Calculating Arithmetic Mean for Grouped Data

The grouped data is the data given as the continuous interval, i.e. in grouped data the class interval is given along with the frequency of each class. There are three different methods which are used to find the arithmetic mean for grouped data, they are

• Direct Method
• Short-Cut Method
• Step-Deviation Method

We can use any of the three methods for finding the arithmetic mean for grouped data depending on the value of frequency and the mid-terms of the interval. Now let’s discuss the three methods for finding the arithmetic mean for grouped data in detail.

Direct Method for Finding the Arithmetic Mean

We can easily find the arithmetic mean using the direct method as,

Let we have to find the mean of n observation say x₁, x₂, x₃ ……xₙ, and their frequency is f₁, f₂, f₃ ……fₙ respectively. Then the formula for arithmetic mean is,

x̄ = (x₁f₁+x₂f₂+……+xₙfₙ) / ∑f_i

where 
x̄ is the arithmetic mean
f₁+ f₂ + ….fₙ = ∑f_i indicates the sum of all frequencies

Example: Find the mean of the following data.

Solution:

For mean,

xi	5	10	15	20	25
fi	5	2	2	3	4
fixi	25	20	30	60	100

 

∑f_i = 5+2+2+3+4 = 16

∑f_ix_i = 25+20+30+60+100 = 235

x̄ = (x₁f₁+x₂f₂+……+xₙfₙ) / ∑f_i

x̄ = 235/16 = 14.6875

Thus, the mean of the given data set is 14.6875

Short-cut Method for Finding the Arithmetic Mean

We can easily find the arithmetic mean using the shortcut method also called the assumed mean method by using the steps discussed below,

Step 1: Find the midpoint of each class interval say x_i

Step 2: Assumed a random number as the assumed mean say A.

Step 3: Find the deviation of each class interval midpoint as, (d_i) = x_i – A

Step 4: Use the formula for finding the arithmetic mean

x̄ = A + (∑f_id_i/∑f_i)

Example: Find the mean of the given data using the short-cut method.

Solution:

For Arithmetic Mean,

Let the assumed mean be 20 

Class Interval (CI)	xi	Frequency(fi)	di = (xi – A)	fidi
5-15	10	4	10 – 20 = -10	-40
15-25	20	12	20 – 20 = 0	0
25-35	30	8	30 – 20 = 10	80
35-45	40	6	40 – 20 = 20	120

 

∑f_i = 4+12+8+6 = 20

∑f_id_i = -40+0+80+120 = 160

Using the Formula,

x̄ = A + (∑f_id_i/∑f_i)

x̄ = 20 + 160/20

   = 20 + 8

   = 28

Thus, the Arithmetic mean is, 28

Step-Deviation Method for Finding the Arithmetic Mean

We can easily find the arithmetic mean using the step-deviation method also called the scale method by using the steps discussed below,

Step 1: Find the midpoint of each class interval say x_i

Step 2: Assumed a random number as the assumed mean say A.

Step 3: Find the u_i = (x_i-A)/h, where, h is the class interval.

Step 4: Use the formula for finding the arithmetic mean

x̄ = A + h(∑f_iu_i/∑f_i)

Example: Find the mean of the given data using the short-cut method.

Solution:

For Arithmetic Mean,

Let the assumed mean be 20 

The class interval is 10.

Class Interval (CI)	xi	Frequency(fi)	ui = (xi-A)/h	fiui
5-15	10	4	-1	-4
15-25	20	12	0	0
25-35	30	8	1	8
35-45	40	6	2	12

 

∑f_i = 4+12+8+6 = 20

∑f_iu_i = -4+0+8+12 = 16

Using the Formula,

x̄ = A + h(∑f_id_i/∑f_i)

x̄ = 20 + 10(16/20)

   = 20 + 8

   = 28

Thus, the Arithmetic mean is, 28

Arithmetic Mean is defined as the ratio of all the values or observations to the total number of values or observations. Arithmetic Mean is one of the fundamental formulas used in mathematics and it is highly used in various solving various types of problems. We can understand it with some examples, if in a family the husband earns 35,000 rupees and his wife earns 40,000 rupees then what is their average salary? This average is also called the arithmetic mean of 35,000 rupees and 40,000 rupees, which is calculated by adding these two salaries and then dividing it by 2.

Average Salary (Arithmetic Mean of Salary) = (35000 + 40000)/2 = 37500. 

Thus, the arithmetic mean is used in various scenarios such as in finding the average marks obtained by the student in marks, the average rainfall in any area, etc.