वर्गीकृत आँकड़ों का बहुलक

The mode is one of the values of the measures of central tendency. This value gives us a rough idea about which of the items in a data set tend to occur most frequently.

Mode of grouped data can be calculated using the mode formula. In the case of grouped frequency distribution, mode can't be obtained just by looking into the frequency, we first need to find out the modal class, in which lies the mode of the given data.

Mode for Ungrouped Data

Data that does not appear in groups are called ungrouped data. Let us take an example to understand how to find the mode of ungrouped data. Let us say a garment company manufactured Shirts with the sizes as mentioned in the frequency distribution table:

Size of the shirts	Total number of shirts
$38$	$33$
$39$	$11$
$40$	$22$
$42$	$55$
$43$	$44$
$44$	$11$

We can clearly see that size $42$ has the greatest frequency. Hence, the mode for the size of the shirts is $42$. However, the same does not hold good for grouped data.

What is Mode of Grouped Data?

Mode is one of the measures of the central tendency of a given dataset which demands the identification of the central position in the data set as a single value. In the case of ungrouped data, the mode is simply the item having the greatest frequency. For grouped data, the mode is calculated using the formula,

Mode = $l+\left({\frac {f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}}}\right)h$

Where

$l=$ Lower limit of the modal class

$h=$ Size of the class interval

$f_{1}=$ Frequency of the modal class

$f_{0}=$ Frequency of the class preceding the modal class

$f_{2}=$ Frequency of the class succeeding the modal class

Example: A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household

Family size	Number of families
$1-3$	$7$
$3-5$	$8$
$5-7$	$2$
$7-9$	$2$
$9-11$	$1$

Find the mode of this data.

Solution:

Here the maximum class frequency is $8$, and the class corresponding to this frequency is $3-5$. So, the modal class is $3-5$.

Now

modal class = $3-5$

lower limit $(l)$ of modal class = $3$

class - interval size $(h)$ = $2$

frequency $(f_{1})$ of the modal class = $8$

frequency $(f_{0})$ of class preceding the modal class = $7$

frequency $(f_{2})$ of class succeeding the modal class = $2$

Mode = $l+\left({\frac {f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}}}\right)h$

$=3+\left({\frac {8-7}{2\times 8-7-2}}\right)\times 2$

$=3+\left({\frac {1}{16-7-2}}\right)\times 2$

$=3+\left({\frac {1}{7}}\right)\times 2$

$=3.286$

Therefore mode of the above data is $3.286$