वर्गीकृत आँकड़ों का बहुलक

बहुलक केंद्रीय प्रवृत्ति के मापों में से एक मान है। यह मान हमें एक मोटा अंदाज़ा देता है कि आँकड़ों के समुच्चय में कौन सी वस्तुएँ सबसे अधिक बार आती हैं।

वर्गीकृत आँकड़ों के बहुलक की गणना बहुलक सूत्र का उपयोग करके की जा सकती है। वर्गीकृत बारंबारता बंटन के मामले में, केवल बारंबारता को देखकर बहुलक प्राप्त नहीं किया जा सकता है, हमें पहले बहुलक वर्ग का पता लगाना होगा, जिसमें दिए गए आँकड़ों का बहुलक निहित है।

अवर्गीकृत आँकड़ों के बहुलक

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.

वर्गीकृत आँकड़ों का बहुलक क्या है?

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.

हल:

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 $3.286$