Mode

Mode is that value of a variety that occurs most often. More precisely, the mode is the value of the variable at which the concentration of the data is maximum. 

Modal Class: In a frequency distribution, the class having the maximum frequency is called the modal class.

Mode Formula for Grouped Data

The formula for Calculating Mode:  

M_o = x_k + h{(f_k – f_k-1)/(2f_k – f_k-1 – f_k+1)} 

Where,

• x_k = lower limit of the modal class interval.
• f_k = frequency of the modal class. 
• f_k-1= frequency of the class preceding the modal class.
• f_k+1 = frequency of the class succeeding the modal class.
• h = width of the class interval.

Example 1: Calculate the mode for the following frequency distribution.

Solution: 

Class 40-50 has the maximum frequency, so it is called the modal class.

x_k = 40, h = 10, f_k = 28, f_k-1 = 12, f_k+1 = 20

Mode, M_o= x_k + h{(f_k – f_k-1)/(2f_k – f_k-1 – f_k+1)}

= 40 + 10{(28 – 12)/(2 × 28 – 12 – 20)}

= 46.67

Hence, mode = 46.67

Important Result

Relationship among mean, median and mode,

Mode = 3(Median) – 2(Mean)

Example 2: Find the mean, mode, and median for the following data,

Solution:

We have,

Class	Mid Value xi	Frequency fi	Cumulative Frequency	fi . xi
0-10	5	8	8	40
10-20	15	16	24	240
20-30	25	36	60	900
30-40	35	34	94	1190
40-50	45	6	100	270
		∑fi=100		∑fi. xi=2640

Mean = ∑(f_i.x_i)/∑f

= 2640/100

= 26.4

Here, N = 100 ⇒ N / 2 = 50. 

Cumulative frequency just greater than 50 is 60 and corresponding class is 20-30. 

Thus, the median class is 20-30. 

Hence, l = 20, h = 10, f = 36, c =  c. f. of preceding class = 24 and N/2=50

Median, M_e = l + h{(N/2 – c_f)/f}

= 20+10{(50-24)/36}

Median = 27.2. 

Mode = 3(median) – 2(mean) = (3 × 27.2 – 2 × 26.4) = 28.8.

Check: Mean, Median and Mode

Suppose we want to compare the age of students in two schools and determine which school has more aged students. If we compare on the basis of individual students, we cannot conclude anything. However, if for the given data, we get a representative value that signifies the characteristics of the data, the comparison becomes easy. 

A certain value representative of the whole data and signifying its characteristics is called an average of the data. Three types of averages are useful for analyzing data. They are: 

• Mean
• Median
• Mode

In this article, we will study three types of averages for the analysis of the data.