Question

The marks obtained out of $50$ by $102$ students in a physics test are given in the frequency table below:
Marks ${\text{ }}15{\text{ 20 22 24 25 30 33 38 45}}$
Frequency $5{\text{ 8 11 20 23 18 13 3 1}}$
Find the average number of the marks.

Answer 1

Here, we are given that the marks of the students and the number of students represent the frequency then the arithmetic mean or average will be given by the formula:
$\dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{f_i}} }}$ and here ${x_i}$ represent the marks of the students and ${f_i}$ as the frequency.

Complete step by step solution:
Here we are given the range of marks which are obtained by the students as given below
Marks ${\text{ }}15{\text{ 20 22 24 25 30 33 38 45}}$
Frequency $5{\text{ 8 11 20 23 18 13 3 1}}$
So here the frequency of $15$ marks is $5$ that means there are five students whose marks are $15$ and similarly we are given the frequency of all the marks obtained, that is the marks are obtained by how many number of students.
The mean is given by the formula $\dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{f_i}} }}$ and here ${x_i}$ represent the marks of the students and ${f_i}$ as the frequency.
So ${x_1} = 15$
${x_2} = 20 \\\ {x_3} = 22 \\\ {x_4} = 24 \\\ {x_5} = 25 \\\ {x_6} = 30 \\\ {x_7} = 33 \\\ {x_8} = 38 \\\ {x_9} = 45 \\\$
$\sum {{f_i}} = 5 + 8 + 11 + 20 + 23 + 18 + 13 + 3 + 1 = 102$
So here mean$= \dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{f_i}} }}$
$= \dfrac{{{x_1}{f_1} + {x_2}{f_2} + {x_3}{f_3} + {x_4}{f_4} + {x_5}{f_5} + {x_6}{f_6} + {x_7}{f_7} + {x_8}{f_8} + {x_9}{f_9}}}{{{\text{ total number of students}}}} \\\ = \dfrac{{15(5) + 20(8) + 22(11) + 24(20) + 25(23) + 30(18) + 33(13) + 38(3) + 45(1)}}{{102}} \\\ = \dfrac{{2660}}{{102}} = 26.07 \\\$

Therefore mean or average mark is $26.07$

Note:
If the question asks for the mode then we need to find which mark is obtained by the maximum number of students and that will be our mode. For example here we have maximum students achieving a particular mark as $23$ so the mode will be $25$