Arithmetic Mean

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What Is Arithmetic Mean?

Arithmetic Mean (AM) is the mathematical term used to refer to the average value of some data values. To find out the arithmetic mean, all the numbers in a given data set are added and then the sum is divided by the total number of items within the set. The arithmetic mean can be calculated by different methods, according to the volume of data and the distribution of the data. But the simplest and most widely used method is to take the sum of values of all data and then divide that sum by the count of the number of data. 

Therefore, Arithmetic mean = (Sum of Observation)/ (Total numbers of Observations)

For example, the marks obtained by a student in four consecutive tests are 76, 83, 78, 91 so the average mark obtained or arithmetic mean value will be = (76+83+78+91)/4= 82.

Properties of Arithmetic Mean

  • If all the data in the given set have the same value, then their arithmetic mean is the same as each of the data values. For example, a data set having five observations of same value 20 will have the arithmetic mean = (20+20+20+20+20)/5 = 100/5 =20
  • If the deviation of each number from the arithmetic mean is added together, the algebraic sum of the deviations will be zero. For example, the deviation of each number in the set of numbers 2, 4, 6, 8 from their arithmetic mean 5 is -3, -1, 1, 3 respectively and their algebraic sum is zero.
  • If each number in the data set increases or decreases by a fixed value, then the arithmetic means of the data set also increases or decreases by the same value. 
  • If each number in the data set is multiplied or divided by a fixed value, then the arithmetic means also gets multiplied or divided by the same value.
  • The arithmetic mean for a series of numbers that are evenly distributed is equal to the middlemost number of the series. For example, the AM of a set of values such as 1, 3, 5, 7, and 9 is 5.

 Geometric Mean

Geometric Mean (GM) represents an average value of a set of data. It’s not the same as the arithmetic mean. In the arithmetic mean, the sum of all data values is taken which is then divided by the total number of values to get the arithmetic mean. But to calculate the geometric mean, we need the following:

  • All the given data values are multiplied, and then the root of the product is calculated taking the index value as the counting number of data. 
  • In other words, if there are two data, the geometric mean will be obtained by taking the square root of the product of those two numbers, or if there are three data, then we have to take the cube root of the product of all three numbers, and so on. 
  • The importance of geometric mean is that it gives a value that indicates the central tendency of a given set of numbers using the product of their values.

For example, for two data values of 8 and 2 the arithmetic mean is AM= (8+2)/2 = 5 and the geometric mean GM= square root of (8 x 2) = 4

Properties of Geometric Mean

  • The geometric mean of a given data set is always less than the arithmetic mean of the data set.
  • If each value in the data set is replaced by the geometric mean, then the product of the new values will be the same as the product of the original values. 

Geometric mean has a variety of uses in statistics and can be applied in many areas to analyze a set of data. It is used in financial calculations for finding average growth rates on investments and is also used in biological studies like cell division.