Calculation of Mode in Continuous Series | Formula of Mode

What is Mode?

The ‘word’ mode gets its roots from the old French word ‘La Mode’, which means most popular phenomenon, and from the Latin word ‘Modus’, meaning measurements, quantity, way or manner. However, in English, the literal meaning of mode is ‘the most frequent value in a set of values’. Mode, therefore, in statistics, refers to the variable that occurs most of the time in the given series. In simple words, a mode is a variable that repeats itself most frequently in a given series of variables (say X). We can determine the mode in two series; viz., individual and discrete series. 

Mode is denoted as ‘Z‘.

What is Continuous Series?

A discrete series cannot take any value in an interval; therefore, in cases where it is essential to represent continuous variables with a range of values of different items of a given data, Continuous Series is used. In this series, the measurements are only approximations and these approximations are expressed in the form of class intervals. The classes are formed from beginning to end, without any breaks. Mode in a continuous series belongs to a specific class or group that is known as the modal class. 

Example of Continuous Series

If 10 students of a class score marks between 50-60, 8 students score marks between 60-70, 12 students score marks between 70-80, and 5 students score marks between 80-90, then this information will be shown as:

Marks	No. of Students
50-60	10
60-70	8
70-80	12
80-90	5

Methods of Calculating Mode in Continuous Series

There are two methods of calculating Mode in Continuous Series:

• Observation Method or Inspection Method
• Grouping Method

1. Observation Method

The observation method can be used to identify mode if the frequencies are uniform, homogeneous, and have only one maximum frequency. When the frequencies of a continuous series rise and fall in any systematic order, the modal class can be determined simply by inspection of the series.

Steps to calculate mode using Observation Method in case of Continuous Series

Step 1: Identify the modal class, which means the class with the highest frequency.

Step 2: The exact value of the mode can be calculated by using the following formula:

[Tex] M_o=l_{1}+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}[/Tex]

Where,  

M_o = Mode

l₁ = Lower limit of modal class

f₁ = Frequency of modal class

f₀ = Frequency of class preceding the modal class

f₂ = Frequency of the class succeeding the modal class

i = Class interval of the modal class

The formula can also be expressed as:

[Tex]M_o=l_{1}+\frac{f_1-f_0}{(f_1-f_0)+(f_1-f_2)}\times{i}[/Tex]

Note:

If the frequency of the pre-modal class or post-modal class is higher than the modal class; i.e., if (f₁ – f₂) is negative or (2f₁ – f₀ – f₂) is zero, then the formula for calculating mode will be:

[Tex]M_o=l_1+\frac{|f_1-f_0|}{|f_1-f_0|+|f_1-f_2|}\times{i}[/Tex]

The formula and the meanings of the symbols are same, the only difference is that we have taken absolute values in this formula after ignoring negative signs.

Example: Find out the mode of the following series.

Solution: By looking at the data, it is evident that the modal class is 20-30 because the frequency of this class is the maximum; i.e., 15.

[Tex] M_o=l_{1}+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}[/Tex]

Where, l₁ = 20, f₁ = 15, f₂ = 8, f₀ = 10, and i =10

[Tex]M_o=20+\frac{15-10}{2(15)-10-8}\times{10}[/Tex]

[Tex]M_o=20+\frac{5}{30-18}\times{10}[/Tex]

[Tex]M_o=20+\frac{50}{12}[/Tex]

M_o = 20 + 4.16

Mode (Z) = 24.16

2. Grouping Method

The inspection method can be used if the frequencies are uniform, homogeneous, and have only one maximum frequency. But in case of irregularity, Grouping Method is preferable. In the grouping method, frequencies are grouped to get a unique pattern and two tables are prepared to determine the mode, viz., The Grouping Table and The Analysis Table.

Calculation of  Mode using Grouping Method in case of Continuous Series

Grouping Table

 Prepare 6 columns in addition to the column of Class Interval (X) and then form groups according to the following instructions: 

1. Firstly, take the given frequencies in column 1.
2. Then take the sum of frequencies in two(s) in column 2. 
3. Now in column 3, take the sum of frequencies in two(s), starting from the second value of the given frequencies. 
4. Take the sum of frequencies in three(s) in column 4.
5. In column 5, take the sum of frequencies in three(s), starting from the second value of the given frequencies.
6. Lastly, in column 6, take the sum of frequencies in three(s), starting from the third value of the given frequencies.

After preparing all six columns, underline, circle, or highlight the maximum frequency (maximum total) of each column.

Analysis Table

1. First, prepare another table showing all six columns vertically and all the given values of the Class Interval (X) horizontally.
2. Now according to the Grouping Table, mark (✓) under that class interval which is part of the maximum total of the column under consideration.
3. Repeat the process for each of the columns and mark (✓) under the concerned class interval.
4. Now count the mark (✓) of each column in the Analysis Table.
5. The Class Interval with a maximum number of ticks (✓) is determined to be the Modal Class for the given series, i.e., the modal class forming most of the highest grouped frequencies is determined as the modal class, which is then used in the formula to determine the mode.

Steps in Grouping Method

Step 1: First of all, determine the modal class by the process of grouping as stated above.

Step 2: Calculate the exact value of the mode by using the following formula:

[Tex] M_o=l_{1}+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}[/Tex]

Example: Find out the mode of the following series using the Grouping Method.

Solution: The modal class is not clear upon closer investigation. Although the largest frequency (26) is in the 10-15 class, the greatest concentration of items is around the 20–25 class (with a frequency of 21). As a result, an analysis table and grouping table are created to calculate the mode.

Grouping Table

• Column (I) shows the frequency of the given series, and 26 is marked as the highest value.
• Column (II) shows the sum of frequencies in two(s), i.e., [5+12=17, 26+13=39, 24+21=45, and 11+4=15]. In this column, the highest value marked is 45.
• Column (III) shows the sum of frequencies in two(s), starting from the second value of given frequencies, i.e., [12+26=38, 13+24-37, and 21+11=32]. The highest value marked is 38.
• Column (IV) shows the sum of frequencies in three(s), i.e., [5+12+26=43 and 13+24+21=58]. In this column, the highest value marked is 58.
• Column (V) shows the sum of frequencies in three(s), starting from the second value of given frequencies, i.e., [12+26+13 and 24+21+11]. The highest value marked is 56.
• Column (VI) shows the sum of frequencies in three(s), starting from the third value of given frequencies, i.e., [26+13+24=63 and 21+11+4=36]. The highest value marked is 63.        

Analysis Table

• 26 is the highest value in the first column, corresponding to the class interval 10-15. So, we have marked (✓) under 10-15 in the analysis table. 
• In the second column, the highest value marked is 45, and it is the sum of 24 and 21, i.e., corresponding to the class intervals 20-25 and 25-30. Therefore, we have marked (✓) under 20-25 and 25-30.
• In the third column, the highest value marked is 38, and it is the sum of 12 and 26, i.e., corresponding to class intervals 5-10 and 10-15. Therefore, we have marked (✓) under 5-10 and 10-15.
• In the fourth column, the highest value marked is 58, and it is the sum of 13, 24, and 21, i.e., corresponding to class intervals 15-20, 20-25, and 25-30. Therefore, we have marked (✓) under 15-20, 20-25, and 25-30.
• In the fifth column, the highest value marked is 56, and it is the sum of 24, 21, and 11, i.e., corresponding to class intervals 20-25, 25-30, and 30-35. Therefore, we have marked (✓) under 20-25, 25-30, and 30-35.
• In the sixth column, the highest value marked is 63, and it is the sum of 26, 13, and 24, i.e., corresponding to class intervals 10-15, 15-20, and 20-25. Therefore, we have marked (✓) under 10-15, 15-20, and 20-25.

According to the Analysis Table, the highest number of ticks (✓) is against the class interval 20-25; therefore, the modal class of the given series is 20-25.

Now, with the help of the following formula, the mode will be:

[Tex] M_o=l_{1}+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}[/Tex]

Where, l₁ = 20, f₁ = 24, f₂ = 21, f₀ = 13, and i = 5

[Tex]M_o=20+\frac{24-13}{2(24)-13-21}\times{5}[/Tex]

[Tex]M_o=20+\frac{11}{48-34}\times{5}[/Tex]

[Tex]M_o=20+\frac{55}{14}[/Tex]

M_o = 20 + 3.92

Mode (Z) = 23.92