1. Mode − Raw Data 2. Mode for Ungrouped Frequency Distribution 3. Mode – Grouped Frequency Distribution: 4. An Empirical Relationship between Mean, Medan and Mode

**Mode**

(i) The votes
obtained by three candidates in an election are as follows:

Who will
be declared as the winner? Mr. Y will be the winner, because the number of votes
secured by him is the highest among all the three candidates. Of course, the votes
of Mr. Y do not represent the majority population (because there are more votes
against him). However, he is declared winner because the mode of selection here
depends on the highest among the candidates.

(ii) An Organisation
wants to donate sports shoes of same size to maximum number of students of class
IX in a School. The distribution of students with different shoe sizes is given
below.

If it places
order, shoes of only one size with the manufacturer, which size of the shoes will
the organization prefer?

In the above
two cases, we observe that mean or median does not fit into the situation.

We need another
type of average, namely the Mode.

The mode
is the number that occurs most frequently in the data.

When you
search for some good video about Averages on You Tube, you look to watch the one
with maximum views. Here you use the idea of a mode.

For an individual
data mode is the value of the variable which
occurs most frequently.

**Example 8.14**

In a rice
mill, seven labours are receiving the daily wages of ₹500,
₹600, ₹600, ₹800, ₹800, ₹800
and ₹1000,
find the modal wage.

*Solution*

In the given
data ₹800
occurs thrice.Hence the mode is ₹ 800.

**Example 8.15**

Find the
mode for the set of values 17, 18, 20, 20, 21, 21, 22, 22.

*Solution*

In this example,
three values 20, 21, 22 occur two times each. There are three modes for the given
data!

**Note**

• A distribution having only one mode is called unimodal.

• A distribution having two modes is called bimodal.

• A distribution having Three modes is called trimodal.

• A distribution having more than three modes is called multimodal.

In a ungrouped
frequency distribution, the value of the item having maximum frequency is taken
as the mode.

**Example 8.16**

A set of
numbers consists of five 4’s, four 5’s, nine 6’s,and six 9’s. What is the mode.

*Solution*

6 has the
maximum frequency 9. Therefore 6 is the mode.

** **

In case of
a grouped frequency distribution, the exact values of the variables are not known
and as such it is very difficult to locate mode
accurately. In such cases, if the class intervals are of equal width, an appropriate
value of the mode may be determined by

The class
interval with maximum frequency is called the modal
class.

Where

* l*
− lower limit of the modal class;

* f − *frequency
of the modal class* *

* f*_{1}* *– frequency of the class just preceding the modal class

* f*_{2}* − *frequency of the class succeeding the
modal class

* c − *width
of the class interval

**Example 8.17**

Find the
mode for the following data.

*Solution*

Modal class
is 16 -20 since it has the maximum frequency.

* l*
= 15.5, *f* = 32, *f*_{1} = 16, *f*_{2}=24,* c *= 20.5–15.5 = 5

= 15.5 + (16/24) × 5 = 15.5 + 3.33 =18.83.

We have seen
that there is an approximate relation that holds among the three averages discussed
earlier, when the frequencies are nearly symmetrically distributed.

Mode ≈ 3 Median
– 2 Mean

**Example 8.18**

In a distribution,
the mean and mode are 66 and 60 respectively. Calculate the median.

*Solution*

Given, Mean = 66 and Mode = 60.

Using,

Mode ≈ 3Median
– 2Mean

60 ≈ 3Median
– 2(66)

3 Median
≈ 60 – 132

Therefore,
Median ≈ 192/3 ≈ 64

Tags : Formula, Example Solved Problems | Statistics | Maths , 9th Maths : UNIT 8 : Statistics

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9th Maths : UNIT 8 : Statistics : Mode | Formula, Example Solved Problems | Statistics | Maths

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