic, the geometric mean can be more useful.