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Arithmetic Mean Formula

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  • Last Updated : 12 Jan, 2022
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Arithmetic is a subject of mathematics that is being used for calculation for the longest time known till now. Arithmetic alone carries an ancient history in the development of human civilization and different systems adapted for day-to-day based calculations and operations.

While looking into the history of arithmetic the word itself is derived from the Greek word ‘arithmos’ which means numbers. Brahmagupta the Indian mathematician of the 17th century is known as the “father of arithmetic”. And, the Fundamental theory of number theory was given by Carl Friedrich Gauss in 1801. Arithmetic is a subject of mathematics that deals with the study of numeric figures, their properties, and operations associated with them like summation, subtraction, multiplication, and division.

Arithmetic Mean Formula

Arithmetic means the formula is used to determine the mean or average of given entire data. Arithmetic mean is used to determine the central tendency that is to derive an average value by calculation from the large set of data given. The symbol used to denote the arithmetic mean is ‘X’. The calculation of the mean is totally based on observations.

In statistics arithmetic mean is used to determine the central tendency. The formula is derived by dividing the sum of a number of data by the number of observations. The numbers can be denoted as n1, n2, n3, n4, n5, ……..nn. And, the number of values will be n. Mathematically,

A.M. = (n1 + n2 + n3 + n4 + … + nn)/n

And, if the frequency is given for the given set of numbers that is f1, f2, f3, f4, f5, …, fn for the numbers n1, n2, n3, n4, n5, … nn.

A.M. = \frac{f_1n_1 + f_2n_2 + f_3n_3 + f_4n_4 + ... + f_nn_n}{f_1 + f_2 + f_3 + f_4 + ... + f_n}

While in arithmetic, the arithmetic mean formula is given by,

A.M={\frac{1}{n}\sum_{i=1}^{n}a_{i}}

Where,

n is the number of items

A.M is the arithmetic mean

ai are set values.

Derivation of the arithmetic mean formula 

Let n be the number of observations in the operation and n1, n2, n3, n4, …, nn be the given numbers. Now as per the definition, the arithmetic means formula can be defined as the ratio of the sum of all numbers of the group by the number of items.

A.M. = (n1 + n2 + n3 + n4 + … + nn)/n

By solving the equation, the formula of arithmetic mean is obtained which is,

A.M={\frac{1}{n}\sum_{i=1}^{n}a_{i}}

Sample Problems

Question 1: Find the arithmetic mean of the first five prime numbers.

Solution:

The arithmetic mean of the first five prime numbers will be given by,

The first prime numbers are 2, 3, 5, 7 and 11.

Number of observations (n) is 5.

Now, 

X = sum of numbers/ number of observations

X = (2 + 3 + 5 + 7 + 11)/5

X = 28/5

X = 5.6

Hence, the arithmetic mean of the first five prime numbers is 5.6.

Question 2: Find the arithmetic mean of the first five natural numbers.

Solution:

The first five natural numbers are 1, 2, 3, 4 and 5.

The number of observations is 5.

Now,

X = sum of numbers/ number of observations

X = (1 + 2 + 3 + 4 + 5)/5

X = 15/5

X = 3

Hence, the arithmetic mean of the first five natural numbers is 3.

Question 3: If the arithmetic mean of five observations 5, 6, 7, x, and 9 is 6. Find the value of x.

Solution:

The five observations are 5, 6, 7, x, and 9.

The number of observations is 5.

Now, 

X = sum of numbers/ number of observations

6 = (5 + 6 + 7 + x + 9)/5

30 = 27 + x

x = 30 – 27

x = 3

Hence, the arithmetic mean of five observations is 3.

Question 4: If the arithmetic mean of five observations 10, 20, 30, x, and 50 is 30. Find the value of x.

Solution:

The five observations are 10, 20, 30, x and 50.

The number of observations is 5.

Now, 

X = sum of numbers/number of observations

30 = (10 + 20 + 30 + x + 50)/5

150 = 110 + x

150 – 110 = x

x = 40

Hence, the arithmetic mean of five observations is 40.

Question 5: What will be the arithmetic mean between 10 and 30.

Solution:

The observations are 10 and 30.

The number of observations is 2.

Now,

 X = sum of numbers/ number of observations

X = (10 + 30)/2

X = 40/2

X = 20

Hence, the arithmetic mean of the two observations is 20.

Question 6: If the arithmetic means of two numbers x and 40 is 30. What is the value of x?

Solution:

The observations are x and 40.

The number of observations is 2.

The arithmetic mean is 30.

X = sum of numbers/number of observations

30 = (x + 40)/2

30 × 2 = x + 40

x = 60 – 40

x = 20

Hence, the value of x is 20.

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