Arithmetic mean is one of the measures of central tendency that is widely used by all of us in our daily lives. From calculating simple averages to finding complex solutions, we use arithmetic mean. Some examples of the arithmetic mean include average runs scored by Virat Kohli in test cricket format, average rainfall of a city, the average salary of an employee in an organisation, and so on. The word ‘average’ is typically referred to as arithmetic mean. Another important measure of central tendency is the geometric mean. In contrast to the arithmetic mean, it has been a relatively rare measure in computing social statistics. In this article, we will discuss both the measures and differentiate between them.
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Arithmetic Mean Definition
In simple words, arithmetic mean can be defined as the sum total of all the observations of a given data divided by the total number of observations. Arithmetic mean is also widely known as the arithmetic average. Arithmetic mean of a given data is the same as a similar number of observations if all the observations are of the same number. Apart from being used in the subject of statistics, it is also used in subjects like economics, history, and sociology. For example, the per capita income of a state or a country is calculated using the concept of arithmetic mean. It is used to measure the average temperature of the earth to measure global warming and also used in areas like the calculation of average rainfall of a particular city or a town.
How to Calculate Arithmetic Mean?
To calculate the arithmetic mean for ungrouped data, we can use the formula given below:
Arithmetic mean (x̄) = Total of all observations / Total number of observations
To calculate the arithmetic mean for grouped data, we can use three formulas, namely, direct formula, short-cut formula, and step-deviation formula. We will see how to compute the arithmetic mean using the direct formula.
Arithmetic mean (x̄) = (f1x1 + f2x2 + f3x3 + ……. + fnxn) / f1 + f2 + f3+ …… + fn
Geometric Mean
Geometric mean for a given set of n positive observations of data can be defined as the nth root of the product of the observations. Although geometric mean is not very popular among the common sections of society, it is employed by important organisations like the United Nations to calculate the United Nations Human Development Index. It is also used widely in the field of geometry and in calculating the portfolio returns in finance.
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How to Calculate Geometric Mean?
For a simple frequency distribution, we can calculate the geometric mean with the help of the formula given below:
G.M = (x1 × x2 × x3 × x4 × ……… × xn) 1/n
where x1, x2, etc are the different values of the given data and n is the total number of observations.
For a grouped frequency distribution, we can calculate the geometric mean with the help of the formula given below:
G.M = (x1f1 × x2f2 × x3f3 × x4f4 × ……… × xnfn) 1/N
Where N = ∑f1
Difference Between Geometric Mean and Arithmetic Mean
Arithmetic mean is the sum total of all the observations of a given data divided by the total number of observations. On the other hand, geometric mean for a given set of n positive observations of data can be defined as the nth root of the product of the observations. Arithmetic mean is a widely used measure of central tendency whereas geometric mean has limited applications. Arithmetic mean is easy to comprehend and compute while geometric mean is difficult to comprehend and compute. Commonly, arithmetic mean is regarded as the best measure of central tendency. If you want to learn more about these concepts in detail and in a fun and interesting way, follow Cuemath.
