# Mode of Grouped Data

The mode is the "most popular" number in a given data. The expression comes from the French "a la mode" meaning "fashionable".

In vital statistics we study the numerical records of marriages, births, sickness, deaths, etc. With the help of these records, the health and the growth of a community may be studied for vital statistics which constitutes a large part of the subject known as demography.

In demography we are concerned with the growth of human population.

Growth of population is divided into three main components.
(i) Fertility (positive component)
(ii) Mortality (negative component)
(iii) Migration (positive and negative component)

In these components we will be able to get the maximum value which gives us the modal value.

The mode or modal value is that value in a series of observations which occur with the greatest frequency e.g. the mode of the series 3, 5, 8, 5, 4, 5, 9, 3 would be 5,since this value occurs most frequently than any of the other values.
The mode of a distribution is the value at the point around which the items tend to be most heavily concentrated. It is regarded as the most typical value of distribution.
The following diagram shows the modal value:

The value of the variable for which the curve reaches the maximum is called the mode. It is the value around which the items tend to be greatly concentrated.
There are many situations in which arithmetic mean and median fail to reveal the true characteristics of data.

Example : When we talk of the most common wage, Income, height, size of the shoe or readymade garment we have to find out mode, because in these cases Arithmetic mean and median cannot represent the data.

For example if the data are

 Size of   shoes 5 6 7 8 9 10 11 No. of    persons 10 20 25 40 22 15 6

Now in this case if we have to find out the average shoe size. Arithmetic mean cannot represent it. This series can be represented by mode.
The modal size is 8, since more persons are wearing this size compared to others.

Calculation of Mode
For determining mode, count the number of times the various values repeat themselves. The value that occurs the maximum number of times is the modal value. The more the modal value appears relatively, the more valuable it is, as an average to represent mode.

Example

Calculate the mode from the following data

 Roll   No 1 2 3 4 5 6 7 8 9 10 Marks 10 27 24 12 27 27 20 18 15 28
Solution

Calculation of mode.

 Marks No. of times it occur 10 1 12 1 15 1 18 1 20 1 24 1 27 3 28 1

Since the item 27 occurs the maximum number of times, i.e. 3, the modal marks are 27.

Example
 Size of garment No. of persons 28 10 29 20 30 40 Modal size 31 65 32 50 33 15
Solution

From the above, we can clearly see the model size is 31, because the value 31 has occurred the maximum number of times. So generally, the maximum number of garments a manufacturer will be producing is of size 31 because that will be in the maximum demand.

A set of data may be having a single mode, in which case it is said to be unimodal, or data having two modes which makes it bimodal and data having several mode are called multimodal.

Example :

For example the series
110, 120, 130, 120, 110, 140, 130, 130, 120,140.

 Size of item No. of items 110 2 120 3 130 3 140 2

In this case mode is ill-defined.
When the mode cannot be found immediately, its value is calculated using the following formula.
Mode = 3 median - 2 mean.

Example

The following is the distribution of height of students of a certain class in a certain city:

 Height (in cm) 160-162 163-165 166-168 169-171 171-173 No. of students 15 118 142 127 18

Find the average height of maximum number of students.

Solution

If the intervals are not continuous, 0.5 should be subtracted from the lower limit and 0.5 should be added to the upper limit. Then the distribution table is

 Height (in cm) 159.5-162.5 162.5-165.5 165.5-168.5 168.5-171.5 171.5-173.5 No. of students 15 118 142 127 18

Here

Mode=

Average height of maximum number of students = 167.35.

Merits of Mode

1. By definition, mode is the most typical or representative value of a distribution. The mode is the most frequently occurring value. If the modal wage in a factory is Rs. 1916, then it means more workers receive Rs. 1916 than any other wage.

2. Mode is not unduly affected by extreme values.

3. It can be used to describe qualitative phenomenon. For example, if we want to compare consumer preference for different types of products say soaps, we will find out modal preferences expressed by different groups of people.

Demerits

1. The value of mode cannot always be determined. In some cases, we have Bi-modal or Multi-modal series.

2. The value of mode is not based on each and every item of the series.

In vital statistics we study the numerical records of marriages, births, sickness, deaths, etc. With the help of these records, the health and the growth of a community may be studied for vital statistics whichconstitutes a large part of the subject known as demography.

In demography we are concerned with the growth of human population.

Growth of population is divided into three main components.
(i) Fertility (positive component)
(ii) Mortality (negative component)
(iii) Migration (positive and negative component)
Now we shall give a brief description of some concepts of vital statistics that we often come across.