Question

The marks scored in an examination of 400 marks is given in the following frequency distribution table. Find the mean of the distribution

Marks	\[200 - 240\]	\[240 - 280\]	\[280 - 320\]	\[320 - 360\]	\[360 - 400\]
Frequency	10	20	30	24	16

Answer 1

Hint: Since, the given frequency distribution has intervals of constant class width, we can directly use the formula to find mean as \[\overline x = \dfrac{{\sum {xf} }}{{\sum f }}\], where \[\overline x \] is the mean of the distribution, \[x\] is the middle value of an interval and \[f\] is the frequency of the interval. First we will find the class mark of each interval, then multiply the class mark with their corresponding frequencies and find the sum of both the frequencies and the product of frequency and class mark. Finally we can substitute these values and get the mean.

Complete step-by-step answer:
Class marks is the average of the upper and lower limit of an interval. For the interval \[200 - 240\], the class mark will be \[\dfrac{{200 + 240}}{2} = \dfrac{{440}}{2} = 220\]. Similarly we can find the class mark of other intervals.
Now, as we have to find the class mark of each interval and then multiply it with the frequency we will organize the data in the form of a table.
The table formed will be as follows.

Marks	Frequency(\[f\])	Class mark(\[x\])	\[xf\]
\[200 - 240\]	10	220	2200
\[240 - 280\]	20	260	5200
\[280 - 320\]	30	300	9000
\[320 - 360\]	24	340	8160
\[360 - 400\]	16	380	6080
	\[\sum f = 100\]		\[\sum {xf} = 30640\]

Now, we have the values of \[\sum {xf} \] and \[\sum f \], which we can substitute in the formula to get the mean as.
\[
\Rightarrow \overline x = \dfrac{{\sum {xf} }}{{\sum f }} \\
\Rightarrow \overline x = \dfrac{{30640}}{{100}} = 306.4 \\
\]
Thus, the mean for the given frequency distribution will be 306.4 marks per student.


Note: As we know that the mean of a data set represents the most likely number in the data set, we can observe that here easily. Maximum number of students are scoring in the range of \[280 - 320\], and the mean also comes out to be 306.4, which is very close to the class mark of this interval thus verifying its definition.