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Grouping Method of Calculating Mode in Discrete Series | Formula of Mode
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Calculation of Mode in Discrete Series | Formula of Mode

Last Updated : 06 Jun, 2024
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The word mode comes from the Latin word 'Modus', meaning measurements, quantity, way, or manner. In statistics, Mode refers to the variable that occurs most of the time or repeats itself most frequently in a given series of variables (say X). It is a maximum occurrence at a particular point or a value, and one of the three measures of central tendency in statistics that aims to analyze and examine data given in categories like the colour of cars, design of handbags, etc. The mode can be determined in two series, viz., individual series and discrete series. It is denoted as 'Z'. 

Methods of Calculating Mode in Discrete Series

There are two Methods of Calculating Mode in Discrete Series:

  •  Inspection Method
  •  Grouping Method

Table of Content

  • 1. Inspection Method 
  • 2. Grouping Method

1. Inspection Method 

Under the Inspection method, the value of mode is simply calculated by observing the given series of the variable. The variable that repeats or occurs the maximum number of times is considered to be the Mode value for that series. It is one of the easiest and simplest methods of determining mode when the data set is not large. 

Calculation of  Mode using Inspection Method in case of Discrete Series

The frequency value given beside each variable represents the number of times that variable occurs in a given series. Under this method, we have to carefully observe the given series to find out the variable/variables with the highest frequency. Variable or variables with the highest frequency value will be the Mode for the given series.

Example: Find the mode for the given discrete series:

It is visible that variable 20 has the highest frequency value of 8. Therefore, 

Mode (Z) = 20

2. Grouping Method

Sometimes it becomes impossible to determine the mode by inspection method when the frequency distribution is not regular, or the concentration of frequencies does not follow a unique pattern, or where more than one variable shows the highest frequency value. Under such a situation, the Grouping Method of calculating mode is used. 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 Discrete Series

 Grouping Table

 Prepare 6 columns in addition to the column of variable (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 the six columns vertically and all the given values of the variable (X) horizontally.
  2. Now according to the Grouping Table, mark (✓) under that variable which is the part of the maximum total of the column under consideration.
  3. Repeat the process for each of the columns and mark (✓) under the concerned variable.
  4. Now count the mark (✓) of each column in the Analysis Table.
  5. The variable with a maximum number of ticks (✓) is determined to be Mode for the given series, i.e., the variable forming most of the highest grouped frequencies is determined as a value of mode.

Example: Find the mode for the given discrete series:

Grouping Table
  • Column (I) shows the frequency of the given series, and 17 is marked as the highest value.
  • Column (II) shows the sum of frequencies in two(s), i.e., [3+8=11, 10+12=22, 16+14+30, 10+8=18, 17+5=22, and 4+1=5]. In this column, the highest value marked is 30.
  • Column (III) shows the sum of frequencies in two(s), starting from the second value of given frequencies, i.e., [8+10=18, 12+16=28, 14+10=24, 8+17=25, and 5+4=9]. The highest value marked is 28.
  • Column (IV) shows the sum of frequencies in three(s), i.e., [3+8+10=21, 12+16+14=42, 10+8+17=35, and 5+4+1=10]. In this column, the highest value marked is 42.
  • Column (V) shows the sum of frequencies in three(s), starting from the second value of given frequencies, i.e., [8
    +10+12=30, 16+14+10=40, and 8+17+5=30]. The highest value marked is 40.
  • Column (VI) shows the sum of frequencies in three(s), starting from the third value of given frequencies, i.e., [10+12+16=38, 14+10+8=32, and 17+5+4=26]. The highest value marked is 38.                                                                                                          
Analysis Table
  • 17 is the highest value in the first column, corresponding to variable value 10. So, we have marked (✓) under 10 in the analysis table. 
  • In the second column, the highest value marked is 30, and it is the sum of 16 and 14, i.e., corresponding to variable values 6 and 7. Therefore, we have marked (✓) under 6 and 7.
  • In the third column, the highest value marked is 28, and it is the sum of 12 and 16, i.e., corresponding to variable values 5 and 6. Therefore, we have marked (✓) under 5 and 6.
  • In the fourth column, the highest value marked is 42, and it is the sum of 12, 16 and 14, i.e., corresponding to variable values 5, 6 and 7. Therefore, we have marked (✓) under 5, 6 and 7.
  • In the fifth column, the highest value marked is 40, and it is the sum of 16, 14 and 10, i.e., corresponding to variable values 6, 7  and 8. Therefore, we have marked (✓) under 6, 7 and 8.
  • In the sixth column, the highest value marked is 38, and it is the sum of 10, 12 and 16, i.e., corresponding to variable values 4, 5 and 6. Therefore, we have marked (✓) under 4, 5, and 6.   

According to the analysis table, the highest number of ticks(✓) is against the variable 6. Therefore,

Mode (Z) = 6.

What is Discrete Series?

Discrete series shows the number of times each variable is being repeated, known as the frequency of variable. In simple, discrete series shows the variables along with their corresponding frequency. A discrete series does not involve class intervals, rather, in discrete series, variables are always presented corresponding to their frequency value. Data in discrete series are presented in tabular form.

Example of Discrete Series

If 3 students score 60 marks, 9 students score 70 marks, 5 students score 80 marks, and 2 students score 90 marks, then this information will be shown as:

Marks

Number of Students

60

3

70

9

80

5

90

2

Also Read:

  • Mode: Meaning, Formula, Merits, Demerits, and Examples
  • Calculation of Mode in Individual Series | Formula of Mode
  • Grouping Method of Calculating Mode in Discrete Series | Formula of Mode
  • Calculation of Mode in Continuous Series | Formula of Mode
  • Calculation of Mode in Special Cases
  • Calculation of Mode by Graphical Method

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Grouping Method of Calculating Mode in Discrete Series | Formula of Mode

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    The extent to which the values of a distribution differ from the average of that distribution is known as Dispersion. The measures of dispersion can be either absolute or relative. The Measures of Absolute Dispersion consist of Range, Quartile Deviation, Mean Deviation, Standard Deviation, and Loren
    3 min read
    Quartile Deviation in Discrete Series | Formula, Calculation and Examples
    What is Quartile Deviation?Quartile Deviation (absolute measure) divides the distribution into multiple quarters. Quartile Deviation is calculated as the average of the difference of the upper quartile (Q3) and the lower quartile (Q1).Quartile~Deviation=\frac{Q_3-Q_1}{2} Where,Q3 = Upper Quartile (S
    2 min read
    Quartile Deviation in Continuous Series | Formula, Calculation and Examples
    What is Quartile Deviation?Quartile Deviation (absolute measure) divides the distribution into multiple quarters. Quartile Deviation is calculated as the average of the difference of the upper quartile (Q3) and the lower quartile (Q1).Quartile~Deviation=\frac{Q_3-Q_1}{2} Where,Q3 = Upper Quartile (S
    3 min read
    Mean Deviation: Coefficient of Mean Deviation, Merits, and Demerits
    Range, Interquartile range, and Quartile deviation all have the same defect; i.e., they are determined by considering only two values of a series: either the extreme values (as in range) or the values of the quartiles (as in quartile deviation). This approach of analysing dispersion by determining t
    5 min read
    Calculation of Mean Deviation for different types of Statistical Series
    What is Mean Deviation?The arithmetic average of the deviations of various items from a measure of central tendency (mean, median, or mode) is known as the Mean Deviation of a series. Other names for Mean Deviation are the First Moment of Dispersion and Average Deviation. Mean deviation is calculate
    3 min read
    Mean Deviation from Mean | Individual, Discrete, and Continuous Series
    Mean Deviation of a series can be defined as the arithmetic average of the deviations of various items from a measure of central tendency (mean, median, or mode). Mean Deviation is also known as the First Moment of Dispersion or Average Deviation. Mean Deviation is based on all the items of the seri
    4 min read
    Mean Deviation from Median | Individual, Discrete, and Continuous Series
    What is Mean Deviation from Median?Mean Deviation of a series can be defined as the arithmetic average of the deviations of various items from a measure of central tendency (mean, median, or mode). Mean Deviation is also known as the First Moment of Dispersion or Average Deviation. Mean Deviation is
    5 min read
    Standard Deviation: Meaning, Coefficient of Standard Deviation, Merits, and Demerits
    The methods of measuring dispersion such as quartile deviation, range, mean deviation, etc., are not universally adopted as they do not provide much accuracy. Range does not provide required satisfaction as in the entire group, range's magnitude is determined by most extreme cases. Quartile Deviatio
    6 min read
    Standard Deviation in Individual Series
    A scientific measure of dispersion that is widely used in statistical analysis of a given set of data is known as Standard Deviation. Another name for standard deviation is Root Mean Square Deviation. Standard Deviation is denoted by a Greek Symbol σ (sigma). Under this method, the deviation of valu
    3 min read
    Standard Deviation in Discrete Series
    A scientific measure of dispersion that is widely used in statistical analysis of a given set of data is known as Standard Deviation. Another name for standard deviation is Root Mean Square Deviation. Standard Deviation is denoted by a Greek Symbol σ (sigma). Under this method, the deviation of valu
    5 min read
    Standard Deviation in Frequency Distribution Series
    A scientific measure of dispersion that is widely used in statistical analysis of a given set of data is known as Standard Deviation. Another name for standard deviation is Root Mean Square Deviation. It is denoted by a Greek Symbol σ (sigma). Under this method, the deviation of values is taken from
    3 min read
    Combined Standard Deviation: Meaning, Formula, and Example
    A scientific measure of dispersion, which is widely used in statistical analysis of a given set of data is known as Standard Deviation. Another name for standard deviation is Root Mean Square Deviation. Standard Deviation is denoted by a Greek Symbol σ (sigma). Under this method, the deviation of va
    2 min read
    Coefficient of Variation: Meaning, Formula and Examples
    What is Coefficient of Variation? As Standard Deviation is an absolute measure of dispersion, one cannot use it for comparing the variability of two or more series when they are expressed in different units. Therefore, in order to compare the variability of two or more series with different units it
    2 min read
    Lorenz Curveb : Meaning, Construction, and Application
    What is Lorenz Curve?The variability of a statistical series can be measured through different measures, Lorenz Curve is one of them. It is a Cumulative Percentage Curve and was first used by Max Lorenz. Generally, Lorenz Curves are used to measure the variability of the distribution of income and w
    4 min read

    Chapter 9: Correlation

    Correlation: Meaning, Significance, Types and Degree of Correlation
    The previous statistical approaches (such as central tendency and dispersion) are limited to analysing a single variable or statistical analysis. This type of statistical analysis in which one variable is involved is known as Univariate Distribution. However, there are instances in real-world situat
    9 min read
    Methods of Measurements of Correlation
    What is Correlation?A statistical tool that helps in the study of the relationship between two variables is known as Correlation. It also helps in understanding the economic behaviour of the variables. However, correlation does not tell anything about the cause-and-effect relationship between the tw
    4 min read
    Scatter Diagram Correlation | Meaning, Interpretation, Example
    What is a Scatter Diagram?A simple and attractive method of measuring correlation by diagrammatically representing bivariate distribution for determination of the nature of the correlation between the variables is known as the Scatter Diagram Method. This method gives the investigator/analyst a visu
    6 min read
    Spearman's Rank Correlation Coefficient in Statistics
    Spearman's Rank Correlation Coefficient or Spearman's Rank Difference Method or Formula is a method of calculating the correlation coefficient of qualitative variables and was developed in 1904 by Charles Edward Spearman. In other words, the formula determines the correlation coefficient of variable
    6 min read
    Karl Pearson's Coefficient of Correlation | Assumptions, Merits and Demerits
    What is Karl Pearson's Coefficient of Correlation?The first person to give a mathematical formula for the measurement of the degree of relationship between two variables in 1890 was Karl Pearson. Karl Pearson's Coefficient of Correlation is also known as Product Moment Correlation or Simple Correlat
    9 min read
    Karl Pearson's Coefficient of Correlation | Methods and Examples
    What is Karl Pearson's Coefficient of Correlation?The first person to give a mathematical formula for the measurement of the degree of relationship between two variables in 1890 was Karl Pearson. Karl Pearson's Coefficient of Correlation is also known as Product Moment Correlation or Simple Correlat
    6 min read
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