Mean is a concept in mathematics that is used to find the average collection of numbers. Mean is also known as the expected value. In general, mean refers to the addition of the largest value and the smallest value and dividing them by two. Mean is used in Statistics where the set of values have a vast difference or they are closely related to each other. Mean will be the center point among the set of numbers.

Mean is the sum of all given data divided by the total number of data given in the set.

Types of Mean

There are three types of mean, arithmetic mean, geometric mean, and harmonic mean. These three are well-known sequences and therefore, their mean is also well known and widely used. The mean taken out from these three types is different from each other in formulae. Let’s learn about these types and the way to find the mean,

How to find the mean of a dataset?

In order to find the mean, it is important to first learn the type of sequence and then the formula for the respective sequence is applied,

Arithmetic Mean

Arithmetic mean is calculated when the values have more differences between them. Some of the values can be closer to each other but most of the other values have a large difference between them.

Arithmetic Mean = (x₁ + x₂ + x₃ +… x_n) / n

Calculating the mean

Example:

Step 1: First, add all the numbers given in the set to find the overall sum.

If the given set is 7, 18, 45, 4, 21.

Then the sum of these numbers is 95.

Step 2: Second, count how many numbers are there in a given set. Here, the total numbers given in the set are 5. Then divide the total sum of the numbers by the total numbers given. Therefore, 95 is divided by 5, which is equal to 19. This is the arithmetic mean.

Geometric Mean

The geometric mean refers to the average of the set. It is also known as the ‘nth root of n numbers’.

Geometric Mean =  2 √(a₁ × a₂  × a₃ × …. a_n)

An arithmetic mean of two numbers is the number when added to itself equals the sum of the two numbers and geometric mean is the number, when multiplied by itself, is equal to the product of the two numbers.

Example:

The geometric mean of 10 and 10 is 10 because √(10 × 10) = 10. If there are 3 numbers, we have to find the cube root of 3 numbers.

If there are ‘n’ numbers we have to find the nth root of the product of all ‘n’ numbers. 

Harmonic mean

Harmonic mean is a type of average that is calculated by dividing the number of values in a series by the sum of the reciprocals (1/x) of each value in the series.

Harmonic Mean= n / (∑1/x_i)

Example 1:

If two people are doing same work. 1st person takes 3 hours for completing that work and 2nd person takes 4 hours for completing the same work. Then, the rate of doing work is 1/3 and 1/4 respectively. If they work together then the rate of work will be 1/3 + 1/4 = 7/12.

Therefore, the time taken by both of them working together is 12/7 hours.

Example 2:

Finding the harmonic mean of 4, 7, 5.

Then, Harmonic Mean = 3 / (1/4 + 1/7 + 1/5) = 420/83 = 5.06

Sample Problems

Question 1: Find the arithmetic mean of the numbers 8, 64, 27, 48, 33.

Solution: 

Arithmetic Mean = (x₁ + x₂ + x₃ +…. x_n) / n

Arithmetic Mean = (8 + 64 + 27 + 48 + 33) / 5

Arithmetic Mean = 180/5

Arithmetic Mean = 36

Question 2: Find the arithmetic mean of the numbers 5, 12, 26.

Solution:  

Arithmetic Mean = (x₁ + x₂ + x₃ + … x_n) / n  

Arithmetic Mean = (5 + 12 + 26) / 3

Arithmetic Mean = 43/3

Arithmetic Mean = 14.3333

Question 3: Find the geometric mean of 15, 12.

Solution:  

Geometric Mean = n√(a₁ × a₂ × a₃ × … a_n)

Geometric Mean = 2√(15 × 12)

Geometric Mean = 2√180

Geometric Mean = 13.42

Question 4: Find the geometric mean of 6, 18, 10.

Solution:

Geometric Mean = n√(a₁ × a₂ × a₃ × …. a_n)

Geometric Mean = 3√(6 × 18 × 10)

Geometric Mean = 3√1080

Geometric Mean = 10.25

Question 5: Find the harmonic mean of 2, 3, 4, 5.

Solution: 

Harmonic Mean = n / (∑1/x_i)

Harmonic Mean = 4/(1/2 + 1/3 + 1/4 + 1/5)

Harmonic Mean = 4/(77/60)

Harmonic Mean = 240/77

Harmonic Mean = 3.12  

Question 6: Find the harmonic mean of 7, 6, 9.

Solution: 

Harmonic Mean = n / (∑1/x_i)

Harmonic Mean = 3/(1/7 + 1/6 + 1/9)

Harmonic Mean = 3/(53/126)

Harmonic Mean = 378/53

Harmonic Mean = 7.13