Question

Find the mode of the following distribution of marks obtained by 80 students

Marks Obtained	0-10	10-20	20-30	30-40	40-50
Number of students	6	10	12	32	20

A) 28.42
B) 34.82
C) 36.25
D) 34.86

Answer 1

Hint:
Here in this question we have to use the concept and formula of the mode to find the value of mode. Mode is the most common or most repeating numbers in the given data. So we have to select the modal class from the given data and modal class is the class with the maximum frequency. So then we have to apply the formula of the mode to find its value.

Complete step by step solution:
The given data is for the 80 students.
Formula of the mode \[{\rm{ = L}} + {\rm{h}}\left( {\dfrac{{{{\rm{f}}_{\rm{m}}} - {{\rm{f}}_1}}}{{{\rm{2}}{{\rm{f}}_{\rm{m}}} - {{\rm{f}}_1} - {{\rm{f}}_2}}}} \right)\]where,
L is the lower limit of the modal class.
h is the size of the class interval.
\[{{\rm{f}}_{\rm{m}}}\]is the frequency of the model class.
\[{{\rm{f}}_1}\]is the frequency of the class preceding the modal class
\[{{\rm{f}}_2}\] is the frequency of the class succeeding the modal class.
Modal is the class with the maximum frequency in the given data. So from the data we can see that class 30-40 is the class with the maximum frequency i.e. 32. So, modal class is 30-40.
Lower limit of the modal class is 30. Therefore, \[{\rm{L}} = 30\]
Size of the class interval in the given data is 10. Therefore, \[{\rm{h}} = 10\]
Frequency of the modal class is 32. Therefore,\[{{\rm{f}}_{\rm{m}}} = 32\]
Frequency of the class preceding the modal class i.e. class 20-30 is 12. Therefore,\[{{\rm{f}}_1} = 12\]
Frequency of the class succeeding the modal class i.e. class 40-50 is 20. Therefore,\[{{\rm{f}}_2} = 20\]
Now we have to put all the values in the formula of mode to find its value, we get
Mode \[{\rm{ = 30}} + {\rm{10}}\left( {\dfrac{{32 - 12}}{{{\rm{2(32)}} - 12 - 20}}} \right){\rm{ = 30}} + {\rm{10}}\left( {\dfrac{{20}}{{{\rm{64}} - 32}}} \right){\rm{ = 30}} + {\rm{10}}\left( {\dfrac{{20}}{{32}}} \right){\rm{ = 30}} + 6.25 = 36.25\]
Therefore, the mode of the data is 36.25.

So, option C is correct.

Note:
Statistics is the science of collecting some data in the form of the number and studying it to forecast or predict its future possibility. Some definitions we should know
Mean is equal to the ratio of sum of the total numbers and total count of the numbers. Mean is also known as the average of the numbers.
Median is the middle value of the given list of numbers or it is the value which is separating the data into two halves i.e. upper half and lower half.