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Mode – Measures of Central Tendency

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  • Last Updated : 10 Jun, 2022

The word mode is derived from the French word ‘La Mode’, meaning anything that is in fashion or vogue. A measure of central tendency in statistical series that determines the value occurring most frequently in a given series is known as mode. In other words, the modal value of the series has the highest frequency in the given series. For example, if in a class of 100 students, 20 students have opted for Mathematics, 50 students have opted for IP, and the rest 30 have opted for Physical Education, then the modal opted subject of the class would be IP. 

Mode is represented by the letter Z.

According to Kenny, “The value of the variable which occurs most frequently in a distribution is called the mode.”

According to Croxton and Cowden, “The mode may be regarded as the most typical of a series of value.”

Mode is calculated in three different types of series; viz., Individual, Discrete and Continuous Series. 

Individual Series

The mode of an individual series is the value that occurs the most frequently and can be determined by two methods, the Inspection Method and by converting Individual Series into Discrete Frequency Series.  

For example,

The age of a class of 12 students is given below. Find the modal age of the class. 

Age (Years) 13 10 15 11 13 13 12 9 11 13 13 14

Solution:

By Inspection Method,

The age 13 occurs most frequently in the given series (5 times). 

Therefore, Mode (Z) = 13

 Discrete Series

In a discrete series, values are given with their respective frequencies. The mode of discrete series is the value whose frequency has occurred most frequently in a given series. It can be determined by two methods, the Inspection Method and Grouping Method. 

For example, 

The table below shows the number of sixes in the cricket taken by a batsman in 12 matches. Determine the mode of the provided data set.

Matches 1 2 3 4 5 6 7 8 9 10 11 12
Six’s 2 3 3 1 2 3 3 1 2 4 0 1

Solution:

By Inspection Method:

Here 3 is repeated the most. 

Therefore, Mode (Z) = 3

Continuous Series

In the case of a continuous frequency distribution, the modal class is the one with the highest frequency. It can be determined by two methods, the Inspection Method and Grouping Method. The formula for calculating mode by both the methods is

Z=l_1+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}

Here,

Z = Value of Mode

l_1   = lower limit of the modal class

f_1   = frequency of modal class

f_0   = frequency of pre-modal class

f_2   = frequency of the next higher class or post-modal class

i = size of the modal group

Modal class is the class interval with the highest frequency in the series. 

For example,

Calculate the value of mode for marks obtained by the students from the table below.

Marks Obtained Students
10-20 10
20-40 5(f_0)
40-60 20 (f_1)
60-80 15(f_2)
80-100 5

Solution:

By Inspection method:

The Class Interval 40-60 has the highest frequency, i.e., 20. Therefore, the modal class of the given series is 40-60, which means that 20 is the frequency of the modal class, 5 is the frequency of the pre-modal class, and 15 is the frequency of the post-modal class. 

Z=l_1+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}

=40+\frac{20-5}{2(20)-5-15}\times{20}

=40+\frac{15}{40-20}\times{20}

=40+\frac{15}{20}\times{20}

= 55

Therefore, Mode (Z) = 55

Now, there are four cases in which the mode of a series can be determined. The first one is Uni-Model. In this case, the variable which occurs the maximum number of times is considered to be the value of mode. The second case is Bi-Model. In the given series, if two variables repeat themselves an equal number of times, then both variables will be considered as the value of Mode. The third case is Multi-Model. If in the series, more than two variables repeat themselves an equal number of times, then the value of Mode will be all such variables. The last case is Without Mode. When no variable in a series repeats itself, then the value of Mode is ill determined. Also, when most of the variables in a given series repeat themselves an equal number of times, then also the value of Mode is ill-determined. 

Merits of Mode

  1. Simple and Popular: Mode is a simple measure of central tendency, as one can easily determine the modal value by just glancing at the given series. Its simplicity makes it a very popular measure of central tendency. 
  2. Graphic Location: One can even locate the mode of a given series through the graph. it can be done with the help of a histogram. 
  3. No need to know all the items or frequencies: To calculate or determine the mode of a given series, one does not require to have knowledge of all the items or frequencies of a series. In simple series, the value with the highest frequency is enough for determining the mode. 
  4. Least effect of marginal values: As compared to the mean, the mode is the measure of central tendency that carries with it the least effect of the marginal values of a series. It is because the mode of a series is determined only through the highest frequencies. 
  5. Best representative value: Mode is that value of a series that occurs most frequently. Therefore, it shows the best representative value of the given series. 

Demerits of Mode

  1. Uncertain and vague: Amongst the other measures of central tendency, the mode is vague and uncertain.
  2. Difficult: When the frequency of all items in a statistical series is identical, it is difficult for the investigator to identify the modal value of that series.
  3. Not capable of algebraic treatment: Unlike mean, one cannot perform further algebraic treatment with mode.  
  4. Complex procedure of grouping: Calculating the mode of a statistical series sometimes also includes the grouping of data, which can be cumbersome, making it difficult to determine the modal value. Also, if the extent of the group changes, it will also change the modal value of the series. 
  5. Ignores extreme marginal frequencies: Calculation of mode does not consider extreme marginal frequencies. It means that mode is not a representative value of all the values in a given series. 

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