Question

Calculate the median from the following data:

Marks	\[0 - 10\]	\[10 - 30\]	\[30 - 60\]	\[60 - 80\]	\[80 - 90\]
No. of students	5	15	30	8	2

Answer 1

Hint: To find the median from the following data given in the form of intervals, we need to find the cumulative frequencies. Also here the class width are unequal, hence we don’t have to adjust the frequencies of the intervals. We will find the median class using the cumulative frequency then apply the formula to find median of a data which is \[Median = l + \dfrac{{\dfrac{n}{2} - c.f}}{f} \times h\].

Complete step-by-step answer:
The formula to find median is \[Median = l + \dfrac{{\dfrac{n}{2} - c.f.}}{f} \times h\], where \[l\] is the lower limit of the median class, \[n\] is the total cumulative frequency, \[c.f.\] is the cumulative frequency of the preceding class, \[f\] is the frequency of the median class and \[h\] is the class width of the median class.
Now to find the median class, let us first draw a table and organize all the required data in it.
The table formed will have three columns for class interval, frequency and cumulative frequency.
Thus, the table formed is as follows.

Class interval (Marks)	Frequency	Cumulative frequency
\[0 - 10\]	5	5
\[10 - 30\]	15	20
\[30 - 60\]	30	50
\[60 - 80\]	8	58
\[80 - 90\]	2	60

Thus, the cumulative frequency (\[n\]) comes out to be 60. Now the median class will the one which has cumulative frequency greater than \[\dfrac{n}{2}\]. As we know, \[\dfrac{n}{2} = \dfrac{{60}}{2} = 30\], thus by observing the table we can say that 50 is the cumulative frequency which is greater than 30 and thus the median class is \[30 - 60\].
As the median class is \[30 - 60\], thus the value of its lower limit and \[l\] will be equal to 30, value of class width and \[h\] will be \[60 - 30 = 30\] and the value of frequency and \[f\] will be 30. Also the value of \[c.f.\] will be 20 as it is the value of cumulative frequency at preceding class.
Now, as we have all the values, we can easily substitute these in the formula and get the value of median.
Thus, the value of median after substituting these values will be
\[
  Median = l + \dfrac{{\dfrac{n}{2} - c.f.}}{f} \times h \\
\Rightarrow Median = 30 + \dfrac{{\dfrac{{60}}{2} - 20}}{{30}} \times 30 \\
\Rightarrow Median = 30 + \dfrac{{30 - 20}}{{30}} \times 30 \\
\Rightarrow Median = 30 + 10 \\
\Rightarrow Median = 40 \\
\]
Thus, the value of median will be 40 for the given data.
Note: Here, it is important to note that formulas for finding mean and median look very similar in their basic structure and thus students must make sure to know both these formulae very well. Also it is important to draw a table and organize the data into it as it helps to decrease the calculations involved and also the chances of making mistakes.